Sébastien Guisset

Devant

Professeur M Alain Bachelot

Professeur M Bruno Després

Professeur M Francis Filbet

Professeur M Christophe Chalons

Professeur M Philippe Helluy

M Mohammed Lemou

Expert Sénior CEA/DAM M Bruno Dubroca

Professeur Invité M Vladimir Tikhonchuk

Maître M Stéphane Brull

M Emmanuel

Modélisation et méthodes numériques pour l'étude du transport de particules dans un plasma chaud

Keywords: Plasma physics modelling, particle transport, angular moments models, Asymptotic-preserving schemes, Godunov-type schemes, approximate Riemann solvers

la commission d'examen composée de :

Titre Modélisation et méthodes numériques pour l'étude du transport de particules dans un plasma chaud.

Résumé Les modèles aux moments angulaires constituent des descriptions intermédiaires entre les modèles cinétiques et les modèles uides. Dans ce manuscrit, les modèles aux moments angulaires basés sur un principe de minimisation d'entropie sont étudiés pour des applications en physique des plasmas. Ce mémoire se découpe en trois parties. La première est une contribution à la modélisation en physique des plasmas à travers le formalisme des modèles aux moments angulaires. Dans celle-ci, le domaine de validité de ces modèles est étudié en régimes non-collisionels.

Il est également montré que les opérateurs de collisions proposés pour le modèle M 1 permettent de retrouver des coecients de transport plasma précis. La deuxième partie de ce document concerne la dérivation de méthodes numériques pour l'étude du transport de particules en temps long. Dans ce cadre, des schémas numériques appropriés pour le modèle M 1 , préservant l'asymptotique, sont construits et validés numériquement. La troisième partie représente un premier pas signicatif vers la modélisation multi-espèces. Ici, le modèle aux moments angulaire M 1 , construit dans un référentiel mobile, est appliqué à la dynamique des gaz raréés. Les propriétés de ce modèle sont détaillées, un schéma numérique est proposé et une validation numérique est menée.

Mots-clés Modélisation en physique des plasmas, transport de particules, modèles aux moments angulaires, schémas préservant l'asymptotique, schémas de type Godunov, solveurs de Riemann approchés.

Introduction Intention

This manuscript is a contribution to the modelling and numerical methods for the transport of charged particles in dense plasmas. We are interested in hot plasmas created by lasers and the general context is the understanding of the processes leading to ignition of the fusion reactions. This issue constitutes the main motivation of this work and is followed as a general research direction. However, many other research areas are closely related to this work since they present a similar physics.

They extend to hypersonic ows [START_REF] Anderson | Hypersonic and High-Temperature Gas Dynamics[END_REF], radiotherapy [11,12,13], magnetic connement fusion (MCF) [START_REF] Eriksson | Simulation of runaway electrons during tokamak dis-ruptions[END_REF][START_REF] Eriksson | Current Dynamics during Disruptions in Large Tokamaks[END_REF], astrophysics [START_REF] Chièze | Gas and Dark Matter Spherical Dynamics[END_REF], studies of lighting and ames, engines for space propulsion [START_REF] Goebel | Fundamentals of electric propulsion: ion and Hall thrusters[END_REF] or plasma remediation processes.

Context

Nowadays, everybody knows the emergency to tackle the issues related to energetic resources exhaustion and global warming. Several research axes are dealing with measures permitting signicant energy savings. Other approaches consider new energy production methods based on nuclear fusion reactions. Stars are powered by nuclear fusion in their cores and the control of such processes on earth represents a great perspective for abondant energy production. A nearly inexhaustible combustible in addition to a small quantity of long-life time radioactive waste and greenhouse gas make this approach very attractive.

Nuclear fusion reactions are based on the fusion of light atomic nuclei, which releases large quantities of energic particles or radiation. In the case of the fusion of deuterium and tritium (D-T), an alpha particle and a neutron are produced. A large kinetic energy is carried out by these two reaction products (3.5 MeV for the alpha particle and 14.1 Mev for the neutron). The use of deuterium and tritium (D-T) nuclei is currently the nominal way chosen to achieve fusion with a minimum of energy provided to the system. The cross section of D-T reaction fusion is indeed several times larger compared to the other fusion reactions (D-3 He, D-Dp, D-Dn, etc). The D-T reaction, producing the 14 MeV neutrons, is the following D + T → 4 He (3.52 MeV) + n (14.06 MeV).

The main diculty in the initiation of the fusion reaction lies in the fact that both parent nuclei have the same positive charge and repulse each other. In order to achieve the fusion of two nuclei one has to overcome the Coulomb repulsive forces between them. The amount of energy released and the number of reactions in the volume of plasma depend on the density of particles and their temperature. The reaction gain becomes higher than one when the energy released by fusion reactions becomes greater than the one invested in the plasma heating and connement. This condition can be expressed in terms of the density n, temperature T of the plasma and its connement time τ through the Lawson criterion [START_REF] Lawson | Some Criteria for a Power Producing Thermonuclear Reactor[END_REF]. For a Deuterium-Tritium plasma heated to the temperature of 10 keV or 10 8 K, this criteria writes as nτ > 10 20 m -3 s.

There exist dierent ways to satisfy this criterion. An enormous mass insures through the gravitation forces a very large connement time in stars. The connement time is the major factor of success. On Earth, two approaches are considered to achieve a large gain in fusion reactions: the magnetic and inertial connement. The rst method consists in using strong magnetic elds to maintain the combustible at a very high temperature. The plasma can be stabilised and conned away from the walls by the magnetic eld, to obtain a continuous production of the fusion energy:

this is the process of magnetic connement fusion [START_REF] Eriksson | Simulation of runaway electrons during tokamak dis-ruptions[END_REF][START_REF] Eriksson | Current Dynamics during Disruptions in Large Tokamaks[END_REF]. The devices considered to study the feasibility of magnetic connement fusion are the tokamaks, stellarators and other magnetic traps, which conne a plasma mixture of hydrogen isotopes in a magnetic eld produced by supra-conducting coils. The connement is achieved by choosing a toroidal geometry for the magnetic fusion reactor.

The second approach consists in bringing to very high density and temperature a small volume of fuel within an extremely short time by the use of very powerful lasers. In this case the connement is achieved by the inertia of imploding mass:

this is the process of inertial connement fusion [START_REF] Duderstadt | Inertial Connement Fusion[END_REF][START_REF]La fusion thermonuclaire inertielle par laser, Partie 1: L'interaction laser-matière[END_REF][START_REF] Lindl | Inertial Connement Fusion[END_REF]. The idea of inertial connement appeared in 1960 with the invention of the laser. This new source of radiation brought new attractive perspectives for fusion since high energy uxes could be reached (of the order of 10 14 -10 15 W/cm 2 ). One of the rst ideas was to transfer the laser beams energy to a small spherical target, in order to heat it suciently and to ignite a thermonuclear ame. In this context, Russian, American and French laboratories started to study laser-matter interactions in the sixties. However, the idea to directly use the laser energy to ignite the target was quickly given up and in 1972 an "ablative" implosion scheme was proposed. This scheme, based on the action-reaction principle, consists in the irradiation of a deuterium-tritium spherical shell by the use of powerful laser beam carefully set to obtain a symmetric illumination. Under the eect of the laser-radiation, the external face of a spherical shell is vaporised and the plasma produced expands toward the exterior: this is called the ablation process. By action-reaction, the external plasma expansion thrusts the internal part of the shell toward the center. The ablative pressure compresses and heats the shell while the gas contained inside the shell creates the hot spot in the center where the ignition conditions can be created. In that case, a nuclear combustion ignited in the hot spot propagates radially and consumes the fuel in the compressed Introduction shell. To access such a laser energy, two facilities were built: the Laser Mégajoule (LMJ) in Bordeaux and the National Ignition Facility (NIF) in Livermore (USA).

The NIF is operational since 2009. The ignition process described here supposes a perfectly isotropic irradiation with a high number of laser beams perfectly balanced and synchronised, which makes the direct drive approach particularly challenging.

Also, in order to overcome the contradictory conditions of heating and compressing the target at the same time, another ignition scheme called the fast ignition scheme was proposed in 1994. The idea is to separate the fuel compression and heating. One starts by compressing the target by using the direct drive scheme, then launches a short highly energetic beam of particles to ignite the compressed fuel.

Another way to ensure an isotropic illumination is to use the indirect drive approach.

It consists in the irradiation, of a hollow cylindrical metallic cavity (usually made of gold) of a few millimetres in diameter and one centimer long called "hohlraum" from inside by using many intense laser beams. The energy deposited in the holraum is converted in X-rays and creates an isotropic illumination of the target inside the cavity.

Figure 1: Irradiation scheme of a target by intense laser light in the case of the direct drive approach (left) and the indirect drive approach (right).

The understanding of the processes involved in inertial connement fusion requires a deep and challenging study of the collisional kinetic transport of multiple species of charged particles. In addition, a rigorous mastering of the coupling with other phenomena involved in the ignition process such as radiative transfer [START_REF] Berthon | Mathematical models and numerical methods for radiative transfer[END_REF][START_REF] Gonzáles | Contribution a l'étude numérique de l'hydrodynamique radiative: des expériences de chocs radioatifs aux jets astrophysiques[END_REF][START_REF] Mihalas | Foundations of Radiation Hydrodynamics[END_REF], neutron production [START_REF] Dellacherie | Contribution à l'analyse et a la simulation numérique des equations cinétiques décrivant un plasma chaud[END_REF] or laser-plasma absorption processes [190] is also required. The present document mainly focuses on the modelling and the numerical study of the transport of charged particles created in the zone of laser plasma interaction at the outer part of the target [START_REF] Dautray | La fusion thermonucléaire Inertielle par Laser. Collection du CEA[END_REF].

In order to take into account dierent aspects of the particle transport numerous contributions have been made. We refer here to [START_REF] Toro | Riemann Solvers and Numerical Methods for Fluids dynamics[END_REF] for a review of Vlasov-Fokker-Planck numerical modelling for inertial connement fusion plasmas. We mention here, the work of N. Crouseilles and F. Filbet [START_REF] Crouseilles | Numerical approximation of collisional plasmas by high order methods[END_REF] who developed in 2004 a Maxwell-Fokker-Planck-Landau numerical code integrating important developments dealing with the discretisation of collisional operators [START_REF] Buet | Numerical analysis of conservative and entropy schemes for the Fokker-Planck-Landau equation[END_REF][START_REF] Buet | Fast algorithm for numerical, conservative and entropy approximation of the Fokker-Planck-Landau equation[END_REF][START_REF] Buet | Comparison of numerical scheme for Fokker-Planck-Landau equation[END_REF][START_REF] Degond | An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory[END_REF][START_REF] Degond | The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case[END_REF]. We also mention [ [START_REF] Buet | Comparison of Numerical Schemes for Fokker-Planck-Landau Equation[END_REF][START_REF] Filbet | A Numerical Method for the Accurate Solution of the Fokker-Planck-Landau Equation in the Non Homogeneous Case[END_REF][START_REF] Lemou | Fast implicit schemes for the Fokker-Planck-Landau equation[END_REF][START_REF] Lemou | Implicit schemes for the Fokker-Planck-Landau equation[END_REF]. These work were largely pursued in [START_REF] Duclous | Modélisation et Simulation Numérique multi-échelle du transport cinétique électronique[END_REF][START_REF] Mallet | Contribution à la modélisation et à la simulation numérique multi-échelle du transport cinétique électronique dans un plasma chaud[END_REF] permitting to include the contribution of magnetic elds and explore relativistic regimes.

Study of particle transport in plasmas

Research directions and objectives

Kinetic descriptions are known to be very accurate to describe the transport of charged particles in a plasma. However, they are also known to be computationally expensive to describe most realistic physical applications. An alternative way consists in considering uid descriptions based on averaged physical quantities. However, such a macroscopic description is not suciently accurate. For example, in the context of inertial connement fusion, the plasma particles may have an energy distribution far from the thermodynamic equilibrium so that the uid description is not adapted. Moreover kinetic eects like the non local transport [START_REF] Brantov | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF][START_REF] Marocchino | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF], wave damping or the development of instabilities [START_REF] Drake | Parametric instabilities of electromagnetic waves in plasmas[END_REF] can be important over time scales shorter than the collisional time so that uid simulations are insucient and kinetic codes have to be considered to capture the physical processes. Kinetic approaches are usually limited to times and lengths much shorter than those studied with uid simulations. It is therefore an important challenge to describe kinetic eects using reduced kinetic codes operating on uid time scales [START_REF] Filbet | A hierarchy of hybrid numerical methods for multi-scale kinetic equation[END_REF][START_REF] Helluy | Reduced Vlasov-Maxwell modeling[END_REF].

The angular moments models represent an alternative method situated in between the kinetic and the uid models. They require computational times shorter than kinetic models and provide results with a higher accuracy than uid models. They originate from an angular moments average [START_REF] Minerbo | Maximum entropy Eddigton Factors[END_REF][START_REF] Pomraning | Maximum entropy Eddington factors and ux limited diffusion theory[END_REF] of the kinetic equations. The idea is to keep the velocity modulus (denoted ζ in this work) as a variable. That allows to consider the particle distributions in energy far from equilibrium, while using a simplied description of particle angular distribution. Such models are obtained by integration of the kinetic equation in angle (integration on the unit Introduction sphere). Thus a hierarchy of moments equations can be obtained. There exist several moment models whose dierences come from the choice of the closure relation. In this document we consider the angular moments models [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF] based on an entropy minimisation principle. The entropy minimisation problems have been widely studied in [START_REF] Alldredge | High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem[END_REF][START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF][START_REF] Minerbo | Maximum entropy Eddigton Factors[END_REF][START_REF] Muller | Rational Extended Thermodynamics[END_REF][START_REF] Subramaniam | Statistical representation of a spray as a point process[END_REF]. The underlying distribution function is

given by an exponential of a polynomial function depending on the particle energy and it is therefore non negative. Moreover, these closures verify the fundamental mathematical properties [START_REF] Groth | Towards physically-realizable and hyperbolic moment closures for kinetic theory[END_REF][START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF][START_REF] Mcdonald | Towards realizable hyperbolic moment closures for viscous heat-conducting gas ows based on a maximum-entropy distribution[END_REF] such as hyperbolicity and entropy dissipation. However, their solutions could be rather dierent from the solution of the full kinetic equation. Moreover, from the numerical point of view, even if the closure is well dened, computational challenges remain. In particular, the resolution of the entropy minimisation problem can be very computationally costly and we refer to [START_REF] Alldredge | High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem[END_REF] for a specic treatment.

The angular M 1 model is largely used in the context of radiative transfer [START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF][START_REF] Charrier | Multigroup model for radiating ows during atmospheric hypersonic re-entry[END_REF][START_REF] Dubroca | Entropic moment closure hierarchy for the radiative transfert equation[END_REF][START_REF] Ripoll | An averaged formulation of the M1 radiation model with presumed probability density function for turbulent ows[END_REF][START_REF] Ripoll | A factored operator method for solving coupled radiation-hydrodynamics models[END_REF][START_REF] Turpault | Multigroup half space moment appproximations to the radiative heat transfer equations[END_REF]202], however it is relatively new in applications to the transport of charged particles in plasma [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]. Therefore the rst objective of this work is to provide a better understanding of the advantages and the limits of these reduced models for plasma physics modelling. Since angular moments models can be seen as a compromise between kinetic and uid descriptions, they represent attractive natural candidates for capturing kinetic eects on large time scale. Therefore, another important objective is to propose appropriate numerical methods for computing the numerical solutions of angular moments models in long time regimes. Indeed, in the case where the characteristic quantities of the problem become large compared to the plasma parameters or the collisional parameters, stinesses appear in the considered set of equations and the model degenerates. In general, classical numerical schemes used for angular moments models are not able to correctly capture the asymptotic limit under suitable conditions on the time and space step. Also, in addition to correctly behaving in asymptotic regimes, the numerical methods also should preserve the fundamental properties of the angular moments models such as the preservation of the admissible sets for instance. This point is investigated in details in this document. Another objective concerns the simulation of multi-species particle transport. The electronic transport studies are often performed considering immobile ions [START_REF] Duclous | Modélisation et Simulation Numérique multi-échelle du transport cinétique électronique[END_REF][START_REF] Mallet | Contribution à la modélisation et à la simulation numérique multi-échelle du transport cinétique électronique dans un plasma chaud[END_REF]. Indeed, because of their large mass compared to electrons, the ion motion is often neglected considering small time intervals. However, working on long time scales requires to take the ion motion into account. This will give access to a more general and interesting physics related to inertial connement fusion applications. A signicant work is then required for the use of the angular moments models to the multi-species particle transport studies.

Structure of the manuscript

This work is situated in between the plasmas physics modelling, applied mathematics and numerical analysis. This manuscript is organised in three parts. The rst part is a contribution to plasma modelling through the scope of angular moments Study of particle transport in plasmas models. The second part of this document deals with the derivation of numerical methods for the description of the long time particle transport. Common schemes usually used for angular moments models preserve the admissibility of the numerical solutions. However, such schemes are in general not able to capture the correct solution when considering asymptotic regimes. The aim is to propose numerical schemes which handle the asymptotic limit under resonable constraints on the space and time steps. The third part is a contribution to multi-species modelling. The nal goal is the study of the electron and ion dynamics with an accurate model which is also numerically aordable for applications. In this work a simpler problem is investigated considering non-charged particles. The angular model is derived in a moving framework where the mean velocity is dened by the rareed gas dynamics.

Chapter 1. In this rst chapter we introduce the basic concepts essential for plasma physics modelling. These elements are used in the following chapters.

Chapter 2. In this chapter, we start recalling the principle of the angular moments closures. Angular moments models are widely used in numerical solutions of kinetic equations. While in the strongly collisional limit they are providing a good approximation of the full kinetic equation, their validity domain in the weakly collisional limit is unknown. The work of this chapter is devoted to dene the validity domain of the M 1 model and its extensions, the two populations M 1 and the M 2 angular moments models for the collisionless kinetic physics applications. Three typical kinetic plasmas eects are considered, which are the charged particle beams interaction, the Landau damping and the electromagnetic wave absorption in an overdense semi-innite plasma. For each case, a perturbative analysis is performed and the dispersion relation is established using the moments models. These relations are compared with those computed by considering the Vlasov equation. The validity limits of each model are demonstrated.

Chapter 3. In this chapter, the electronic M 1 model introduced in chapter 1 is applied here for electron transport studies in a hot collisional plasma. The moment extraction of the electron-electron collision operator from the kinetic collision operator, for this angular moments model, is challenging and some approximations are required. Firstly, we recall the collisional operators used for the electronic M 1 model proposed in [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF][START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF]. Then, a characterisation of the electron-electron and electron-ion collision operators is given and following [START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF] the electron plasma transport coecients are derived. It is shown that in the high Z limit the electronic Introduction tems are recalled following the ideas introduced in [START_REF] Chalons | Modied Suliciu relaxation system and exact resolution of isolated shock waves[END_REF][START_REF] Godlewski | Numerical approximation of hyperbolic system of conservation laws[END_REF][START_REF] Godunov | A dierence method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF][START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF][START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF]. These methods directly apply to angular moments model and are used in the next chapters.

Chapter 5. This section deals with the numerical resolution of the M 1 -Maxwell system in the quasi-neutral regime [START_REF] Crispel | An asymptotic preserving scheme for the two-uid Euler-Poisson model in the quasi-neutral limit[END_REF][START_REF] Degond | Numerical approximation of the Euler-Maxwell model in the quasineutral limit[END_REF]. In this regime the stiness of the stability constraints of classical schemes implies long calculation times. That is why we introduce a stable numerical scheme consistent with the transitional and limit models. This scheme is able to handle the quasi-neutrality limit regime without any restrictions on time and space steps. This approach can be easily applied to angular moment models by using a moments extraction. Finally, two physically relevant numerical test cases are presented for the asymptotic-preserving scheme in dierent regimes. The rst one corresponds to a regime where electromagnetic eects are predominant. The second one on the contrary shows the eciency of the asymptotic-preserving scheme in the quasi-neutral regime. In the latter case the illustrative simulations are compared with kinetic and hydrodynamic numerical results.

Chapter 6. This chapter is devoted to the derivation of an asymptotic-preserving scheme for the electronic M 1 model in the diusive regime. In the rst part of this section, the case without electric eld and the homogeneous case are studied. The derivation of the scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modied to enable the derivation of a numerical scheme which also satises the admissible conditions and is well-suited for capturing steady states. Moreover, it enjoys asymptotic-preserving properties and handles the diusive limit recovering the correct diusion equation. Numerical tests cases are presented, in each case, the asymptotic-preserving scheme is compared to the classical HLL [START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] scheme usually used for the electronic M 1 model. It is shown that the new scheme gives comparable results with respect to the HLL scheme in the classical regime. On the contrary, in the diusive regime, the asymptotic-preserving scheme coincides with the expected diusion equation, while the HLL scheme suers from a severe lack of accuracy because of its unphysical numerical viscosity. The second part of this section is devoted to the extension of the proposed numerical scheme proposed to the general case. The goal is to deal with the mixed derivatives which arise in the diusive limit leading to an anisotropic diusion. The derived numerical scheme preserves the realisibility domain and enjoys asymptotic-preserving properties correctly handling the diusive limit recovering the relevant limit equation. In addition, the cases with electric eld and varying collisional parameter are naturally taken into account with the present approach. Numerical test cases validate the considered scheme in the non-collisional and diusive limits.

Chapter 7. This study is a rst step towards the multi-species charged particles modelling. Before considering complex congurations dealing with charged Study of particle transport in plasmas particles interactions, in the present chapter, we only consider non-charged particles and apply the angular M 1 model in a moving frame to rareed gas dynamics. In the present work, the velocity framework is centered on the particle mean velocity.

First of all, the derivation of the angular M 1 moments model in the mean velocity frame is introduced. The choice of the mean velocity framework in order to enforce the Galilean invariance property of the model is highlighted. In addition, it is shown that the model rewritten in terms of the entropic variables is Friedrichs-symmetric.

Also, the derivation of the associated conservation laws and the zero mean velocity condition are detailed. Secondly, a suitable numerical scheme, preserving the realisable requirement of the numerical solution for the angular M 1 moments model in the mean velocity frame is proposed. Thirdly, some numerical results obtained considering several test cases in dierent collisional regimes are displayed.

Conclusion

We present our conclusions and some short and long time perspectives.

Introduction Résumé en français

Ce manuscrit est une contribution à la modélisation et aux méthodes numériques pour l'étude du transport de particules dans un plasma dense. Nous sommes intéressés par des plasmas chauds crées par lasers et le contexte général est la compréhension des processus conduisant à l'allumage des réactions de fusion. Cette thématique constitue la principale motivation de ce travail et est suivie comme direction de recherche générale. Cependant, de nombreux thèmes de recherche sont étroitement reliés à ce travail puisqu'ils présentent une physique similaire. Nous pouvons citer par exemple l'étude des écoulements hypersoniques [START_REF] Anderson | Hypersonic and High-Temperature Gas Dynamics[END_REF], la radiothérapie [11,12,13], la fusion par connement magnétique [START_REF] Eriksson | Simulation of runaway electrons during tokamak dis-ruptions[END_REF][START_REF] Eriksson | Current Dynamics during Disruptions in Large Tokamaks[END_REF], l'astrophysique [START_REF] Chièze | Gas and Dark Matter Spherical Dynamics[END_REF],

l'étude des éclairs et des ammes, la propulsion pour engin spatiaux [START_REF] Goebel | Fundamentals of electric propulsion: ion and Hall thrusters[END_REF] ou les processus de décontamination plasma.

Directions de recherches et objectifs

Les descriptions cinétiques sont connues pour être très précisent pour décrire le transport de particules chargées dans un plasma. Cependant, elles sont aussi connues pour êtres particulièrement coûteuse en terme de ressources informatiques lorsqu'elles sont utilisées pour décrire la plupart des applications physiques. Une approche alternative consiste à considérer une description basée sur des quantités physique moyennées. Cependant ce type de description macroscopique peut ne pas être susamment précise. Par exemple, dans le contexte de la fusion par connement inertielle, les particules constituant le plasma peuvent posséder une distribution énergétique éloignée d'une distribution énergétique à l'équilibre thermodynamique telle qu'une description uide n'est pas adaptée. De plus, les eets cinétiques comme le transport non-local [START_REF] Brantov | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF][START_REF] Marocchino | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF], l'amortissement d'ondes ou le développement d'instabilités [START_REF] Drake | Parametric instabilities of electromagnetic waves in plasmas[END_REF] peuvent être important sur des échelles de temps plus courtes que les périodes de collisions. Ainsi les simulations uides sont insusamment précisent et des codes cinétiques doivent êtres considérés pour capturer correctement le processus physique. Les approches cinétiques sont souvent limitées à des échelles de temps et de longueurs bien plus petites que celles étudiées avec des simulations uides. La description d'eets cinétiques par l'utilisation de modèles réduits opérant sur des échelles uides constitue un dé considérable [START_REF] Filbet | A hierarchy of hybrid numerical methods for multi-scale kinetic equation[END_REF][START_REF] Helluy | Reduced Vlasov-Maxwell modeling[END_REF].

Study of particle transport in plasmas 25

Chapter 1

Basic concepts for plasma physics modelling

In this rst chapter, we introduce some basics concepts necessary for plasma physics modelling. These notions will be used in the next chapters.

Kinetic description of a plasma

The rigorous derivation of kinetic models to describe the transport of particles is particularly challenging and is still an active eld of research. Indeed, two main problems remain opens in kinetic theory. The rst one deals with the rigorous derivation of the Boltzmann equation [START_REF] Boltzmann | Leçons sur la théorie des gaz[END_REF] starting from a set of particles evolving according to Newton's laws. This important problem has been partially solved (only for short times) by Lanford [START_REF] Lanford | Time evolution of large classical systems. Dynamical Systems, theory and applications[END_REF] considering the Boltzmann-Grad limit [START_REF] Grad | On the kinetic theory of rareed gases[END_REF] for hard spheres. Important related results have been obtained in [START_REF] Cercignani | The Mathematical Theory of Dilute Gases[END_REF][START_REF] Gallagher | From Newton to Boltzmann : the case of hard spheres and short-range potentials[END_REF][START_REF] Illner | Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum. Erratum and improved result[END_REF][START_REF] Illner | Global validity of the Boltzmann equation for two and three dimensional rare gas in vacuum[END_REF][START_REF] King | BBGKY hierarchy for positive potentials[END_REF][START_REF] Pulvirenti | On the validity of the Boltzmann equation for short range potentials[END_REF].

The second problem deals with the rigorous derivation of the Vlasov-Poisson system starting from a set of charged particles interacting through a Coulombian potential.

This issue has been solved in the case of a regular potential by Braun and Hep [START_REF] Braun | The Vlasov dynamics and its uctuations in the 1/n limit of interacting classical particles[END_REF] and Dobrushin [START_REF] Dobrushin | Vlasov equations[END_REF] and for singular potentials by Hauray and Jabin [START_REF] Hauray | N-particles approximation of the Vlasov equation with singular potential[END_REF] but the case of Coulombian potentials remains open.

In combination with the Vlasov equation, the Landau kinetic equation [START_REF] Landau | Kinetic equation for the case of Coulomb interaction[END_REF] is the most important kinetic model in the theory of collisional plasma physics. This equation is often called the Fokker-Planck-Landau equation because the Landau equation was derived in the Fokker-Planck form [START_REF] Rosenbluth | Fokker-Planck equation for an inverse-square force[END_REF] in 1957. The rst formal derivation of the Landau equation from the BBGKY hierarchy was performed by Bogolyubov in 1946 [START_REF] Bogoliubov | Problems of Dynamical Theory in Statistical Physics[END_REF]. The Landau collisional term can be seen as an approximation of the Boltzmann collision integral or the Balescu-Lenard collision integral [START_REF] Balescu | [END_REF]158]. Therefore, it is reasonable to consider the Vlasov or the Fokker-Planck-Landau equation to describe the transport of charged particles in a plasma.

In this chapter following [START_REF] Nicholson | Introduction to plasma theory[END_REF] a formal derivation of a kinetic equation is presented. This approach enables the derivation of the Klimontovich equation and the 1.1. Kinetic description of a plasma kinetic equation thereafter. Connections can be made between this approach and the Liouville approach [START_REF] Nicholson | Introduction to plasma theory[END_REF]196]. We refer to [204] for a review of Mathematical topics in collisional kinetic theory and the references therein for more rigorous derivations of kinetic equations.

The most complete microscopic description of a gas, considering a system of N particles in a volume V , is to describe the coordinates r i (t) and the momenta p i (t) of the N th particles over time. One can introduce a microscopic distribution function f micro (t, r, p) characterising the number of particles at the time t, in the phase volume d 3 rd 3 p dN = f micro (t, r, p)d 3 rd 3 p.

(1.1)

When the positions and velocities of each particle are known, the microscopic distribution function is completely dened by an exact expression. The microscopic distribution can be seen as a product of Dirac functions of all particles coordinates

f micro (t, r, p) = N i=1 δ( r -r i (t))δ( p -p i (t)), (1.2)
with δ( r) = δ( x)δ( y)δ( z) the Dirac function in three dimensions. Here in the phase space the distribution function is singular, it represents the coordinates of all particles in the phase space.

In order to derive a more employable model one introduces a continuous description considering a spatial average of the microscopic distribution function. The following continuous distribution is dened

f (t, r, p) = f micro (t, r, p) Va , (1.3)
where the operation

Va represents the spatial average on a volume V a . Obviously, the volume V a must be suciently large compared to mean volume attributed to each particle V a >> V /N . According to the statistical theory, the uctuation amplitudes of the average value is of order

N -1/2 a
, where N a is the number of particles in the volume V a . However, the volume V a must also be small compared to the total volume V, V a << V , in order to describe the system with sucient accuracy. In practice, the volume V a is dened by the spatial resolution of the measurement techniques and numerical simulations. The temporal evolution equation of the distribution function is now derived. According to the denition (1.2), the temporal evolution of the distribution function is due to the motion of all the particles. Therefore, the temporal derivative of the microscopic distribution function writes (1.4) where ∇ = ∂/∂ r is the spatial gradient and ∂ p = ∂/∂ p is the partial derivative in term of the momentum. The derivative of d t r i is by denition the velocity of the particle v i = p i /m and the derivative of p i satises the Newton law

∂ t f micro = d t N i=1 δ( r -r i (t))δ( p -p i (t)),
∂ t f micro = - N i=1 d t r i . ∇δ( r-r i (t))δ( p-p i (t))- N i=1 d t p i .∂ p δ( p-p i (t))δ( r-r i (t)),
∂ t p i = F i ,
where F i is the force applied on the particle i. Here we can drop the indexes of the velocity v i and the force F i in equation (1.4) using the property of the Dirac function aδ(x -a) = xδ(x -a). From equation (1.4), it follows that

∂ t f micro + v. ∇f micro + F micro .∂ p f micro = 0. (1.5)
This kinetic equation is known as the Klimontovich kinetic equation [START_REF] Klimontovich | The kinetic theory of electromagnetic processes[END_REF]. This equation is microscopic, describing a set a discrete particles and enables to derive an equation for the mean distribution function. In order to derive a continuous description for the mean distribution function (1.3), the Klimontovich equation (1.5) is averaged over a spatial volume V a as prescribed above. The average of the Klimontovich equation (1.5) reads

∂ t f + v. ∇f + F micro .∂ p f micro = 0. (1.6)
One can remark a problem arising with the average of the last term of the previous equation because of its nonlinearity. In order to develop this term a hypothesis is made called the weak particle correlation. It is supposed that the distance between particles is suciently large compared to the characteristic interaction distance, so that they are almost free. Their trajectories are almost regular, dened by mean forces with perturbations induced by the chaotic motion of others particles, which are of an inferior order. Here, the quantities are developed as an average quantity and a uctuation

f micro = f + δf micro , F micro = F + δ F micro , (1.7)
where the uctuations are small δf micro << f , with a zero mean value < δf micro >= 0. These denitions enable to expand the last nonlinear term of (1.6).

In a plasma, several species of particles are present. The minimum number is two:

the ions and the electrons. Considering the distribution function of the species α and retaining only the term of rst order one obtains the following classical kinetic equation

∂ t f α + v. ∇f α + F α .∂ p f α = 0.
(1.8)

Study of particle transport in plasmas

Maxwell's equations and macroscopic quantities

In a plasma, F α is called the Lorentz force which is induced by self-consistent electric and magnetic elds

F α = q α ( E + v ∧ B).
(1.9)

This kinetic equation with Lorentz self-consistent force is called the Vlasov equa-

∂ t f α + v. ∇f α + F α .∂ p f α = -δ F micro α .∂ p β δf micro β = β C αβ .
(1.10)

The new term in the right side of the equation is called the collision integral. The form of this integral will be investigated in section 1.4.

Maxwell's equations and macroscopic quantities

The kinetic equation (1.10) must be completed with equations for the mean elds,

E =< E micro > and B =< B micro >. The microscopic elds satisfy the Maxwell's equations [START_REF] Maxwell | A Dynamical Theory of the Electromagnetic Field[END_REF] ∇

∧ E micro = -∂ t B micro , (1.11)
∇.

E micro = ε -1 0 ρ micro , (1.12)
∇ ∧ B micro = µ 0 j micro + c -2 ∂ t E micro , (1.13)
∇. B micro = 0.

(1.14)

These elds are generated by the charged particles in the plasma. The microscopic charge density and the current density read

ρ micro (t, r) = α q α f micro α (t, r, p)d p, j micro (t, r) = α q α f micro α (t, r, p) vd p.
The previous microscopic Maxwell's equations can be averaged directly to obtain the following set of equations

∇ ∧ E = -∂ t B, (1.15)
∇.

E = ε -1 0 (ρ + ρ ext ), (1.16) ∇ ∧ B = µ 0 ( j + j ext ) + c -2 ∂ t E micro , (1.17)
∇. B = 0.

(1.18)

1. Basic concepts for plasma physics modelling where ρ ext and j ext are the external charge and current density. The charge density and the current density are also obtained from the distribution functions of particles

ρ(t, r) = α q α f α (t, r, p)d p, j(t, r) = α q α f α (t, r, p) vd p.
Electromagnetic forces are long range interactions, each particle follows a collective electromagnetic eld created by many others particles. This collective behaviour is the main dierence between plasmas and neutral gas. As for the charge and current density, average quantities can be dened. For each particles species we dene -the particle density

n α (t, r) = f α (t, r, p)d p,
-the mean velocity

u α (t, r) = 1 n α (t, r) f α (t, r, p) vd p, -the mean energy E α (t, r) = 1 n α (t, r) f α (t, r, p) m α v 2 2 d p, -the heat ux q α (t, r) = f α (t, r, p)m α ( v -u α ) 2 2 ( v -u α )d p.
(1.19)

Fluid Models

The kinetic description is particularly accurate. However, this description can be dicult to use and numerical calculations are only possible considering small plasma volumes and short time scales. Therefore, in many applications, one prefers to use reduced models considering that the distribution functions of particles remain close to Maxwellian distribution functions. In this case, the plasma is described with macroscopic quantities: the density, the mean velocity and the mean energy (or the temperature). This hypothesis is not always valid but can apply on very collisional plasmas. In this section, following [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF][START_REF] Krall | Principles of Plasma Physics[END_REF] the development of hydrodynamic models is introduced.

Study of particle transport in plasmas 1.

Fluid Models

The rst macroscopic equation is the particles conservation equation. This equation is derived integrating in velocity the kinetic equation and using the denitions of the density and the mean velocity, it comes that

∂n α ∂t + ∇.(n α u α ) = 0.
∂n α m α u α ∂t + ∇.( Pα ) -q α n α ( E + u α ∧ B) = β R αβ . (1.21)
The term in the right side comes from the integration of the collisions and is called the friction force

R αβ = p α C αβ d p α .
In the case of a collision between particles of the same species, the conservation of momentum implies that R αα = 0. In addition the conservation of the total momentum implies that α,β R αβ = 0. Therefore, it follows that

R αβ = -R βα .
In general, the friction force R αβ is proportional to the dierence of mean velocities of particles α and β and is dened using the collision frequency ν αβ such that

R αβ = -ν αβ m α n α ( u α -u β ).
The tensor pressure is split in three parts. First, we consider the pressure of the uid ow and the part linked with the thermal agitation

P αij = n α m α u i u j + m α w i w j f α (t, r, w)d w.
Then the kinetic part is split into a diagonal part and a symmetric part without trace

P αij = n α m α u i u j + δ ij p α + Π αij ,
where δ ij is the Kronecker symbol and p α the scalar pressure dened by

p α = 1 3 m α ( v -u α ) 2 f α d v.
The term Π α is called the stress tensor and is dened by

Π αij = m α (w i w j - 1 3 w 2 δ ij )f α (t, r, w)d w.

Basic concepts for plasma physics modelling

In the case of an isotropic plasma in the centre of mass framework, the stress tensor is zero and the pressure is a scalar quantity. A more detailed analysis shows that the stress tensor comes from the internal friction of the uid and leads to the viscosity phenomenon. In the case of a plasma without magnetic eld, the stress tensor reads

Π αij = -η α ∇ i u αj + ∇ j u αi - 2 3 δ ij ∇. u α , (1.22)
where η α is coecient of viscosity. In the case where the distribution function are close to a Maxwellian distribution function, one has the following relation between pressure and temperature p α = n α k B T α . Therefore, an additional equation is required to obtain the temperature.

The equation on the temperature is derived by multiplying the kinetic equation by the particle energy 1 2 m α v 2 and integrating over the momentum. One obtains

∂ t 1 2 n α m α u 2 α + 3 2 n α k b T α + ∇.( 1 2 n α m α u 2 α u α + 5 2 n α k b T α u α + q α ) = j α . E + β W αβ . (1.23)
The rst term in the right hand side describes the energy deposition due to the Ohmic heating. The second term is due to the exchange of energy due to collisions between particles

W αβ = m α v 2 α 2 C αβ d p α .
(1.24)

The conservation of the energy in elastic collisions implies the reciprocity condition W αβ = -W βα and W αα = 0. The second term in the left side of the equation describes the convective energy transport (terms proportional to u α ) and diusive energy transport due to the heat ux (1.19). Similarly to the stress tensor, the heat ux exists only in an inhomogeneous plasma. In the case of collisional plasma without magnetic eld and for small deviations from equilibrium one can show the heat ux is dened by the Fourier law [START_REF] Struchtrup | Macroscopic Transport Equations for Rareed Gas Flows[END_REF]

q α = -κ α ∇T α , (1.25)
where κ α is the thermal conductivity. One remarks that in each equation a superior order moment appears: in (1.20) the density is linked with the velocity. Equation

(1.21) links the velocity and the pressure tensor. Finally, equation (1.23) links the energy to the heat ux. One must cut somewhere this innite chain expressing the highest order moment as a function of inferior order ones. This procedure is called the closure of the uid equations. In this case q α and Πα must be determined. The formulae (1.25) and (1.22) are examples of closure for q α and Πα . However, this choice is not unique and other closure can be considered.

Study of particle transport in plasmas

C αβ (f α , f β ) = q 2 α q 2 β ln Λ 8πε 2 0 ∂ ∂p αi (| u| 2 δ ij -u i u j ) | u| 3 ( ∂f α ∂p αj f β -f α ∂f β ∂p βj )d v β , (1.26)
where u = v α -v β is the relative velocity.

C ei (f e ) = Z 2 n i e 4 ln Λ 8πε 2 0 m 2 e ∂ ∂v j v 2 δ js -v s v s v 3 ∂f e ∂v s ,
where n i is the ion density.

H(f α ) = α (f α ln f α -f α )d v α .
We multiply the kinetic equation (1.10) by ln f α , and integrate in velocity. The rst term rewrites

ln f α ∂ t f α d v α = ∂ t (f α ln f α -f α )d v α .
The right collisional term is integrated by part, and one obtains

- αβ q 2 α q 2 β ln Λ 8πε 2 0 ∂ ln f α ∂ p α Kαβ G αβ f α f β d v α d v β ,
where the vector G is given by

G αβ = ∂ ln f α ∂ p α - ∂ ln f β ∂ p β ,
and the tensor

K = (u 2 Id -u ⊗ u)/u 3 , (1.27)
is the integral kernel of the Landau collision operator (1.26). One remarks, this operator is symmetric therefore this property enables to switch the indices α and β in this formula and to rewrite it under the following form

αβ q 2 α q 2 β ln Λ 8πε 2 0 ∂ ln f β ∂ p β Kαβ G αβ f α f β d v α d v β ,
where we used the fact that G is anti-symmetric. These two forms are equivalent and summing the two expressions one obtains

∂ t H(f α ) + div x (F (f α )) = - 1 2 αβ q 2 α q 2 β ln Λ 8πε 2 0 G αβ Kαβ G αβ f α f β d v α d v β ,
Study of particle transport in plasmas 1.4. Collisions between particles: the Landau collisional operator where the term G K G = (u 2 G 2 -( u. G) 2 )/u 3 is positive and the entropy ux is dened by

F (f α ) = (f α ln f α -f α ) v α d v α .
Therefore, the integral is negative and the total entropy decreases in time.

Conservation of the total energy

An important feature of the Maxwell-Fokker-Planck-Landau system is the conservation of the total energy [START_REF] Hinton | Collisional transport in plasma[END_REF]. More precisely, under conditions at the boundaries of the domain the sum of the electromagnetic energy and the plasma energy is conserved over time. The demonstration is given if only electrons are considered.

The total energy E tot is dened by

E tot = ε 0 ( E) 2 + (c B) 2 2 + m 2 v 2 f d v.
Proposition 1.1. The Maxwell-Fokker-Planck system conserves the total energy under the following conditions on the frontier ∂Ω of Ω

ε 0 c 2 ( E ∧ B). n + m 2 | v| 2 ( v. n)f d v = 0.
This property is derived multiplying the Maxwell equations (1.17 ∂ ∂t

E 2 c 2 + B 2 + ∇ x ( E ∧ B) = - 1 c 2 ε 0 j. E.
(1.28)

Now, multiplying the electron kinetic equation by m v 2 2

and integrating in velocity one obtains

∂ ∂t m v 2 2 f d v + ∇ x m v 2 2 f vd v + q ∇ v (( E + v ∧ B)f ) v 2 2 d v = 0. (1.29)
The third term of the right side of this equation rewrites

∇ v .(( E + v ∧ B)f ) v 2 2 = ∇ v .(( E + v ∧ B)f v 2 2 ) -f ( E + v ∧ B). ∇ v v 2 2 .
Therefore, equation (1.29) simplies into

∂ ∂t m v 2 2 f d v + ∇ x m v 2 2 f vd v -j. E = 0.
(1.30)

1. Basic concepts for plasma physics modelling Finally, the total energy equation is obtained summing equations (1.30) and (1.28)

1 2 ∂ ∂t ε 0 ( E 2 + (c B) 2 ) + m v 2 f d v + ∇ x . ε 0 c 2 .( E ∧ B) + m v 2 f vd v = 0. (1.31)
The rst and second terms in the temporal derivative correspond to the electromagnetic energy and the plasma energy. The rst and second term in the spatial derivative are the electromagnetic energy ux (the term ε 0 c 2 ( E ∧ B) is called the Poynting vector) and the plasma energy ux. Then integrating over the space domain, equation (1.31) gives

1 2 ∂ ∂t Ω ε 0 ( E 2 +(c B) 2 )+ m v 2 f d v d Ω+ ∂Ω ε 0 c 2 .( E∧ B). n+ m v 2 ( v. n)f d v dσ = 0,
which gives proposition (1.1).

Study of particle transport in plasmas

Introduction

The aim of this chapter is to introduce the principle of the angular moment closure and to dene the validity domain of the M 1 , the two populations M 1 and the M 2 angular moments models for kinetic plasma physics applications. The purpose is to investigate if these three moments models are able to capture and describe correctly the basic phenomena occurring in a collisionless plasma. It has been shown

in [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF] that the M 1 model is very accurate in the case of isotropic congurations or with congurations where one direction is dominant. However the model loses precision in the case of an anisotropic conguration and in the limit where the mean free path is larger than the characteristic length of the problem. The accuracy can be improved by considering the two populations M 1 model or the M 2 model [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF].

However, their respective domains of validity are not dened either.

We consider here three classical kinetic eects, which are the interaction of charged particle beams, the Landau damping of a Langmuir plasma wave and the absorption of a electromagnetic wave incident normally on the boundary of an overdense plasma. Historically, the two beams instability was one of the rst studied plasma physics problems [START_REF] Buneman | Dissipation of Currents in Ionized Media[END_REF][START_REF] Krall | Principles of Plasma Physics[END_REF]. A beam of charged particles propagates in a plasma generating an oscillating electric eld exponentially increasing in time, and reducing the beam kinetic energy. The collisionless damping of plasma waves was rst discovered theoretically by Landau [START_REF] Landau | On the vibration of the electronic plasma[END_REF] then demonstrated in laboratory [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF][START_REF] Malmberg | [END_REF].

The latter physical phenomenon corresponds to the collisionless absorption of an electromagnetic wave incident on an overcritical plasma. A part of the wave energy is absorbed and transferred to the plasma while the other part is reected [190].

For these three phenomena, a perturbative analysis is performed and the dispersion show that the M 1 model is not able to describe the absorption phenomenon, while the two populations M 1 model and the M 2 model capture it qualitatively. In order to perform an explicit calculation of the absorption rate, the two limiting cases of a cold and hot electron plasma are studied corresponding to the low and high frequency skin eect [START_REF] Lifshitz | Plasma Kinetics[END_REF]. We show that in the cold plasma limit the two populations M 1 and the M 2 moments models give inaccurate absorption coecients. In the opposite limit the two populations M 1 model fails in describing correctly the phenomenon while the M 2 model provides an accurate result. Some conclusions are given in Section 2.7.

Principle of the angular moment closure

The purpose of angular moment models is to reduce the computational cost of the kinetic descriptions as introduced in Chapter 1. The electronic M 1 model [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF][START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF]

f 0 (ζ) = ζ 2 S 2 f ( Ω, ζ)d Ω, f 1 (ζ) = ζ 2 S 2 f ( Ω, ζ) Ωd Ω, f2 (ζ) = ζ 2 S 2 f ( Ω, ζ) Ω ⊗ Ωd Ω.
(2.1)

The complete angular integration of the Fokker-Planck-Landau equation, as performed in [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF][START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF][START_REF] Turpault | A consistent multigroup model for radiative transfer and its underlying mean opacity[END_REF] is detailed in Appendix A. In this Section we directly give the result of this angular moment extraction and detail the closure procedure. The 2. Angular moments models angular integration of the Fokker-Planck-Landau equation reads

     ∂ t f 0 + ∇ x .(ζ f 1 ) + q m ∂ ζ ( f 1 . E) = Q 0 (f 0 ), ∂ t f 1 + ∇ x .(ζ f2 ) + q m ∂ ζ ( f2 E) - q mζ (f 0 E -f2 E) - q m ( f 1 ∧ B) = Q 1 ( f 1 ), (2.2)
where the collisional operators Q 0 and Q 1 are given by

Q 0 (f 0 ) = 2α ee 3 ∂ ζ ζ 2 A(ζ)∂ ζ ( f 0 ζ 2 ) -ζB(ζ)f 0 , (2.3)
Q 1 ( f 1 ) = - 2α ei ζ 3 f 1 .
(2.4)

The coecients A(ζ) and B(ζ) write

A(ζ) = ∞ 0 min( 1 ζ 3 , 1 ω 3 )ω 2 f 0 (ω)dω, (2.5)
B(ζ) = ∞ 0 min( 1 ζ 3 , 1 ω 3 )ω 3 ∂ ω ( f 0 (ω) ω 2 )dω. (2.6)
The fundamental point of the moments models is the denition of a closure, which writes the highest moment as a function of the lower ones. This closure relation corresponds to an approximation of the underlying distribution function, which the moments system is constructed from. In the M 1 model (2.2), we need to dene f2 as a function of f 0 and f 1 . The closure relation originates from an entropy minimisation principle [START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF][START_REF] Minerbo | Maximum entropy Eddigton Factors[END_REF]. The underlying distribution function f is obtained as a solution of the following minimisation problem

min f ≥0 { H(f ) / ∀ζ ∈ R + , ζ 2 S 2 f ( Ω, ζ)d Ω = f 0 (ζ), ζ 2 S 2 f ( Ω, ζ) Ωd Ω = f 1 (ζ) }, (2.7)
where H(f ) is the angular entropy dened by

H(f ) = ζ 2 S 2 (f ln f -f )d Ω.
(2.8)

The solution of (2.7) writes [START_REF] Dubroca | Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif[END_REF][START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF] f

( Ω, ζ) = exp( a 0 (ζ) + a 1 (ζ) . Ω ), (2.9)
where a 0 (ζ) is a scalar and a 1 (ζ) a real valued vector. An important parameter is the anisotropy parameter α dened by

α = f 1 f 0 , ( 2
f 0 = 4π exp(a 0 ) sinh(|a 1 |) |a 1 | , f 1 = 4π exp(a 0 ) sinh(|a 1 |)(1 -|a 1 | coth(|a 1 |)) |a 1 | 3 a 1 .
These relations can be combined to give

α = 1 -|a 1 | coth(|a 1 |) |a 1 | 2 a 1 ,
then by taking the modulus of the previous expression

|α| = |a 1 | coth(|a 1 | -1) |a 1 | . (2.11)
The relation (2.11) cannot be inverted explicitly by hand. However, this relation determines a unique solution which can be computed numerically. Then the moment f2 can be computed assuming we know a 1

f2 = f 0 1 -χ 2 Īd + 3χ -1 2 f 1 | f 1 | ⊗ f 1 | f 1 | , (2.12)
where

χ = |a 1 | 2 -2|a 1 | coth(| a 1 |) + 2 | a 1 | 2 .
The χ factor can be computed as a function of the anisotropy parameter α χ( α) ≈ 1 + α 2 + α 4 3 .

(2.13)

The denition (2.12) enables to close the problem (2.2). We note here that the choice χ( α) ≈ 1 3 .

(2.14) corresponds to the P 1 closure largely used in the context of radiative transfer [START_REF] Franck | Approximate models for radiative transfer[END_REF].

Using the denitions of the angular moments (2.1), one remarks that f 0 is nonnegative as the integral of a non-negative distribution function. Similarly, taking the absolute value of f 1 with the denition (2.1), one shows that |f 1 | ≤ f 0 . Therefore we consider the following set of admissible states [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF] dened by

A = (f 0 , f 1 ) ∈ R 2 , f 0 ≥ 0, |f 1 | ≤ f 0 .
(2.15)

2. Angular moments models

Two populations M 1 and M 2 angular moments models

The M 1 model is well adapted (2.2)-(2.12)-(2.13) to the case of a near-isotropic conguration, where |f 1 | << f 0 (| α| << 1). In this case it is equivalent to the P 1 model (2.2)-(2.12)- (2.14). It provides also a good approximation in the case of one dominant direction (| α| ≈ 1) [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]. However, for the other values of α, the M 1 model may be not suciently accurate [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]. In order to improve the accuracy of the model in intermediate cases one can consider the two populations M 1 model [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]202] and the M 2 model [START_REF] Alldredge | High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem[END_REF][START_REF] Hauck | High-order entropy-based closures for linear transport in slab geometry[END_REF]. This section provides a description of the two populations M 1 model and the M 2 model.

Two populations M 1 model

In [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]202], it was suggested to decompose the distribution function into two parts. One part for particles with positive velocities and another one for particles with negative velocities. The total distribution function writes

f = f -+ f + ,
where f -= f | vx<0 describes the particle with negative velocities and f + = f | vx>0 the particles with positive velocities. We can now dene the zeroth order angular moments f - 0 and f + 0 . Similarly the rst angular moments are dened as

v y v z v x v θ ϕ
f 1 + (ζ) = ζ 2 2π 0 π/2 0 f (ζ, Ω) Ω sin(θ) dθdϕ, (2.16
)

f 1 - (ζ) = ζ 2 2π 0 π π/2 f (ζ, Ω) Ω sin(θ) dθdϕ, (2.17)
where Ω = (cos θ, sin θ cos φ, sin θ sin φ), see Fig.

(ζ) = ζ 2 S 2 f ( Ω, ζ) Ω ⊗ Ω ⊗ Ω d Ω.
The entropy minimisation principle for the M 2 model [START_REF] Alldredge | High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem[END_REF][START_REF] Hauck | High-order entropy-based closures for linear transport in slab geometry[END_REF] implies that the underlying distribution function writes

f ( Ω, ζ) = exp( a 0 (ζ) + a 1 (ζ) . Ω + ā2 (ζ) : Ω ⊗ Ω ), (2.19)
where a 0 (ζ) is a scalar, a 1 (ζ) a real valued vector and ā2 (ζ) a real valued tensor of order two. The notation ⊗ represents the tensor product and : is the two times contracted product. The equations of the M 2 model write

         ∂ t f 0 + ∇ x .(ζ f 1 ) + q m ∂ ζ ( f 1 . E) = 0, ∂ t f 1 + ∇ x .(ζ f2 ) + q m ∂ ζ ( f2 E) -q mζ (f 0 E -f2 E) = 0, ∂ t f2 + ∇ x .(ζ f3 ) + q m ∂ ζ ( f3 E) -q mζ ( f 1 ⊗ E + 2 f3 E + E ⊗ f 1 ) = 0.
(2.20)

Angular moments models

For clarity, the contribution of the magnetic eld has been removed. The selfconsistent magnetic eld leads to superior order terms in the perturbative analysis performed in the next sections and can be neglected because the unperturbed distribution function is isotropic.

The aim of the next sections is to dene the validity domain of the M 1 , the two populations M 1 and the M 2 moments models. The purpose is to investigate if these three moments models are able to capture and describe correctly the basic phenomena occurring in a collisionless plasma.

Particle beam interaction

In this section we study the interaction of electron beams using the M 1 model. We demonstrate that the dispersion relation obtained from the M 1 model agrees exactly with the one obtained from the Vlasov equation.

Dispersion relation for the M 1 model in the one-dimensional electrostatic case

In the electrostatic case, only one component of the electric eld is considered (E x ). The system of equations (2.2) and the Poisson equation read as follows

   ∂ t f 0 + ∂ x (ζf 1x ) -∂ ζ (E x f 1x ) = 0, ∂ t f 1x + ∂ x (ζf 2xx ) -∂ ζ (E x f 2xx ) + (f 0 -f 2xx )Ex ζ = 0, ∂ x E x = 1 - ∞ 0 f 0 (ζ)dζ, (2.21)
where the time is normalised to the inverse of the electron plasma frequency ω pe = e 2 n 0 /mε 0 , the velocity is normalised to the thermal velocity v th = T /m, the length to the Debye length λ D = v th /ω pe , the electric eld is normalised to E p = mv th ω pe /e and 0 is the vacuum dielectric permittivity. Only one component of the closure relation (2.12) is non zero. According to equation (2.10)

f 2xx = χ(α x )f 0 .
Let us consider a perturbation of the electric eld δE x and the corresponding perturbation of the zeroth and rst moment δf 0 and δf 1x

   E(t, x) = 0 + δE x (t, x), f 0 (t, x, ζ) = F 0 (ζ) + δf 0 (t, x, ζ), f 1x (t, x, ζ) = F 1x (ζ) + δf 1x (t, x, ζ),
where F 0 , F 1x correspond to the homogeneous stationary solution of system (2.21). For the sake of clarity, we omit in the following the arguments t, x and ζ in the Study of particle transport in plasmas 2.4. Particle beam interaction equations. The linearised system (2.21) reads

   ∂ t δf 0 + ∂ x (ζδf 1x ) -∂ ζ (F 1x δE x ) = 0, ∂ t δf 1x + ∂ x ((χ(F) -χ (F)F)ζ δf 0 + χ (F)ζ δf 1x ) -∂ ζ (F 2xx δE x ) + (F 0 -F 2xx )δEx ζ = 0, ∂ x δE x = - ∞ 0 δf 0 (ζ)dζ, (2.22)
where F 2xx = χ(F)F 0 and F = F 1x /F 0 . We dene the Fourier transform f of a function f as

f (ω, k) = 1 2π +∞ -∞ +∞ -∞ f (t,
x)e i(ωt-kx) dxdt.

The Fourier transform of the rst and second equations of (2.22) results in

-iωδ f0 + ikζδ f1x = ∂ ζ (F 1x δ Êx ), (2.23)
-iωδ f1x + ikζ(χ(F) -χ (F)F)δ f0 + ikζχ (F)δ f1x = (2.24) ∂ ζ (χ(F)F 0 δ Êx ) - (1 -χ(F))F 0 δ Ê ζ .
For the sake of simplicity, in the following the quantities δ f are replaced by δf .

Inserting (2.23) into (2.24) gives

δf 0 = - 1 iD [(ω -kζχ (F))∂ ζ F 1 + kζ∂ ζ (χ(F)F 0 ) -k(1 -χ(F))F 0 ] δE, (2.25)
with

D = ω 2 -ωkζχ (F) -k 2 ζ 2 [χ(F) -χ (F)F].
The Fourier transform of the third equation of (2.21) gives ikδE = -∞ 0 δf 0 (ζ)dζ.

Then the integration of (2.25) leads to

1 + ∞ 0 1 Dk [(ω -kζχ (F))∂ ζ F 1 + kζ∂ ζ (χ(F)F 0 ) -k(1 -χ(F))F 0 ] dζ = 0. (2.26)
This equation is the general formulation of the dispersion relation for the M 1 model in the one dimensional electrostatic case. It is applied to the electron beams in the next subsection and to the Landau damping in the next section.

Electron beams

Let us consider the electron distribution function as a sum of n beams of particles aligned along the x-axis. The distribution function writes

f (x, v) = 1 n n l=1 δ(v -v l ),

Angular moments models

where v l = l |v l | = l ζ l , with l = ±1 depending on the direction of propagation of electrons. Now, the corresponding zeroth and rst moments F 0 , F 1 are given by,

F 0 (x, ζ) = ζ 2 1 n n l=1 δ(ζ -ζ l ), F 1 (x, ζ) = ζ 2 1 n n l=1 l δ(ζ -ζ l ).
After a simple computation using (2.13) and the denition of F, we obtain that F = l , χ(F) = 1 and χ (F) = 2

1 - 1 n n l=1 v 2 l (ω -kv l ) 2 = 0,
which agrees exactly with the dispersion relation obtained from the Vlasov equation [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF].

In this part we have shown that the M 1 model (2.2)-(2.12)-(2.13) correctly describes the particle beams interaction. In the case of dierent energy beams, the dispersion relation obtained using the M 1 model coincides exactly with the one obtained from the Vlasov equation. It is then evident that more accurate models such as the two populations M 1 model or the M 2 model give the same dispersion equation.

We study in the next part, the Landau damping. It is shown that even if the M 1 model captures qualitatively the phenomenon, it is not accurate enough to describe it quantitatively.

Dispersion of an electron plasma wave

Landau damping is a well-known process in plasma physics, which also presents a large interest in some other elds such as galaxy dynamics [START_REF] Lynden-Bell | Statistical mechanics of violent relaxation in stellar systems[END_REF]. The aim of this part is to study if the M 1 model (2.2)-(2.12)-(2.13) is able to describe electron plasma waves including the Landau damping eect. We suppose that the equilibrium solution to the Vlasov equation is given by a Maxwellian function

f (ζ) = (2π) -3/2 exp(-ζ 2 /2).
(2.27)

The dispersion relation is established from the Vlasov equation in [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF] ω

= √ 1 + 3k 2 - i k 3 π 8 exp(- 1 2k 2 ), (2.28)
for small k << 1. The negative imaginary part corresponds to the Landau wave damping. In the following, we perform the dispersion analysis of the Landau wave damping using the three moments models.

Study of particle transport in plasmas 2.5.1 M 1 model applied to an electron plasma wave

In this case, the two rst moments are given by,

F 0 (ζ) = ζ 2 2 π 1 2 exp(- ζ 2 2 ), F 1x (ζ) = 0, (2.29)
with F = 0, χ(F) = 1/3 and χ (F) = 0.

1 + ∞ 0 ζ∂ ζ F 0 (ζ) -2F 0 (ζ) 3ω 2 -k 2 ζ 2 dζ = 1 - 2 π 1 2 ∞ 0 ζ 4 exp(-ζ 2 2 ) 3ω 2 -k 2 ζ 2 dζ = 0.
Using the Landau theory [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF] we obtain an approximate dispersion relation assuming a large phase velocity ω/k >> 1 and a weak damping Im(ω) << Re(ω) ≈ 1. The pole ω/k lies near the real ζ axis, and by using a contour prescribed by Landau with a small semicircle around the pole, the residue formula makes the previous equation equal to

1 = - √ 2 √ πk 2   P ∞ 0 ζ 4 exp(-ζ 2 2 ) ζ - √ 3ω k ζ + √ 3ω k dζ + iπ ζ 4 exp(-ζ 2 2 ) ζ + √ 3ω k ζ= √ 3ω k   , (2.30)
where P stands for the Cauchy principal value. As in the case of plasma waves, the main contribution to the integral comes from velocities ζ << ω/k, we perform a

Taylor expansion for the rational fraction in

1 ζ 2 - √ 3ω k 2 = - k √ 3ω 2 1 1 -ζ 2 √ 3ω k 2 ≈ - k √ 3ω 2    1 + ζ 2 √ 3ω k 2    .
Equation (2.30) then reads

1 = 1 ω 2 + 5 3 k 2 ω 4 - i √ 2 π k 2 ζ 4 exp(-ζ 2 2 ) ζ + √ 3ω k ζ= √ 3ω k . (2.31)
We consider the imaginary part of ω as a small perturbation and write

ω = ω 0 + iδω, (2.32)
with δω << ω 0 . Inserting (2.32) into (2.31), neglecting the terms of order (δω) 2 leads to

ω 2 0 + 2iδωω 0 = 1 + 5k 2 3ω 2 0 (1 - 2iδω ω 0 ) -if (ω 0 + iδω, k), (2.33)
2. Angular moments models where

f (ω 0 + iδω, k) = 3 √ 6π(ω 0 + iδω) 5 k 5 exp( -3ω 2 0 k 2 ) exp(- 6iδωω 0 k 2 ).
Considering the following linearisation

f (ω 0 + iδω, k) = f (ω 0 , k) + iδωf (ω 0 , k)
into (2.33) and using the fact δω << ω 0 gives

ω 2 0 = 1 + 5k 2 /
f ± (v) = 1 (2π) 3 2 exp(- v 2 2 )H(± cos(θ)),
where H is the Heaviside function. The corresponding reduced distribution functions are given by,

F ± 0 (ζ) = ζ 2 1 2π 1 2 exp(- ζ 2 2 ), F ± 1x (ζ) = ± 1 2 F ± 0 (ζ).
(2.35)

The anisotropic coecients are calculated using (2.13), χ(

F -) = χ(F + ) = 7/16 and χ (F -) = -χ (F + ) = -1/2. The dispersion relation (2.26) writes as, 0 = 1 + ∞ 0 1 β + k (ω -kζχ (F + ))∂ ζ F + 1 + kζ∂ ζ (χ(F + )F + 0 ) -k(1 -χ(F + ))F + 0 dζ + ∞ 0 1 β -k (ω -kζχ (F -))∂ ζ F - 1 + kζ∂ ζ (χ(F -)F - 0 ) -k(1 -χ(F -))F - 0 dζ, = 1 + ∞ 0 1 k 0.661 ω 2 ζ 2 + 0.079 ζ 6 k 2 + 0.063 k 2 ζ 4 -0.887 ζ 4 ω 2 (ω 2 -ω 2 1 k 2 ζ 2 )(ω 2 -ω 2 2 k 2 ζ 2 ) F + 0 dζ, ω = √ 1 + 2.916k 2 -i 0.19 k 3 + 0.085 k 5 exp(- 0.88 k 2 ),
which is close to the dispersion relation (2.28) obtained from the Vlasov equation.

The real part of the dispersion relation is almost exact. Considering the imaginary part, the pre-exponential factor varies in (0.19/k 3 + 0.085/k 5 ) instead of 0.1398/k 3 and the coecient in the exponential is 0.88/2k 2 instead of 3/2k 2 . The representation of the dissipation coecient in Fig. 2.2 shows that the two populations M 1 model gives a more accurate result than the previous model for k < 0.6. The two populations M 1 model is then a good candidate to model the Landau damping.

M 2 model

In this part the dispersion relation is established using the M 2 model (2.20) and compared to the one obtained with the Vlasov equation. It is shown that the M 2 model gives more accurate results than the two populations M 1 model.

In the one dimensional electrostatic case, after normalisation the M 2 model (2.20)

writes    ∂f 0 + ζ∂ x (f 1x ) -E x ∂ ζ (f 1x ) = 0, ∂ t f 1x + ζ∂ x (f 2xx ) -E x ∂ ζ (f 2xx ) + Ex ζ (f 0 -f 2xx ) = 0, ∂ t f 2xx + ζ∂ x (f 3xxx ) -E x ∂ ζ (f 3xxx ) + Ex ζ (2f 1x + 2f 3xxx ) = 0.
(2.36)

The derivation is similar to the one performed for the M 1 model with an additional equation. The term f 2xx needs to be developed with the perturbative analysis

f 2xx = F 2xx + δf 2xx .
(2.37)

In this case F 2xx can be calculated by using the equilibrium state (2.27)

F 2xx = F 0 /3,
with F 0 dened in equation (2.29). The term f 3xxx must be expressed as a function of the other terms. As opposed to the M 1 model closure (2.12), the M 2 model closure cannot be given explicitly. Nevertheless, using [START_REF] Alldredge | High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem[END_REF][START_REF] Groth | Towards physically-realizable and hyperbolic moment closures for kinetic theory[END_REF][START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF] and the equilibrium state (2.27), the rst terms of the development of f 3 are determined. The linearisation of f 3xxx results in

f 3xxx = 0 + 3 5 δf 1x .
52 Sébastien GUISSET

Angular moments models

Then the linearisation of (2.36) gives

   ∂ t δf 0 + ζ∂ x (δf 1x ), ∂ t δf 1x + ζ∂ x (δf 2xx ) -1 3 ∂ ζ (δE x F 0 ) + 2 3ζ δE x F 0 = 0, ∂ t δf 2xx + 3 5 ζ∂ x (δf 1x ) = 0. (2.38)
Following a development similar to the one performed for the M 1 model, the dispersion relation for the M 2 model writes

1 = 1 3 2 π +∞ 0 ζ 4 ω 2 -k 2 ζ 2 λ exp(- ζ 2 2 ) dζ,
where the coecient λ = 3/5.

Using a contour prescribed by Landau with a small semicircle around the pole, the residue formula applied to the previous equation leads to,

1 = 1 3 2 π P ∞ 0 ζ 4 exp(-ζ 2 2 ) ω 2 -k 2 ζ 2 λ dζ - iπ 2ωkλ ζ 4 exp(- ζ 2 2 ) ω k √ λ . (2.39)
As the phase velocity ω/k >> ζ, the rational fraction is expanded with a Taylor series

1 ω 2 -k 2 ζ 2 λ = 1 ω 2 1 + k 2 ζ 2 λ ω 2 .
(2.40)

Then the dispersion relation for the M 2 model reads.

ω = √ 1 + 3k 2 -i 0.123 k 5 exp(- 1.667 2k 2 ).
The real part of the dispersion relation is the same as the one obtained with the Vlasov equation. The imaginary part is dierent, the pre-exponential factor varies in 0.123/k 5 instead of 0.1398/k 3 and the coecient in the exponential is 1.667/2k 2 instead of 3/2k 2 but its representation in Fig. 2.2 shows a good accuracy of the model.

In conclusion, the dispersion and dissipation of the plasma wave found by using the M 1 model are shown to be inaccurate. One notices in Fig.

Collisionless skin eect

In contrast to the electrostatic plasma waves, the electromagnetic waves are not damped in a homogeneous collisionless plasma. However, the dissipation appears if the plasma is inhomogeneous. We consider here the case of a plane electromagnetic wave, which is normally incident on a semi-innite, overcritical plasma. Here the wave absorption is due to the electrons reecting from the plasma boundary in a skin layer. The aim of this part is to study how the moments models are able to model such a more complicated situation with an electromagnetic eld. The conductivity and absorption coecient obtained with the M 1 , two populations M 1 and M 2 models are compared to the conductivity and absorption coecient obtained with the Vlasov equation. We consider a low amplitude electromagnetic wave of a frequency ω assuming the linear approach.

Consider a semi-innite plasma with an electronic density n 0 higher than the critical density n c = mε 0 ω 2 /e 2 . The electromagnetic wave is reected at the vacuum plasma interface. We propose here to compute the fraction of wave energy absorbed in the plasma [190]. There are two components of the electromagnetic elds E y and B z . We suppose the Debye length λ De much smaller than the penetration depth and 54

Sébastien GUISSET 2. Angular moments models then the electrons are reected specularly at x = 0. In order to apply the Fourier transform we extend the plasma to the whole space by assuming that an electron coming from x > 0, which is reected in x = 0, comes from the ctive region x < 0.

The study is then extended to the entire space. The electrostatic eld E y is extended as an even function

E y (x) = E y (-x).
The Faraday equation gives

∂B z ∂t = - ∂E y ∂x ,
the magnetic eld is then to be extended as an odd function

B z (x) = -B z (-x).
In this model the electric eld is continuous at the surface x = 0 but not its rst derivative nor the magnetic eld. As introduced in [190], the ratio of the electric and magnetic elds at the plasma boundary is characterised by the surface impedance

E(0)/B(0) = Z Z = iω cπ +∞ -∞ dk ω 2 c 2 -k 2 + iωµ 0 σ yy , (2.41)
where σ is the plasma conductivity. Knowing the impedance one can calculate the absorption coecient

A = 4Re(Z) |1 + Z| 2 , (2.42)
which is related to the real part of the impedance. We suppose that the equilibrium solution is given by the Maxwellian function (2.27).

M 1 model for the plasma skin eect

In this part, the conductivity σ and the absorption coecient A are computed with the M 1 model (2.2)-(2.12)-(2.13). We show in this section that the M 1 model is not able to capture the absorption phenomenon.

There is no electromagnetic eld and no electron current in the unperturbed plasma. We consider solutions with the perturbation theory. The angular moments are expanded

f 0 (t, x, ζ) = F 0 (ζ) + δf 0 (t, x, ζ), (2.43) f 1x (t, x, ζ) = F 1x (ζ) + δf 1x (t, x, ζ), (2.44) f 1y (t, x, ζ) = F 1y (ζ) + δf 1y (t, x, ζ).
(2.45)

Study of particle transport in plasmas

where F 0 and F 1x are given in (2.29) and F 1y = 0. The following system corresponds to the M 1 linearised equations

               ∂δf 0 ∂t + ζ ∂δf 1x ∂x = 0, ∂δf 1x ∂t + ζ 3 ∂δf 0 ∂x = 0, ∂δf 1y ∂t + qδE y 3m ∂F 0 ∂ζ - 2qδE y F 0 3mζ = 0.
The Fourier transform of the last equation of (2.6.1) results in an explicit solution for f 1y

δf 1y = iq mω 2F 0 3ζ - 1 3
∂F 0 ∂ζ δE y .

(2.46)

Considering (2.18) and (2.46), the electric current is calculated

j y = iq 2 mω R + 2F 0 3 - ζ 3
∂F 0 ∂ζ dζ δE y .

(2.47)

Introducing the conductivity tensor σ such that j y = σ yy δE y , the integration by part in equation ( 2.47) provides σ yy = ie 2 n 0 mω .

(2.48)

Inserting this expression into equation (2.41) one obtains the impedance without any real part. Correspondingly there is no absorption, A = 0.

(2.49)

Therefore, the M 1 model (2.2)-(2.12)-(2.13) is not able to correctly model the absorption phenomenon. After linearisation of the M 1 model, we note there is no contribution of the space derivative in the third equation of (2.6.1). Then the conductivity (2.48) does not depend on the wave number k and there is no absorption. This is an important result showing a limitation in the M 1 model for collisionless plasma physics applications. More accurate models need to be used for studies of the electromagnetic wave absorption. The aim of the next section is to make a calculation using the two populations M 1 model.

Two populations M 1 model

Here, the conductivity and the absorption coecient are calculated using the two populations M

F + 1y = F - 1y = 0. σ yy = i q 2 2m R + (1+χ)F 0 ζ -(1 -χ) ∂F 0 ∂ζ -F 0 k ω ω + ζ(3χ -1)k + (1+χ)F 0 ζ -(1 -χ) ∂F 0 ∂ζ + F 0 k ω ω -ζ(3χ -1)k ζdζ.
(2.50)

The calculation of the previous equation leads to

σ yy = iω 2 pe ε 0 √ 2πv 3 th R -1 2 + 3 2 χ + kζ 2ω + ζ 2 (1-χ) 2v 2 th ω -ζ(3χ -1)k ζ 2 exp(- ζ 2 2v 2 th
)dζ.

(2.51)

The conductivity (2.51) cannot be evaluated analytically. We consider the two limiting cases ω/k << v th and ω/k >> v th .

Hot electron case

Following the method introduced in the previous section for the calculation of the integral in expression (2.51) in the limit ω/k << v th one obtains the following expression for the plasma conductivity

σ yy = iω 2 pe ε 0 √ 2πv 3 th k 3 - 4v 2 th k 2 (3χ -1) 4 - iπω 2 [(3χ -1) 2 + 1] 2(3χ -1) 4 .
(

It has to be compared to the one obtained with the Vlasov equation [190]

σ V lasov yy σ V lasov yy = iω 2 pe ε 0 √ 2πv th k ω √ 2π kv th -iπ .
(2.53)

In contrast to the M 1 model case, the conductivity in this case depends on the wave number k. The conductivity obtained with the two populations M 1 model is dierent from the one obtained with the Vlasov equation. Indeed, ignoring the constant values, the real part of the conductivity varies in ω 2 pe /v th k instead of ω 2 pe ω/v 2 th k 2 for the Vlasov equation and the imaginary part varies in ω 2 pe ω 2 /v 3 th k 3 instead of ω 2 pe /v th k for the Vlasov equation. Using the fact that ω << kv th the calculation of the impedance Z leads to

Z = - 2iω cπ +∞ 0 dk k 2 -i K k 3 , (2.54) with K = πω 3 ω 2 pe [(3χ -1) 2 + 1] 2 √ 2πv 3 th c 2 (3χ -1) 4
.

The impedance computation results in

Z = 2ωe -i 2π 5 5 sin( π 5 )c 5 K . ( 2
A = K 1 ω ω pe 2 5 v th c 3 5 1 + K 1 4 ω ω pe 2 5 v th c 3 5 2 + K 1 4 ω ω pe 2 5 v th c 3 5 2 , with K 1 = 8 cos( 2π 5 ) 5 sin( π 5 ) 2 √ 2π(3χ -1) 4 π(3χ -1) 2 + 1 1 5 ≈ 0.434.
This absorption coecient has to be compared to the one obtained with the Vlasov equation [190] A

V lasov = K 2 ω ω pe 2 3 v th c 1 3 1 + K 2 4 ω ω pe 2 3 v th c 1 3 2 + K 2 4 ω ω pe 2 3 v th c 1 3 2 , (2.56)
with

K 2 = 16 √ 3 9 2 π 1 6 cos( π 3 ) ≈ 1.428.
The coecient ω/ω pe varies as the power 2/5 instead of 2/3 for the Vlasov equation and v th /c varies as the power 3/5 instead of 1/3 for the Vlasov equation.

Cold electron case

We now explore the limit ω >> kv th , where the conductivity equation (2.51)

gives

σ yy = iω 2 pe ε 0 √ 2πv 3 th k 3 (3χ + 1) √ 2πk 3 v 3 th 2ω - iπω 4 (1 -χ) (3χ -1) 4 v 2 th k 2 exp - ω 2 2v 2 th k 2 (3χ -1) 2 .
(2.57)

This expression has to be compared with the one obtained with the Vlasov equation

σ V lasov yy = iω 2 pe ε 0 v th k √ 2π √ 2πkv th ω -iπ exp - ω 2 2v 2 th k 2 .
(2.58)

Here, the real part of the conductivity varies as ω 2 pe /ω similarly to the Vlasov equation but the imaginary part varies as ω 2 pe ω 4 /v 5 th k 5 instead of ω 2 pe /v th k for the Vlasov equation. We also observe for the two populations M 1 model, a presence of the term (3χ -1) 2 in the exponential factor instead of 1. Inserting equation (2.57) into the impedance equation (2.41) results in

Z = 6ω 6 ω 2 pe β c 3 v 5 th √ 2π √ 2v th (3χ -1) ω 8 - iω ω 2 pe α -ω 2 .

Angular moments models

Using the denition (2.42), the absorption coecient is

A M 1 = K 3 v th c 3 ω pe ω 2 1 + K 3 4 v th c 3 ω pe ω 2 2 + ω 2 pe ω 2 0 α -1 -1 2 , (2.59)
where K 3 and α are given by

K 3 = 384(3χ -1) 4 (1 -χ) √ 2π ≈ 0.822, α = (3χ -1) 2 ≈ 0.156.
This expression has to be compared with the one obtained with the Vlasov equation [190] A

V lasov = K 4 v th c 3 ω pe ω 2 1 + K 4 4 v th c 3 ω pe ω 2 2 + ω 2 pe ω 2 0 -1 -1 2 , (2.60)
with

K 4 = 16 √ 2π ≈ 6.383.
The two expressions of the absorption coecient are similar but the major dierence originates from the parameter α in the denominator of (2.59). The coecient ω/ω pe varies as the power 2 and v th /c varies as the power 3 exactly like in the Vlasov absorption coecient. The parameter α in the denominator of the two populations M 1 model coecient absorption makes a signicant dierence with the Vlasov equation absorption coecient. While a pole is reached for ω/ω pe = 1 for the Vlasov equation, the pole is reached when ωα /ω pe = 1 for the two populations M 1 model.

Even if in both limits the absorption phenomenon is captured qualitatively, the results are not satisfactory. This shows the limits of using the two populations M 1 model for studying laser plasma absorption. The aim of the next part is to see if these results can be improved using the M 2 model (2.20).

M 2 model

In this part the conductivity and the absorption coecient are calculated with the M 2 model (2.20). In this case the rst order moments in the perturbative development F 1x , F 1y , F 2xx , F 2xy and F 2yy are calculated using (2.1)

F 1x = F 1y = 0, Study of particle transport in plasmas F 2xx = F 0 3 , F 2xy = 0, F 2yy = F 0 3 ,
where F 0 is given by equation (2.29). On the contrary to the M 1 model closure (2.12), the M 2 model closure cannot be given explicitly [START_REF] Alldredge | High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem[END_REF]. Nevertheless, only the component f 3xyx of the tensor f 3 is required in this study. Using [START_REF] Alldredge | High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem[END_REF][START_REF] Groth | Towards physically-realizable and hyperbolic moment closures for kinetic theory[END_REF][START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF], one can show that the linearisation of f 3xyx around the equilibrium state (2.27) gives

f 3xyx = 0 + δf 1y 5 .
(2.61)

The linearisation of the M 2 model (2.20) leads to

∂ t δf 1y + ∂ x (ζδf 2xy ) + q 3m ∂ ζ (F 0 δE y ) - 2qδE y F 0 3mζ = 0.
Performing a Fourier transform of the previous equation one nds

δf 1y = -i qδE y 3m ∂F 0 ∂ζ -2 F 0 ζ ω - ζ 2 k 2 5ω
.

Following the method introduced in the two populations M 1 model section, one obtains the conductivity

σ yy = iω 2 pe ε 0 ω 3v 5 th 2 π   P ∞ 0 ζ 4 exp(-ζ 2 2v 2 th ) ω 2 -k 2 ζ 2 λ 1 dζ - iπω 3 2k 5 3 √ λ 1 exp ( -ω 2 2k 2 λ 1 v 2 th )   , (2.62)
where λ 1 = 1/5. The integral in this expression cannot be calculated analytically.

In order to perform the complete calculation, two limiting cases are considered: ω/k << v th and ω/k >> v th .

Hot electron case

Following the method introduced for the two populations M 1 model, the conductivity is

σ yy = iω 2 pe ε 0 ω 3v 5 th √ π - v 3 th k 2 - iπω 3 λ √ 2λk 5 . (2.63)
This expression has to be compared with the one obtained with the Vlasov equation v th c 5 7

1 + K 5 4 ω ω pe 2 7 v th c 5 7 2 + K 5 4 ω ω pe 2 7 v th c 5 7 2 , 60 Sébastien GUISSET 2. Angular moments models with K 5 = 49.651λ 2 √ λ cos( 3π 7 ) π √ 2π ≈ 0.025.
The coecient ω/ω pe varies as the power 2/7 instead of 2/3 for the Vlasov equation and v th /c varies as the power 5/7 instead of 1/3 for the Vlasov equation.

Cold electron case

In the limit ω >> kv th , the conductivity (2.62) reads

σ yy = iω 2 pe ε 0 v 5 th v 5 th ω - iω 4 √ π k 5 3λ 2 √ 2λ exp(- ω 2 2k 2 λv 2 th ) . (2.64)
This expression has to be compared with the one obtained with the Vlasov equation (2.58). In this case the real part of the conductivity varies in ω 2 pe /ω like for the Vlasov equation. This good behavior was already obtained with the two populations M 1 model. The imaginary part varies as ω 2 pe ω 4 /v 5 th k 5 like for the two populations M 1 model, instead of ω 2 pe /v th k for the Vlasov equation but the exponential factor is obtained using the M 2 model contrarily to the two populations M 1 model. The expression for the M 2 model absorption coecient reads

A = K 6 v th c 3 ω pe ω 2 1 + K 6 4 v th c 3 ω pe ω 2 2 + ω 2 pe ω 2 -1 -1 2 , with K 6 = 128 5 √ 10π ≈ 4.567.
This expression is compared with the one obtained from the Vlasov equation (2.60).

The coecient ω/ω pe varies as the power 2 and v th /c varies as the power 3 exactly like the Vlasov equation absorption coecient. As opposed to the two populations M 1 model, the pole is reached at ω/ω pe = 1 like for the Vlasov equation. In this limit, one observes the advantage in using the M 2 model compared to the two populations M 1 model.

The calculation of the impedance Z, has been performed using the conductivity expressions established in the hot and cold electron limits ω/k << v th and ω/k >> v th . However, equation (2.41), implies the integration over all k from minus innity to innity. We can consider that the calculation of the impedance, using equation (2.41), holds if the main contribution of the integral comes from a set of wave numbers k where the limiting expressions for the conductivity are valid. In order to check this assumption, the parameters ω/ω pe and v th /c are xed and the expression in the integral (2.41) is analysed as a function of k. We present here an example with ω/ω pe = 0.1 and v th /c = 0.8 to illustrate how one can validate our approach for these parameters. The same steps can be used for any choice of parameters in order to verify if the calculated absorption is valid.

The modulus integrand of the impedance for the Vlasov equation, the M 2 and M 1 two populations models are displayed in Fig. 2.3, using the expressions derived in the limit ω/k << v th . In this case the dimensionless wave number kc/ω pe must be larger than (ω/ω pe )/(v th /c) = 0.125. Indeed, according to Fig. 2.3 the main contribution to the integral comes from a set of wave numbers where the conductivity expressions are valid. Moreover, the position and the shape of integrand in the case of M 2 model agrees well with the Vlasov result. A second example is displayed in Figure 2.3: Representation of the modulus integrand of the impedance (2.41) as a function of k in the limit ω/k << v th in the case ω/ω pe = 0.1 and v th /c = 0.8. The modulus integrand has been multiplied by a factor 0.1 for the M 2 model and by 0.0025 for the M 1 two populations model. Fig. 2.4, with ω/ω pe = 0.3 and v th /c = 0.1 using the expressions established in the limit ω/k >> v th . In this case, the dimensionless wave number kc/ω pe must be smaller than (ω/ω pe )/(v th /c) = 3. Indeed, one can verify in Fig. 2.4 that the main contribution to the integral comes from a set of wave numbers where the conductivity expressions are valid.

In conclusion, it has been shown that the M 1 model (2.2)-(2.12)-(2.13) is not able to model the skin eect of an electromagnetic wave in an overdense plasma.

In the limit ω/k << v th , the two populations M 1 (2.2)-(2.18) and the M 2 (2.20) moments models both capture the absorption phenomenon qualitatively, but do not describe it correctly. In the opposite limit ω/k >> v th , the absorption phenomenon is not captured correctly by the two populations M 1 model. The M 2 model, on the contrary, correctly captures the phenomenon and the absorption expression obtained is very close to the one followed from the Vlasov equation. This study shows the limits of these three models for studies of laser plasma absorption. Higher moments models must therefore be used to correctly describe this phenomenon. In the hot electron limit, the M 3 model could be tested but the calculation is beyond the scope of this study.

Angular moments models

Conclusion

Particle beams interaction, Landau damping and collisionless skin eect have been studied using the M 1 , the two populations M 1 and M 2 moments models. By analytically deriving the dispersion relations, we have demonstrated that the particle beams interaction is correctly captured by the moments models. Landau damping is also captured by the three models, but the M 1 model is inaccurate while the two populations M 1 and M 2 moments models describe it accurately. The electromagnetic wave absorption coecients in the case of collisionless skin eect have been calculated with the three models. We have shown that the M 1 model is not able Study of particle transport in plasmas 2.7. Conclusion to model the absorption phenomenon. Two limit cases have been considered. In the case ω/k << v th , the two populations M 1 and the M 2 moments models both capture the absorption phenomena results but they are inaccurate. In the second case, ω/k >> v th , the two populations M 1 model does not describe correctly the absorption eect while the M 2 model is suciently accurate. Higher moments models such as the M 3 moments or full kinetic models can be used to correctly describe the absorption phenomenon in both limits. This work demonstrates through the Landau damping and the laser-plasma absorption that angular moments models have to be used carefully. These models do not always behave as a full kinetic model and can suer from a severe lack of accuracy depending on the phenomenon studied.

This study can be extended to other plasma eects and also to take into account collisional processes. In this direction, the next chapter is dedicated to the study of collisional operators for the electronic M 1 model.

Introduction

It was proposed to use laser pulses in order to compress a deuterium-tritium target and ignite nuclear fusion reactions. In this process the laser energy is deposited near the critical surface and than it is transported to denser parts of the plasma by electrons. This process plays a key role in the understanding of plasma phenomena such as, parametric [START_REF] Epperlein | [END_REF]173] and hydrodynamic [START_REF] Drake | Parametric instabilities of electromagnetic waves in plasmas[END_REF][START_REF] Shkarofsky | The Particle Kinetics of Plasmas M[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF] instabilities, laser-plasma absorption [START_REF] Guisset | Limits of the M1 and M2 angular moments models for kinetic plasma physics studies[END_REF]190], wave damping [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF][START_REF] Landau | On the vibration of the electronic plasma[END_REF], energy redistribution and hot spot formation [START_REF] Brantov | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF][START_REF] Marocchino | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF]. High energy, long pulse lasers produce a collisional ionised hot plasma, where the electron-ion mean free path is small compared to the plasma characteristic spatial size and the distribution function is close to the isotropic Maxwellian function. The physics of laser plasma interaction is described within the hydrodynamic plasma model. However, the moment extraction of the electron kinetic equation leads to an unclosed hydrodynamic set of equations. The closure of the system requires to express the uxes in terms of the hydrodynamic variables and electron plasma transport coecients. Spitzer and Härm rst derived the electron plasma transport coecients solving numerically the kinetic Fokker-Planck-Landau equation using the expansion of the electron mean free path over the temperature scale length. Their results have been reproduced in other works [START_REF] Balescu | Transport Processes in Plasma[END_REF][START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF][START_REF] Shrimpton | Statistical treatment of turbulent polydisperse particle systems[END_REF] using the early works of Chapman [START_REF] Chapman | On the Law of Distribution of Molecular Velocities, and on the Theory of Viscosity and Thermal Conduction, in a Non-Uniform Simple Monatomic Gas[END_REF][START_REF] Chapman | The Mathematical Theory of Non-Uniform Gases[END_REF] and Enskog [START_REF] Enskog | Kinetische Theorie der Vorgänge in Mässig Verdünnten Gasen[END_REF] for neutral gases. However, the Spitzer-Härm theory is valid in the local regime where the electron ux is proportional to the temperature gradient. Indeed the electron transport plasma coecients were derived in the case where the electron distribu-3.1. Introduction tion function remains close to the isotropic Maxwellian function. However, in the context of inertial connement fusion, the plasma particles may have the mean free path comparable with the temperature scale length so that the classical transport description is not adapted [179]. Moreover kinetic eects like the non local transport [START_REF] Brantov | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF][START_REF] Marocchino | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF], wave damping or the development of instabilities [START_REF] Drake | Parametric instabilities of electromagnetic waves in plasmas[END_REF] can be important over time scales shorter than the collisional time so that the uid description is insucient. Therefore, a kinetic description is more appropriate for the study of inertial connement fusion plasmas. However such a kinetic description is computationally expensive for describing real physical applications. Kinetic codes are limited to the time and length much shorter than those studied with uid simulations. It is therefore an essential issue to describe kinetic eects by using reduced kinetic codes operating on uid time scales.

It has been seen in Chapter 1, that angular moments models can be seen as a compromise between kinetic and uid models. The collisional electronic M 1 model is derived by integrating with respect to the velocity direction the Fokker-Planck-Landau equation. However, since the electron-electron collision operator is nonlinear, the moments extraction is complex. A possibility could be to approximate the electron-electron collision operator assuming that the main contribution of the distribution function comes from its isotropic part [START_REF] Yu | Conservative nite-dierence schemes for the Fokker-Planck equation not violating the law of increasing entropy[END_REF]. However, as mentioned in [START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF],

the collisional electronic M 1 model obtained by angular integration does not ensure the preservation of the admissibility states, that is, the angular moments derive from a positive underlying distribution function. Therefore, a new electron-electron collision operator was proposed in [START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF]. In this model, the angular integration leads to an electron-electron collision operator for the electronic M 1 model which preserves the admissible states. In this work, we start to recall the main results established in [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF][START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF] and complete them with an important result characterising the equilibrium states of the collision operators. Such fundamental properties make the model interesting for practical applications. In addition, to complete the validation of the considered collisional electronic M 1 model, we derive the electron transport coecients. It is shown that in the high ion charge (Z >> 1) limit the electronic M 1 model and the Fokker-Planck-Landau equation coincide in the close-equilibrium case. The electron transport coecients derived from the electron-electron collision operator used for the electronic M 1 model are compared with the ones obtained using the electron-electron collision operator for the Fokker-Planck-Landau equation. This chapter is organised as follows: rstly the collisional operators for the electronic M 1 model are introduced. Then, their properties are presented and completed by the characterisation of the equilibrium state. In Section 3, the electron transport coecients are derived using the collisional electronic M 1 model and compared with the ones obtained from the Fokker-Planck-Landau equation. The strategy proposed, based on an expansion on the Laguerre polynomials [START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF][START_REF] Chapman | The Mathematical Theory of Non-Uniform Gases[END_REF], is particularly ecient since the stiness in 1/ζ 3 in the electron-ion collision operator is removed. It is shown that accurate electron plasma transport coecients are obtained. Finally, Section 4 presents our conclusions.

3. Classical transport theory for the collisional electronic M 1 model

Collisional operators for the electronic M 1 model

This section provides a detailed description of the collisional operators used for the electronic M 1 model [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF][START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF].

Collisional electronic M 1 model

The derivation of the collisional operators, from the Fokker-Planck-Landau equation (1.10)-(1.26), for the electronic M 1 model is detailed in Annexe 1. However, the moment extraction of the electron-electron Landau collision operator (1.26) is complex because of its non-linearity [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF] and some approximations are required. It has been pointed out in [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF][START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF] that the electron-electron collisional operator derived in Annexe 1, does not preserve the admissible states. Consequently, in order to overcome this major drawback the following collisional electronic M 1 model has been proposed [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF][START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF]

     ∂ t f 0 (ζ) + ∇ x .(ζ f 1 (ζ)) + q m ∂ ζ ( f 1 (ζ). E) = Q 0 (f 0 ), ∂ t f 1 (ζ) + ∇ x .(ζ f2 (ζ)) + q m ∂ ζ ( f2 (ζ) E) - q mζ (f 0 (ζ) E -f2 (ζ) E) = Q 1 ( f 1 ) + Q 0 ( f 1 ), (3.1)
where the collisional operators Q 0 and Q 1 are given by

Q 0 (f 0 ) = 2α ee 3 ∂ ζ ζ 2 A(ζ)∂ ζ ( f 0 ζ 2 ) -ζB(ζ)f 0 , (3.2)
Q 0 ( f 1 ) = 2α ee 3 ∂ ζ ζ 2 A(ζ)∂ ζ ( f 1 ζ 2 ) -ζB(ζ) f 1 , (3.3)
Q 1 ( f 1 ) = - 2α ei ζ 3 f 1 . (3.4)
The coecients A(ζ) and B(ζ) write

A(ζ) = ∞ 0 min( 1 ζ 3 , 1 ω 3 )ω 2 f 0 (ω)dω, (3.5)
B(ζ) = ∞ 0 min( 1 ζ 3 , 1 ω 3 )ω 3 ∂ ω ( f 0 (ω) ω 2 )dω. (3.6)
Remark 3.1. One remarks that contrarily to the M 1 collisional model derived in Annexe 1, the contribution of the electron-electron collisional operator appears in both equations of (3.1). This modication enables to obtain the admissibility requirement.

Next we set,

F 0 (ζ) = f 0 (ζ) ζ 2 , F 1 (ζ) = f 1 (ζ) ζ 2 .

Collisional operators for the electronic M 1 model

As remarked in [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF], inserting expressions (3.5) and (3.6) into (3.2) and (3.4) gives the following equivalent expressions for Q 0 (f 0 ) and Q 0 ( f 1 )

         Q 0 (f 0 ) = ∂ ζ ζ ∞ 0 J(ζ, ζ ) F 0 (ζ ) ζ ∂ ζ F 0 (ζ) - F 0 (ζ) ζ ∂ ζ F 0 (ζ ) ζ 2 dζ , Q 0 ( f 1 ) = ∂ ζ ζ ∞ 0 J(ζ, ζ ) F 0 (ζ ) ζ ∂ ζ F 1 (ζ) - F 1 (ζ) ζ ∂ ζ F 0 (ζ ) ζ 2 dζ , (3.8)
with

J(ζ, ζ ) = 2α ee 3 min( 1 ζ 3 , 1 ζ 3 )ζ 2 ζ 2 .
In this work, both equivalent forms (3.5)-(3.6) and (3.8) are used. In [START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF], instead of using (1.26) the following electron-electron collision operator was proposed

Q ee (f ) = 1 ζ 2 ∂ ζ ζ ∞ 0 J(ζ, ζ ) F 0 (ζ ) ζ ∂ ζ f (ζ) - f (ζ) ζ ∂ ζ F 0 (ζ ) ζ 2 dζ .
This operator satises mass and energy conservation properties and an entropy dissipation property. Also it preserves the realisability domain [START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF]. The angular integration of this operator leads to the denitions (3.8).

Properties of the collisional operators

In this part, we briey recall important results established in [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF][START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF], then we characterise the equilibrium state of the collisional operators (3.2)-(3.4) which is given by an isotropic Maxwellian, similarly to the Landau collision operator. It is pointed out that this property is an important new result for the model. Firstly, it was demonstrated in [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF][START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF] that the realisability domain A is conserved by the collisional operators (3.2)- (3.4). Secondly, the quantity E = α 0 f 0 + α 1 . f 1 is an entropy for the system in the case without electric eld. More precisely, from system (3.1), in the case without electric eld we can derive the following inequality

∂ t E + ∇ x . F ≤ 0,
where F is the entropy ux given by F = α 0 f 1 + f2 α 1 .

Thirdly, the collisional operators (3. Theorem 3.2. The solution (f 0 , f 1 ) of the following system

Q 0 (f 0 ) = 0, Q 0 ( f 1 ) + Q 1 ( f 1 ) = 0, (3.9)
3. Classical transport theory for the collisional electronic M 1 model is given by

f 0 = ζ 2 K 1 exp(-K 2 ζ 2 )
and f 1 = 0 where K 1 and K 2 are two positive real constants.

Proof. We rst start to prove the following intermediate results

+∞ 0 α 0 Q 0 (f 0 )dζ + +∞ 0 α 1 . Q 0 ( f 1 )dζ ≤ 0, (3.10)
and +∞ 0

α 1 . Q 1 ( f 1 )dζ ≤ 0. (3.11)
The denition of Q 1 ( f 1 ) and the fact that α 1 . f 1 ≥ 0, (see [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF]), directly lead to (3.11).

Next, to prove (3.10) we use a Green formula in the expression of +∞ 0

α 0 Q 0 (f 0 )dζ to obtain +∞ 0 ∂ ζ ζ +∞ 0 J(ζ, ζ ) f 0 (ζ ) ζ 2 1 ζ ∂ ζ ( f 0 (ζ) ζ 2 ) - f 0 (ζ) ζ 2 1 ζ ∂ ζ ( f 0 (ζ ) ζ 2 ) (ζ ) 2 dζ α 0 dζ = - +∞ 0 +∞ 0 J(ζ, ζ ) 1 ζ F 0 (ζ )∂ ζ F 0 (ζ) - 1 ζ F 0 (ζ)∂ ζ F 0 (ζ ) ∂ ζ α 0 ζ (ζ ) 2 dζdζ . (3.12)
Next we compute

1 ζ F 0 (ζ )∂ ζ F 0 (ζ) - 1 ζ F 0 (ζ)∂ ζ F 0 (ζ ).
∂ ζ F 0 (ζ) = S 2 ∂ ζ α 0 (ζ) exp(α 0 (ζ) + α 1 (ζ). Ωd Ω (3.13) + S 2 Ω.∂ ζ α 1 (ζ) exp(α 0 (ζ) + α 1 (ζ). Ω)d Ω.
The expressions of F 0 and ∂ ζ F 0 give

1 ζ F 0 (ζ )∂ ζ F 0 (ζ) - 1 ζ F 0 (ζ)∂ ζ F 0 (ζ ) = S 2 S 2 exp(α 0 (ζ) + α 1 (ζ). Ω) exp(α 0 (ζ ) + α 1 (ζ ). Ω ) ∂ ζ α 0 (ζ) ζ + Ω ζ .∂ ζ α 1 (ζ) - ∂ ζ α 0 (ζ ) ζ - Ω ζ .∂ ζ α 1 (ζ ) d Ωd Ω .
Next by setting

K(ζ, ζ , Ω, Ω ) = J(ζ, ζ ) ζ 2 ζ 2 exp(α 0 (ζ) + α 1 (ζ). Ω) exp(α 0 (ζ ) + α 1 (ζ ). Ω ), (3.14) δ(ζ) = ∂ ζ α 0 (ζ) ζ , β(ζ) = ∂ ζ α 1 (ζ) ζ .

Collisional operators for the electronic M 1 model

and by using equality (3.14) in (3.12) we get

- +∞ 0 ζ 2 +∞ 0 ζ 2 J(ζ, ζ )( 1 ζ F 0 (ζ )∂ ζ F 0 (ζ) - 1 ζ F 0 (ζ)∂ ζ F 0 (ζ )) ∂ ζ α 0 (ζ) ζ dζdζ = - +∞ 0 +∞ 0 S 2 S 2 K(ζ, ζ , Ω, Ω ) (δ(ζ) -δ(ζ ))δ(ζ)dζdζ d Ωd Ω + +∞ 0 +∞ 0 S 2 S 2 K(ζ, ζ , Ω, Ω ) Ω. β(ζ) -Ω . β(ζ ) δ(ζ)dζdζ d Ωd Ω . The change of variables (ζ, ζ ) → (ζ , ζ) leads to - +∞ 0 ζ 2 +∞ 0 ζ 2 J(ζ, ζ )( 1 ζ F 0 (ζ )∂ ζ F 0 (ζ) - 1 ζ F 0 (ζ)∂ ζ F 0 (ζ )) ∂ ζ α 0 (ζ) ζ dζdζ = - 1 2 +∞ 0 +∞ 0 S 2 S 2 K(ζ, ζ , Ω, Ω ) (δ(ζ) -δ(ζ )) 2 dζdζ d Ωd Ω + 1 2 +∞ 0 +∞ 0 S 2 S 2 K(ζ, ζ , Ω, Ω ) Ω. β(ζ) -Ω . β(ζ ) (δ(ζ) -δ(ζ )) dζdζ d Ωd Ω . (3.16)
Next, for the remaining term

+∞ 0 Q 0 ( f 1 ). α 1 (ζ)dζ = - +∞ 0 ζ 2 +∞ 0 J(ζ, ζ )( 1 ζ F 0 (ζ )∂ ζ F 1 (ζ) - 1 ζ F 1 (ζ)∂ ζ F 0 (ζ )) . ∂ ζ ( α 1 ) ζ (ζ ) 2 dζdζ ,
we proceed as previously. The expression of F 1 given in (3.7) leads to

∂ ζ F 1 (ζ) = S 2 Ω∂ ζ exp(α 0 (ζ) + α 1 (ζ). Ω)d Ω (3.17) + S 2 Ω 2 ∂ ζ α 1 (ζ) exp(α 0 (ζ) + α 1 (ζ). Ω)d Ω.
Therefore by using expressions (3.13) and (3.17), we get

- +∞ 0 ∞ 0 J(ζ, ζ ) ζ 2 ζ 2 F 0 (ζ ) 1 ζ ∂ ζ F 1 (ζ) -F 1 (ζ) 1 ζ ∂ ζ F 0 (ζ ) . ∂ ζ ( α 1 ) ζ dζdζ = +∞ 0 ∞ 0 S 2 S 2 K(ζ, ζ , Ω, Ω ) (δ(ζ) -δ(ζ )) Ω. β(ζ) dζdζ d Ωd Ω + +∞ 0 ∞ 0 S 2 S 2 K(ζ, ζ , Ω, Ω ) β(ζ ). Ω -β(ζ). Ω Ω. β(ζ) dζdζ d Ωd Ω .

Classical transport theory for the collisional electronic

M 1 model Then the change of variables (ζ, ζ ) → (ζ , ζ) gives - +∞ 0 ∞ 0 J(ζ, ζ ) ζ 2 ζ 2 F 0 (ζ ) 1 ζ ∂ ζ F 1 (ζ) -F 1 (ζ) 1 ζ ∂ ζ F 0 (ζ ) . ∂ ζ ( α 1 ) ζ dζdζ = 1 2 +∞ 0 ∞ 0 S 2 S 2 K(ζ, ζ , Ω, Ω ) (δ(ζ) -δ(ζ )) Ω. β(ζ) -Ω . β(ζ ) dζdζ d Ωd Ω - 1 2 +∞ 0 ∞ 0 S 2 S 2 K(ζ, ζ , Ω, Ω ) β(ζ ). Ω -β(ζ). Ω 2 dζdζ d Ωd Ω . (3.18)
Finally, we add the right-hand sides of (3.16) and (3.18) and by using the inequality

(δ(ζ) -δ(ζ ))( β(ζ). Ω -β(ζ ). Ω ) ≤ 1 2 ((δ(ζ) -δ(ζ )) 2 + ( β(ζ). Ω -β(ζ ). Ω ) 2 ), (3.19)
we obtain (3.10).

Next, multiplying the rst equation of (3.9) by α 0 and projecting the second on α 1 , adding the two equalities and integrating over ζ gives

+∞ 0 α 0 Q 0 (f 0 )dζ + +∞ 0 α 1 . Q 0 ( f 1 )dζ + +∞ 0 α 1 . Q 1 ( f 1 )dζ = 0.
Since, we proved (3.10) and (3.11), it comes

α 1 . Q 1 ( f 1 ) = 0. It follows that f 1 = 0.
Multiplying the rst equation of (3.9) by ln(F 0 ) and integrating over ζ gives

+∞ 0 ∂ ζ (ζ +∞ 0 J(ζ, ζ ) ∂ ζ F 0 (ζ) F 0 (ζ)ζ - ∂ ζ F 0 (ζ ) F 0 (ζ )ζ ζ 2 F 0 (ζ)F 0 (ζ )dζ ln(F 0 (ζ))dζ = 0.
By integration by part, it comes

- +∞ 0 +∞ 0 K(ζ, ζ ) ∂ ζ F 0 (ζ) F 0 (ζ)ζ - ∂ ζ F 0 (ζ ) F 0 (ζ )ζ ∂ ζ F 0 (ζ) F 0 (ζ)ζ dζ dζ = 0. with K(ζ, ζ ) = ζ 2 ζ 2 F 0 (ζ)F 0 (ζ ). The change of variables (ζ, ζ ) → (ζ , ζ) leads to - +∞ 0 +∞ 0 K(ζ, ζ ) ∂ ζ F 0 (ζ ) F 0 (ζ )ζ - ∂ ζ F 0 (ζ) F 0 (ζ)ζ ∂ ζ F 0 (ζ ) F 0 (ζ )ζ dζ dζ = 0.
Summing the two previous equations gives

+∞ 0 +∞ 0 K(ζ, ζ ) ∂ ζ F 0 (ζ ) F 0 (ζ )ζ - ∂ ζ F 0 (ζ) F 0 (ζ)ζ 2 dζ dζ = 0. It follows that F 0 (ζ) = K 1 exp(-K 2 ζ 2 ),
and so

f 0 (ζ) = ζ 2 K 1 exp(-K 2 ζ 2 ).
Since the integral of f 0 in ζ must be positive and nite, K 1 and K 2 are positive real constants.

Study of particle transport in plasmas

Derivation of the electronic transport coecients

These results demonstrate that the electron-electron collisional operator used for the electronic M 1 model satises fundamental properties. In the next section, the derivation of the plasma transport coecients using this operator is investigated in the framework of the classical transport theory.

3.3 Derivation of the electronic transport coecients

Electron collisional hydrodynamics

It has been shown that the equilibrium state of system (3.9) is given by an isotropic Maxwellian distribution function. Therefore, in this analytical derivation we consider a distribution function close to the equilibrium

f (t, x, ζ, Ω) = M f (ζ, T e (t, x), n e (t, x)) + εF (t, x, ζ, Ω), (3.20)
where the Maxwellian distribution function reads

M f (ζ, T e (t, x), n e (t, x)) = n e (t, x) m e 2πT e (t, x) 3/2 exp - m e ζ 2 2T e (t, x) , (3.21)
and the Knudsen number ε = λ ei /L is a small parameter which corresponds to the ratio between the mean free path λ ei and the macroscopic scale length L. The perturbation F is seeked under the form

F (t, x, ζ, Ω) = F 0 (t, x, ζ) + F 1 (t, x, ζ). Ω, (3.22)
According to the Chapman-Enskog approach, the density and temperature macroscopic quantities are dened as

n e (t, x) = 4π +∞ 0 f (t, x, ζ, Ω)ζ 2 dζ, (3.23
)

T e (t, x) = 4πm e 3n e k B +∞ 0 f (t, x, ζ, Ω)ζ 4 dζ. (3.24)
Therefore the isotropic part of the perturbation veries the following relations

∇ x .(ζ f2 ) + q m ∂ ζ ( f2 E) - q mζ (f 0 E -f2 E) = Q 1 ( f 1 ) + Q 0 ( f 1 ).
Using the fact that f2 = f 0 /3 Īd according to equation (3.22), the previous equation leads to

ζ 3 ∇ x (f 0 ) - e E 3m e ∂f 0 ∂ζ + 2e E 3mζ f 0 = Q 1 ( f 1 ) + Q 0 ( f 1 ),
which also rewrites

ζ 3 ∇ x f 0 - e Eζ 2 3m e ∂ ∂ζ f 0 ζ 2 = Q 1 ( f 1 ) + Q 0 ( f 1 ).
Then using in the place of f 0 the Maxwellian distribution (3.21), the previous equation gives

M f ζ e E * T e + 1 2T e ∇ x (T e )( m e ζ 2 T e -5) = - 2α ei ε ζ 3 F 1 + ε ζ 2 Q 0 (ζ 2 F 1 ), (3.27)
with E * = E + (1/en e )∇ x (n e T e ). In the following we note α ei and α ee instead of α ei ε and α ee ε. In the dimensionless case a parameter 1/ε appears in front of the collisional operators, therefore considering the development (3.20), the parameter ε vanishes.

In order to obtain F 1 , one should solve the integro-dierential equation (3.27). The resolution of this equation is challenging, however it is a linear equation in F 1 and the form of the left hand side indicates that the solution is a linear combination of terms proportional to the generalised forces E * and ∇(T e )/T e which can be represented as follows

F 1 = ζ e E * T e φ E + ∇ x (ln T e )φ Q M f , (3.28)

Study of particle transport in plasmas

where φ E and φ Q are dened below. Inserting this expression into (3.26) one obtains the following relations [START_REF] Balescu | Transport Processes in Plasma[END_REF]

j = σ E * + α∇ x T e , (3.29)
q = -αT e E * -χ∇ x T e , (3.30)
where α, σ and χ are called the plasma transport coecients dened by

σ = - 4πe 2 3T e ∞ 0 ζ 4 φ E M f dζ, χ = 2π 3 ∞ 0 ζ 4 (5 - m e ζ 2 T e )φ Q M f dζ, (3.31) α = - 4πe 3T e ∞ 0 ζ 4 φ Q M f dζ = 2πe 3T e ∞ 0 ζ 4 (5 - m e ζ 2 T e )φ E M f dζ. (3.32)
The coecients σ, α and χ are respectively called the electrical conductivity, the thermoelectric coecient and the thermal conductivity. In the case of a homogeneous plasma (with no density nor temperature gradients) relation (3.29) simplies into the Ohm's law j = σ E and equation (3.30) leads to q = -αT e E. One can dene the heat conductivity coecient κ, which is a combination of the other three coecients κ = χ -α 2 T e /σ.

Transport theory in Lorentzian plasma

In the case of a Lorentzian plasma the ions are highly charged therefore one can neglect the electron-electron collision operator in equation (3.27). As explained in the previous section, in this case (Z >> 1), the plasma transport coecients are the same in the collisional electronic M 1 model (3.1) and in the Fokker-Planck-Landau equation (1.10)- (1.26). An explicit expression of F 1 and the basic functions φ E and φ Q are easily derived

F 1 = ζM f e E * T e - ζ 3 2α ei + ∇ x (ln(T e )) ζ 3 4α ei 5 - m e ζ 2 v 2 Te , (3.34)
3. Classical transport theory for the collisional electronic M 1 model

and Here the subscript 0 corresponds to the high Z limit. In Figure 3.1, the electric current and heat ux are displayed in terms of y = v/v Te using the denition (3.34).

φ E = - ζ 3 2α ei , φ Q = ζ 3 4α ei 5 - m e ζ 2

Transport theory with electron-electron collisions

In the case of low Z plasmas the calculation presented in the previous section overestimates the transport coecients because the electron-electron collision operator is not taken into account. In this case, one should solve the full equation (3.27).

Spitzer and Härm [START_REF] Struchtrup | Macroscopic Transport Equations for Rareed Gas Flows[END_REF] solved it numerically in the case of the Fokker-Planck-Landau

Study of particle transport in plasmas 3.3. Derivation of the electronic transport coecients equation (1.10)-(1.26). Braginskii [START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF] derived an approximate analytical solution by expanding F 1 onto a series of the Laguerre polynomials following ideas used in the kinetic theory of neutral gases [START_REF] Chapman | The Mathematical Theory of Non-Uniform Gases[END_REF]. In the present work, we apply the latter method for the case of the electronic M 1 model (3.1). Following (3.28), using a decomposition of f 1 with the two functions φ E and φ Q in equation (3.27) reads

1 ζ 2 Q 0 (ζ 2 ζM f φ A ) - 2α ei ζ 2 M f φ A = ζM f S A , (3.36)
where

S A = e E * T e S E -∇ x ln(T e )S Q , with S E = 1, S Q = 1 2 ζ 2 v 2 Te -5 .
Following Chapman [START_REF] Chapman | The Mathematical Theory of Non-Uniform Gases[END_REF] and Braginskii [START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF], we expand F 1 over the Laguerre polynomials [START_REF] Abramowitz | Handbook of Mathematical functions[END_REF] L

(3/2) n (x), with x = ζ 2 /2v 2
Te . Indeed, the source term in the right hand side of (3.36) is a combination of the two rst Laguerre polynomials

S E = L 3/2 0 (x) and S Q = -L 3/2 1 (x). We represent the basic function φ A as φ A (ζ) = +∞ m=0 φ A m L (3/2) m (ζ 2 /2v
- 2α ei ζ 2 M f φ A ζ 3 L (3/2) n (ζ 2 /2v 2 Te )dζ = -2α ei +∞ m=0 φ A m +∞ 0 M f v 2 Te L (3/2) m (x)L (3/2) n (x)dx.
Using the denition (3.21), it comes

+∞ 0 - 2α ei ζ 2 M f φ A ζ 3 L (3/2) n (ζ 2 /2v 2 Te )dζ = -2α ei n e v Te (2π) 3/2 +∞ m=0 φ A m +∞ 0 L (3/2) m (x)L (3/2) n (x)e -x dx.
The computation for the source term reads

+∞ 0 ζM f S A ζ 3 L (3/2) n ζ 2 2v 2 Te dζ = n e v 2 Te π √ π ∞ 0 x √ xe -x e E * T e + 1 T e ∇ x (T e )(x - 5 2 ) L (3/2) n (x)dx,
3. Classical transport theory for the collisional electronic M 1 model and using the orthogonality of the Laguerre polynomials, the previous equation reads

+∞ 0 ζM f S A ζ 3 L (3/2) n ζ 2 2v 2 Te dζ = n e v 2 Te π 3 4 e E * T e δ 0n - 15 8
1

T e ∇ x (T e )δ 1n L (3/2) n (x)dx.
A similar derivation applies to the electron-electron collision operator

+∞ 0 1 ζ 2 Q 0 (ζ 2 ζM f φ A )ζ 3 L (3/2) n ζ 2 2v 2 Te dζ = n e v 2 Te π √ π +∞ m=0 φ A m +∞ 0 L (3/2) n (x)Q 0 (x √ xe -x L (3/2) m (x))dx.
A direct calculation nally gives the following set of equations

Z -1 +∞ m=0 ce nm φ A m - +∞ m=0 ci nm φ A m = ν -1 ei S A n .
(3.37)

Here, ce nm and ci nm are the matrices of the integrals of the electron-electron and electron-ion collision operators. They are dened by

ci nm = +∞ 0 L (3/2) n (x)L (3/2) m (x)e -x dx, (3.38)
ce nm = 2 (3/2) v 3 Te Y ee +∞ 0 L (3/2) n (x)Q 0 (x √ xe -x L (3/2) m (x))dx, (3.39)
with Y ee = Z -1 Y ei and Y ei = (3π/2)ν ei v 3

Te .

The term S A n reads

S A n = e E * T e δ 0n - 5 2
1 T e ∇ x (T e )δ 1n .

The vector S A n has only two non-zero components. Therefore, only two rst expan- sion coecients φ A 0 and φ A 1 contribute to the transport coecients (3.31)-(3.32)

σ = - e 2 n e m e φ E 0 , α = - en e m e φ Q 0 = 5 2 en e m e φ E 1 , χ = 5 2 n e v 2 Te φ Q 1 , κ = 5 2 n e v 2 Te (φ Q 1 -φ Q 0 φ E 1 /φ E 0 ).
In the limit Z >> 1, the rst term in (3.37) vanishes and the model simplies into the case of a Lorentzian plasma. In this case the rst expansion coecients read

φ E 0 = -32/3πν ei , φ E 1 = 32/5πν ei , φ E 2 = -32/35πν ei , φ Q 0 = φ Q 2 = -16/πν ei and φ Q 1 = 80/3πν ei . Multiplying (3.

Derivation of the electronic transport coecients

The computation of ci nm using (3.38) is straightforward. However, the derivation of ce nm using (3.39) is more challenging. The coecients A(ζ) and B(ζ) in (3.5) and (3.6) are involved in the denition of the electron-electron collision operator Q 0 . Using the variable x = ζ 2 /2v 2 Te a straight calculation gives

A(x) = n e 2 √ πx √ xv Te 3 √ π √ 2 erf ( √ x) -e -x (3 √ 2x + 2 √ 2x √ x) + 2 π n e v Te e -x , (3.40)
B(x) = - 3n e 4 √ πv 3 Te x √ x √ 2πerf ( √ x) -2 √ 2xe -x - 2 π n e v 3 Te e -x , (3.41
L (3/2) n (x) √ x∂ x 2erf ( √ x) - 4 √ x √ π e -x ∂ x g(x) (3.42) + 2erf ( √ x) - e -x √ π [4 √ x - 8 3 x √ x ] g(x) dx, where g(x) = √ xe -x L (3/2) m (x).
Using denitions (3.38) and (3.42), each component of the matrices ci nm and ce nm can be computed numerically and the set of equations (3.37) can be solved.

The accuracy of the solution of (3.37) increases with the number of coecients φ A n chosen. The minimum number is two since the rst two coecients φ 0 and φ 1 contribute to the transport coecients. Such a two polynomial approximation was considered by Braginskii [START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF] for the Fokker-Planck-Landau equation (1.26). The four-polynomial approximation provides results beyond the need of experimental plasma physics. Kaneko [134] used 6 Laguerre polynomials and the high accuracy of transport coecients he obtained was conrmed in [135] and [136]

Classical transport theory for the collisional electronic M 1 model

The velocity-dependent ux functions presented in Figure 3.2 shows that the electronelectron contribution decreases as Z increases. We introduce the following dimensionless coecients γ σ , γ α , γ χ , γ κ dened by

γ σ = σ/σ 0 , γ α = α/α 0 , γ χ = χ/χ 0 , γ κ = κ/κ 0 ,
where the index 0 denotes the case of the Lorentzian approximation (Z >> 1).

The computation of these coecients shows that all of them are inferior to 1, that is, the Lorentzian approximation (Z >> 1) overestimates the electron transport coecients for low-Z plasmas. The coecients γ σ , γ α , γ χ , γ κ are displayed in Figures 3. [START_REF] Alexandre | On the Landau approximation in plasma physics[END_REF]

Conclusion

In this work, the fundamental properties of the electron-electron and electronion collision operators used for the electronic M 1 model have been studied. It is shown that their equilibrium states is given by an isotropic Maxwellian distribution function. In addition, in the Lorentzian approximation, the electronic M 1 model and the Fokker-Planck-Landau equation coincide. The electron transport coecients are derived using the electron-electron collision operators proposed for the electronic M 1 model. Despite, the approximations used, accurate plasma transport coecients have been obtained. The correct χ and κ plasma transport coecients are recovered and the coecient α is very close to the one obtained with the Fokker-Planck-Landau equation. The main error is made with the electric conductivity σ in the case Z = 1.

In spite of this error, these results show that the electron-electron collision operator is a good candidate for physical applications. It may be possible to improve this operator in order to obtain a more accurate σ coecient. However, since the angular extraction of the kinetic electron-electron collision operator is complex, such an issue seems challenging.

Study of particle transport in plasmas

Numerical methods for the study of the long time behavior particle transport Chapter 4 Some basic concepts of numerical methods for nonlinear systems

The next chapters are devoted to the numerical resolution of the Maxwell-M 1 system introduced in the previous chapters. In particular, we are interested in the long time behaviour of the solutions of this system. More precisely, when the characteristic quantities of the problem become large compared to the plasma parameters, the studied model degenerates into a limit system. However, in general, the methods designed for the numerical resolution of the initial model are not able to correctly capture the limit problem. This point will be developed in detail in the next chapters.

The angular M 1 studied model is a nonlinear system. For this purpose, in this chapter, some concepts of numerical methods for nonlinear systems are given following the ideas of [START_REF] Chalons | Modied Suliciu relaxation system and exact resolution of isolated shock waves[END_REF][START_REF] Godlewski | Numerical approximation of hyperbolic system of conservation laws[END_REF][START_REF] Godunov | A dierence method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF][START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF][START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF]. These methods are applied to the M 1 model. These elements will be used in the next chapters.

Godunov-type methods

In [START_REF] Godunov | A dierence method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF], Godunov proposed a numerical method for solving the Euler equations.

The method is based on the fact that, even for nonlinear systems, the Riemann problem with piecewise constant initial data can be solved and the solution consists of a nite set of waves travelling at constant speeds. The Godunov method represents a major breakthrough for computational uid dynamics. The wave structure is determined by the solution of the Riemann problem and shock waves can be correctly handled. In this section, we recall the derivation of Godunov-type methods.

We consider a given set of hyperbolic equations

∂ t u + ∂ x f (u) = 0, (4.1)

Godunov-type methods

where u ∈ U ∈ R l and f : R l → R l , C 1 with l > 1. We suppose there exists a strictly convex entropy-entropy ux pair (η, q) for (4.1)

∀u ∈ U, η (u).f (u) = q (u).

We look for a weak entropy solution of (4.1), that is a weak solution of (4.1) such that

∂ t η(u) + ∂ x q(u) ≤ 0. (4.2)
The eigenvalues of this system are written λ k (u) and we consider the associated initial condition u 0 . We dene a constant space step ∆x and a constant time step ∆t, the cell center coordinates x i = (i -1 2 )∆x and the mesh interfaces coordinates x i+1/2 = i∆x for i ∈ Z. At each time t n , in the i th cell interval [x i-1/2 , x i+1/2 ], j ∈ Z, we compute u n j a numerical approximation of the solution of (4.1). Consequently, we dene a piecewise constant approximate solution

u h (x, t n ) = u n i for all x ∈ [x i-1/2 , x i+1/2 [, i ∈ Z, n ∈ N. (4.3)
At the initial time, in the i th cell, we dene

u 0 i = 1 ∆x x i+1/2 x i-1/2 u 0 (x)dx for all i ∈ Z.
Now, assuming a known numerical solution at time t n we will detail the Godunov method to compute the numerical solution at time t n+1 .

Firstly, we solve the following Cauchy problem

∂ t w + ∂ x f (w) = 0, x ∈ R w(x, 0) = u h (t n , x), (4.4)
where u h is dened by (4.3). Considering the classical CFL condition

∆t ≤ ∆x 2 max k,u (|λ (k) (u)|) ,
it is known that the solution of (4.4) is given by

w(t, x) = u( x -x i+1/2 t -t n , u n i , u n i+1 ) for all (x, t) ∈ [x i , x i+1 ]×]t n , t n+1 ], (4.5)
where (t, x) → u(x/t, u L , u R ) is the unique weak entropic self-similar solution of the Riemann problem 4. Some basic concepts of numerical methods for nonlinear systems

     ∂ t u + ∂ x f (u) = 0, x ∈ R u(0, x) = u L if x < 0, u R if x > 0.
Then in order to compute a piecewise approximate solution on each cell at time t n+1 , we average the solution of (4.4) in each cell

u n+1 i = 1 ∆x x i+1/2 x i-1/2
w(∆t, x)dx for all i ∈ Z.

u n+1 i = u n i - ∆t ∆x (f n i+1/2 -f n i-1/2 ) for all i ∈ Z,
where the numerical uxes are given by

f n i+1/2 = f (u(0, u n j , u n j+1 )) for all i ∈ Z.
It is said that the Godunov method is exact since we consider an exact resolution of the Riemann problem (4.6). However, such a resolution can be challenging and one often prefers to use an approximate Riemann solver. In order to derive approximate Godunov-type method the exact solution of the Riemann problems considered at each interface (4.6) is replaced by an approximate solution. Following [START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for sources[END_REF][START_REF] Chalons | Modied Suliciu relaxation system and exact resolution of isolated shock waves[END_REF][START_REF] Godlewski | Numerical approximation of hyperbolic system of conservation laws[END_REF][START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] we detail the notion of consistency in the integral sense of an approximate Riemann solver.

Instead of solving exactly the Riemann problem (4.6) we consider the following an approximate Riemann solver made of l + 1 constant states separated by l discontinuities which propagate at speed λ k

w(t, x) = u(x/t, u l , u R ) =                    u 0 = u L if x/t < λ 1 , . . . u k if λ k < x/t < λ k+1 , . . . u l = u R if x/t > λ l . (4.8)
We associate the usual CFL condition

max 1≤k≤l |λ k (u L , u R )| ∆t ∆x ≤ 1 2 .
The approximate Riemann solver (4.8) is said to be consistent with the integral form of (4.1) if the integral of w(., ∆t) is equal to the integral of the exact solution on the space interval [-∆x 2 , ∆x 2 ]. There the approximate Riemann solver (4.8) satises

f (u R ) -f (u L ) = l k=1 λ k (u L , u R )(u k -u k-1
).

(4.9)

Study of particle transport in plasmas

Application to angular moment models

Then, the approximate Godunov method reads

   u n+1 i = u n i - ∆t ∆x (f n i+1/2 -f n i-1/2 ) f n i+1/2 = f (u n j , u n j+1 ), (4.10)
with

f (u L , u R ) = 1 2 f (u L ) + f (u R ) - l k=1 |λ k (u L , u R )|(u k -u k-1 )
.

Similarly, applying the same procedure with the entropy inequality (4.2), the approximate Riemann solver is said to be consistent with the integral form of (4.2)

if q(u R ) -q(u L ) ≤ l k=1 λ k (u L , u R )(η(u k ) -η(u k-1 )).
Then, the numerical scheme (4.10) veries a discrete entropy inequality

   η(u n+1 i ) = η(u n i ) - ∆t ∆x (q n i+1/2 -q n i-1/2 ) q n i+1/2 = q(u n j , u n j+1 ), (4.11)
with

q(u L , u R ) = 1 2 q(u L ) + q(u R ) - l k=1 |λ k (u L , u R )|(η(u k ) -η(u k-1 )) .

Application to angular moment models

In this section, we give an example of approximate Riemann solver, the Harten Lax van Leer's one (HLL) [START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF]. Then, we apply it to angular moment models and show that this approach enables to preserve the admissible sets.

HLL approximate Riemann solver

Here, we introduce the HLL scheme [START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF]. This approximate Riemann solver is obtained considering only one constant intermediate state. It writes

u(x/t, u l , u R ) =      u L if x/t < λ 1 , u * if λ 1 < x/t < λ 2 , u R if x/t > λ 2 .
(4.12)

The consistency relation (4.9) gives

u * = λ 2 u R -λ 1 u L λ 2 -λ 1 - f (u R ) -f (u L ) λ 2 -λ 1 . (4.13)
4. Some basic concepts of numerical methods for nonlinear systems

The structure of the approximate Riemann solver is displayed in Figure 4.1. This approximate Riemann solver is the simplest consistent with the integral form of (4.8).

The wavespeeds λ 1 and λ 2 are chosen to satisfy the sub-characteristic condition [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF] λ 1 ≤ λ k ≤ λ 2 for all k.

(4.14)

λ 1 λ 2 t x u R u L u *

Angular moment models

The HLL approximate Riemann solver is particularly interesting considering angular moment models. Indeed, this solver enables the preservation of the admissible sets. More precisely, considering an admissible numerical solution at time t n , one can show that the numerical solution computed at time t n+1 remains admissible. In this section we detail this fundamental property. We note also that this solver is entropic [START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for sources[END_REF]. These two properties are considered as nonlinear stability properties [START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for sources[END_REF].

In the case of angular moments model, in a one dimensional framework, the moment vector u and the ux function f (u) write

u =< f m >, f (u) =< f mµ >, (4.15)
where m = (1, µ, µ 2 , ..., µ k , ...) is the vector of basis functions used to dened the angular moments. The notation < > represents the angular integration in µ dened by < f >=

u * = 1 λ 2 -λ 1 (λ 2 u R -f (u R )) + 1 λ 2 -λ 1 (-λ 1 u L + f (u L )).
We remark that the terms λ 2 u R -f (u R ) and -λ 1 u L + f (u L ) are admissible. Indeed using the denitions (4.15) it follows that

λ 2 u R -f (u R ) =< mf (λ 2 -µ) > and -λ 1 u L -f (u L ) =< mf (-λ 1 + µ) > .
It will be shown that the eigenvalues of angular moment models based on an entropy minimisation principle are in the interval [-1, 1]. Therefore the condition (4.14)

implies that

λ 1 ≤ -1 and λ 2 ≥ 1.
Then it follows that the two moments vectors λ 2 u R -f (u R ) and -λ 1 u L -f (u L ) are moments vectors of two positive distribution functions f (λ 2 -µ) and f (-λ 1 + µ).

Therefore by denition there are admissible. Then since λ 2 is greater than λ 1 it follows that u * is admissible. Finally using the Godunov approach detailed in the previous section, equation (4.7) gives the admissibility property for the numerical solution at time t n+1 .

To complete this explanation, we show that the eigenvalues of angular moment models based on an entropy minimisation principle are in absolute value smaller than 1. It has been seen in the rst part of this manuscript, that in the case of angular moment models based on an entropy minimisation principle the form of the distribution function is given by a exponential of a polynomial function of µ.

Therefore, using the denitions (4.15) it comes that u =< exp(α.m)m >, f (u) =< exp(α.m)mµ > .

Then it follows that

∂ α u =< exp(α.m)m ⊗ m >, ∂ α f (u) =< exp(α.m)m ⊗ mµ > .
Here, we are interested in the eigenvalues of the jacobian matrix J(u) of (4.1) dened by

J(u) = ∂ u f (u).
This equation rewrites

J(u) = B(u)A(u) -1 , (4.16)
where A(u) and B(u) are l × l matrices dened by

B(u) = ∂ α f (u), A(u) = ∂ α u.
We remark here, that A(u) is regular since it is a positive denite symmetric matrix. By denition the eigenvalues of J(u) verify J(u)X = λ(u)X, 90 Sébastien GUISSET 4. Some basic concepts of numerical methods for nonlinear systems with X ∈ R l . Now, setting Y = A -1 (u)X and using (4.16) it comes

B(u)Y = λ(u)A(u)Y. Then B(u)Y.Y = λ(u)A(u)Y.Y. Finally it comes λ(u) = B(u)Y.Y A(u)Y.Y .
We recover the Rayleigh quotients

λ min (u) = min Y ∈R l B(u)Y.Y A(u)Y.Y , λ max (u) = max Y ∈R l B(u)Y.Y A(u)Y.Y .
Using the denitions (4.15) leads to

λ(u) = < exp(α.m)(m.Y ) 2 µ > < exp(α.m)(m.Y ) 2 > ,
which is in absolute value smaller than 1.

In the next sections appropriate numerical schemes are designed for the M 1 angular moments model. In particular, one is interested in computing long time regimes.

Therefore, the framework of asymptotic-preserving schemes is detailed.

Study of particle transport in plasmas

Chapter 5

Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime

The study introduced in this chapter has been published. The reference is:

S. Guisset, S.

Introduction

In this work we assume that the plasma consists of electrons and one ion species considered as immobile. This approximation is relevant due to the large mass of ions compared to the electron mass. This means the model studied is valid on time scales when the ion motion can be neglected.

For the study of collisional processes, the two important physical scales are the mean free path and the electron-ion collision frequency. The mean free path represents the average distance travelled by an electron between two collisions with an ion.

The electron-ion collision frequency represents the number of electron-ion collisions per unit of time. When the electron plasma period is very small compared to the electron-ion collisional time and the Debye length is very small compared to the mean free path, the plasma is considered as quasi-neutral and the Maxwell-Gauss (also called Maxwell-Poisson) and Maxwell-Ampere equations degenerate into algebric equations on collisional time scales. Therefore to handle this type of situation a new class of methods, called Asymptotic-Preserving (AP) methods has been developed. Asymptotic-preserving schemes in the sense of Jin-Levermore [START_REF] Brull | Degenerate anisotropic elliptic problems and magnetised plasma simulations[END_REF][START_REF] Brull | An asymptotic preserving scheme for a bi-uid Euler-Lorentz system[END_REF] are designed to handle multi-scale situations and behave correctly in the asymptotic limit considered. The literature on Asymptotic-preserving schemes is extensive and in this part we only consider the works dealing with the quasi-neutral limit. Consider a system (S α ) depending on a parameter α, and (S 0 ) being the corresponding limit 5.1. Introduction system when α tends to zero. In our case α is the ratio between the Debye length and the mean free path. A numerical scheme with time step ∆t and space step ∆x is called Asymptotic-Preserving in the limit α tends to zero for the system (S α ) if the scheme is stable independently of the values taken by α and if the limit scheme obtained for α = 0 is consistent with the limit problem (S 0 ). In this work the system (S α ) corresponds to the Fokker-Planck-Landau-Maxwell (or the M 1 -Maxwell) system and (S 0 ) corresponds to the Fokker-Planck-Landau-Maxwell system in the quasi-neutral limit.

This regime has been already studied in the context of uid models [START_REF] Crispel | A plasma expansion model based on the full Euler-Poisson system[END_REF][START_REF] Crispel | An asymptotic preserving scheme for the two-uid Euler-Poisson model in the quasi-neutral limit[END_REF][START_REF] Degond | Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit[END_REF].

For example in [START_REF] Crispel | An asymptotic preserving scheme for the two-uid Euler-Poisson model in the quasi-neutral limit[END_REF], the authors considered a two uid isentropic Euler system coupled with the Poisson equation. It is shown that the Maxwell-Poisson equation can be reformulated into an elliptic equation which does not degenerate at the quasineutral limit. In [START_REF] Crispel | An Asymptotic Preserving scheme for the Euler equations in a strong magnetic eld[END_REF], this approach is generalised to the Euler-Maxwell model with a strong magnetic eld. A kinetic model consisting in a two uid Vlasov-Poisson system has also been investigated in [START_REF] Degond | Asymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality[END_REF]. In [START_REF] Degond | Numerical approximation of the Euler-Maxwell model in the quasineutral limit[END_REF], an Asymptotic-Preserving scheme is proposed for the Euler-Maxwell system in the quasi-neutral regime. The Maxwell equations are reformulated to enable the computation of the electrostatic eld even in the limit regime. The development followed the approach for calculation of the electric eld well known in the plasma physics [START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF][START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF].

The present work deals with the construction of an Asymptotic-Preserving scheme or the M 1 -Maxwell system in the quasi-neutral limit. The strategy adopted is similar to the one in [START_REF] Degond | Numerical approximation of the Euler-Maxwell model in the quasineutral limit[END_REF], nevertheless to our knowledge, it is the rst time that such schemes are considered for kinetic models with true collision operators. This fact is very important to deal with collisional plasma because the collision frequency ν must follow the Coulombian interaction law (ν ≈ 1/|v| 3 ). To perform realistic simulations in plasma physics, Coulombian interactions must be used. Therefore, relaxation operators are not relevant from the physical point of view. Moreover up to now, Asymptotic-Preserving schemes for the quasi-neutral limit have been developed either for uid description of plasma or for collisionless plasmas.

The chapter is organised as follows. Section 5.2 introduces the Fokker-Planck-Landau-Maxwell system and its quasi-neutral limit. A reformulation of the Fokker-Planck-Landau-Maxwell system is presented in the case of one dimension in space and one dimension in velocity. The model is considered with electric elds and collision operators. Then, the method is generalised for full multi-dimensions problems with electromagnetic elds and collision operators. Section 5.3 introduces in detail the numerical construction of an Asymptotic-Preserving scheme for the reformulated system of section 5.2. Section 5.4 deals with the construction of an Asymptotic-Preserving scheme for the M 1 moments model from the kinetic one. Finally, section 5.5 presents two physically relevant numerical test cases for the M 1 -Asymptotic-Preserving scheme for dierent regimes. The rst one corresponds to a regime where electromagnetic eects are predominant whereas the second one on the contrary shows the eciency of the Asymptotic-Preserving scheme in the collisional quasi-neutral regime. The numerical results are compared with kinetic and 5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime hydrodynamic limits.

5.2 Fokker-Planck-Landau-Maxwell system in the quasi-neutral limit

In this section the Fokker-Planck-Landau-Maxwell system in the quasi-neutral limit is introduced. We consider a plasma constituted of electrons and of one xed ion species. The description is performed with a non-negative distribution function for electrons f e (x, v, t), x ∈ R n represents the space variable, v ∈ R n is the velocity variable, n = 1, 2 or 3 and t is the time.

5.2.1

Scaling for the analysis of collisional processes.

For the analysis of collisional processes three important parameters are introduced: the mean free path λ ei which represents the average distance travelled by an electron between two collisions, the thermal velocity v th and the electron-ion collision frequency ν ei . They satisfy the relations

v th = k B T m e , ν e,i = v th λ e,i .
These parameters enable us to scale time, space and speed t = ν e,i t, x = x/λ e,i , ṽ = v/v th .

In the same way, we scale the electric eld, the magnetic eld and the distribution function

Ẽ = eE m e v th ν e,i , B = eB m e ν e,i , f = f e v 3 th n 0 .
n 0 is the initial electronic density.

With these dimensionless quantities the Fokker-Planck-Landau-Maxwell system (1.10-1.26-1.15-1.18) becomes the following system where we have omitted the tildes

                                       ∂f ∂t + v.∇ x f -(E + v × B).∇ v f = 1 Z C ee (f, f ) + C ei (f ) , ∂E ∂t - 1 β 2 ∇ x × B = - j α 2 , ∂B ∂t + ∇ x × E = 0, ∇ x .E = 1 α 2 (1 -n), ∇ x .B = 0, (5.1)
Study of particle transport in plasmas 95 5.2. Fokker-Planck-Landau-Maxwell system in the quasi-neutral limit where α = ν e,i ωpe , ω pe represents the electronic plasma frequency, β = v th /c, n = n e /n 0 and Z the charge of the ions. In this work Z is taken equal to 1.

Electrostatic case

In the electrostatic case with only one dimension for space (x ∈ R) and one for velocity (v ∈ R), the system (5.1) can be written in the following form

         ∂f ∂t + v∂ x f -E∂ v f = C ee (f, f ) + C ei (f ), ∂E ∂t = - j α 2 , (5.2)
where Maxwell-Poisson has to be satised at the initial time. Poisson equation is veried at the initial time. The limit system (S 0 ) is obtained when the parameter α tends to 0 and corresponds to the quasi-neutral limit. It can be written in the form

     ∂f ∂t + v∂ x f -E∂ v f = C ee (f, f ) + C ei (f ), j = 0,
with n = 1 at initial time. When α tends to zero the Maxwell-Poisson equation degenerates into the algebraic equation n = 1. This condition has to be satised at initial time. When α is equal to zero we lose the possibility to obtain the electric eld from the Maxwell-Ampere equation on the collisional time scale. This limit is singular, because the Maxwell-Ampere equation degenerates into an algebraic equation.

Reformulation of the Maxwell-Ampere equation in the simplied case

The aim of this part is to provide a reformulation of the Maxwell-Ampere equation that is equivalent and contains explicitly the quasi-neutral limit as a particular case when α = 0 for the electrostatic case with only one dimension for space and one for the velocity.

Multiplying the rst equation of (5.2) by v, integrating in velocity and using the denition of the dimensionless current

j = - R f vdv, (5.3)
5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime we obtain

- ∂j ∂t + ∂ x ( R v 2 f dv) -E R v∂ v f dv = R C ei vdv.
(5.4)

Here we use the fact that

R v C ee (f, f )dv = 0.
Here, it is important to notice that the integral in velocity against v of the electronion collision operator C e,i does not vanish. The derivation in time of the Maxwell-Ampere equation in the electrostatic case leads to

∂j ∂t = -α 2 ∂ 2 E ∂t 2 .
By using (5.4), we get

α 2 ∂ 2 E ∂t 2 -E R v∂ v f dv = -∂ x ( R v 2 f dv) + R C ei vdv. (5.5) As E R v∂ v f dv = -nE, the equation (5.5) becomes α 2 ∂ 2 E ∂t 2 + nE = -∂ x ( R v 2 f )dv + R C ei vdv.
When the parameter α tends to 0, we nd the limit problem

nE = -∂ x ( R v 2 f dv) + R C ei vdv.
So the electrostatic eld writes

E = -∂ x ( R v 2 f dv) + R C ei vdv n .
(5.6)

In this part we have shown that the Fokker-Planck-Landau-Maxwell system (5.2) is equivalent to the Fokker-Planck-Landau-Maxwell reformulated system

     ∂f ∂t + ∂ x (vf ) -∂ v (Ef ) = C ee (f, f ) + C ei (f ) , α 2 ∂ 2 E ∂t 2 + nE = -∂ x ( R v 2 f )dv + R C ei vdv, (5.7)
where Maxwell-Poisson has to be satised at initial time. The limit system when α → 0 is the following one

     ∂f ∂t + ∂ x (vf ) -∂ v (Ef ) = C ee (f, f ) + C ei (f ) , E = -∂ x ( R v 2 f dv) + R C ei vdv n ,
5.2. Fokker-Planck-Landau-Maxwell system in the quasi-neutral limit

where n = 1 and j = 0 at initial time.

The second equation of (5.2) imposes j = 0 when α = 0. This condition has to be satised at initial time.

Reformulation of the Maxwell-Ampere equation in the general case

In this section, we generalise previous method to a non-homogeneous collisional plasma with magnetic eld in multiple dimensions.

Multiplying this rst equation of (5.1) by -v, integrating in velocity and using the denition of the dimensionless current (5.3) we get

- ∂j ∂t + div x ( R n v ⊗ vf dv) - R n v(E + v × B).∇ v f dv = R n C ei (f )vdv. As R n (v × B).∇ v f v dv = j × B,
the same development as in the electrostatic case is performed.

The derivation in time of the Maxwell-Ampere equation in the general case leads to

∂j ∂t = -α 2 ∂ 2 E ∂t 2 + α 2 β 2 ∇ x × ∂B ∂t .
Finally the following form is obtained

α 2 ∂ 2 E ∂t 2 +n e E -j ×B = -div x ( R n v ⊗vf dv)+ α 2 β 2 ∇ x × ∂B ∂t + R n C ei (f )vdv. (5.8)
When α tends to 0 in (5.8) we nd the limit problem

n e E = -div x ( R n v ⊗ vf dv) + R n C ei (f )vdv + j × B.
So the electrostatic eld writes

E = -div x ( R n v ⊗ vf dv) + R n C ei (f )vdv + j × B n e .
In this part we have shown that the Fokker-Planck-Landau-Maxwell system (5.1) is equivalent to the Fokker-Planck-Landau-Maxwell reformulated system 5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime equation is veried at the initial time.

                           ∂f ∂t + v.∇ x f -(E + v × B).∇ v f = C ee (f, f ) + C ei (f ) , α 2 ∂ 2 E ∂t 2 + n e E -j × B = -div x ( R n v ⊗ vf dv) + α 2 β 2 ∇ x × ∂B ∂t + R n C ei (f )vdv, ∂B ∂t + ∇ x × E = 0, (5.9
The limit system of (5.9) when α → 0 is the following one

           ∂f ∂t + v.∇ x f -(E + v × B).∇ v f = C ee (f, f ) + C ei (f ) , n e E -j × B = -div x ( R n v ⊗ vf dv) + R n C ei vdv , ∂B ∂t + ∇ x × E = 0 , (5.10)
where n = 1 and j = 0 have to be satised at initial time.The second equation of (5.10) is called the Generalised Ohm's law.

In this part, a reformulation of the Maxwell-Ampere equation containing the limit case α = 0 has been performed. This derivation enables us to construct an Asymptotic-Preserving numerical scheme for the quasi-neutral regime.

Discrete model

Limitation of the classical numerical scheme

A classical numerical scheme for the Maxwell-Ampere equation in the collisional regime writes

E n+1 l = E n l - j n l ∆t α 2 .
(5.11)

Discrete model

The stability of this scheme depends directly on the parameter α. So, when α tends to 0, (5.11) can not be used to calculate the new electric eld E n+1 i .

The aim of the following part is to establish a numerical scheme which contains explicitly the quasi-neutral case when α = 0. In this way, a new numerical scheme is developed for the reformulated Maxwell-Ampere equation.

Construction of an Asymptotic-Preserving Maxwell-Ampere numerical scheme

In this part the construction of an Asymptotic-Preserving scheme for the Maxwell-Ampere reformulated equation is explained. In this rst part the numerical scheme is derived in the case of a non-homogeneous collisional plasma without magnetic eld. The next part extends the method to the non-homogeneous collisional case with electromagnetic elds.

Case of a non-homogeneous collisional plasma without magnetic eld.

In this part an Asymptotic-Preserving scheme is constructed for the second equation of (5.7). Let us dene the primary mesh M for the velocity variable v, decomposed into a family of rectangles M p+ 1 2 )∆v. In the same way, a primary mesh N is dened for the space variable x, decomposed into a family of rectangles N l+ 1 2 =]x l , x l+1 [ ∀l ∈ {1; l f } where x l = l∆x and l ∈ N represents the number of points which discretize the space domain. ∆x represents the space discretisation step, which is xed. We denote by E its associated dual mesh consisting of cells

E l =]x l-1 2 , x l+ 1 2 [ where x l-1 2 = (l -1 2 )∆x. Let h l,p (resp. h l+ 1 2 ,p+ 1 2
) be an

approximation of h(x l , v p ) (resp h(x l+ 1 2 , v p+ 1 2
)) for all distribution functions h. The velocity grid is chosen large enough to have f l,p f = f l,-p f = 0 ∀l ∈ {1; l f } which means that there are no particles with such velocities.

By using a conservative discretisation for the Fokker-Planck-Landau equation we obtain

f n+1 l,p -f n l,p ∆t + (vf n ) l+ 1 2 ,p -(vf n ) l-1 2 ,p ∆x - (E n+1 f n ) l,p+ 1 2 -(E n+1 f n ) l,p-1 2 ∆v (5.12) = C n ee,l,p + C n ei,l,p ,
where the computation of the numerical uxes is given by

(vf n ) l+ 1 2 ,p = v p ( f n l,p + f n l+1,p 2 ) - |v p | 2 (f n l+1,p -f n l,p ), (5.13)
5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime

(E n+1 f n ) l,p+ 1 2 = E n+1 l ( f n l,p + f n l,p+1 2 ) - |E n+1 l | 2 (f n l,p+1 -f n l,p ), (5.14)
and

C n ei,l,p = 1 ∆v S p+ 1 2 f n l,p+1 -f n l,p ∆v -S p-1 2 f n l,p -f n l,p-1 ∆v , with S p+ 1 2 = K( v p + v p+1 ∆v ).
The expression of K is given by (1.27). The numerical scheme for the operator C ee,l,p is not given, because this term cancels in the calculation. It is important to notice that the electrostatic eld is calculated implicitly. It will be shown that this choice enables the calculation of the electrostatic eld when α → 0. Using the above numerical uxes, (5.12) reads

f n+1 l,p -f n l,p ∆t + v p f n l+1,p -f n l-1,p -|v p | f n l+1,p -2f n l,p + f n l-1,p 2∆x - E n+1 l f n l,p+1 -f n l,p-1 -|E n+1 l | f n l,p+1 -2f n l,p + f n l,p-1 2∆v = C n ee,l,p + C n ei,l,p .
Multiplying the previous equation by -v p ∆v and summing in p leads to

-p v p f n+1 l,p ∆v + p v p f n l,p ∆v ∆t - ∆v 2∆x p v 2 p f n l+1,p -f n l-1,p -v p |v p | f n l+1,p -2f n l,p + f n l-1,p + 1 2 p v p E n+1 l f n l,p+1 -f n l,p-1 -|E n+1 l |v p f n l,p+1 -2f n l,p + f n l,p-1 = - p C n ei,l,p v p ∆v.
Then using the discrete denition of the current

j l = - p v p f l,p ∆v, (5.15)
the computation of the previous equation leads to

j n+1 l -j n l ∆t - ∆v 2∆x p v 2 p f n l+1,p -f n l-1,p -v p |v p | f n l+1,p -2f n l,p + f n l-1,p + 1 2 p v p E n+1 l f n l,p+1 -f n l,p-1 -|E n+1 l |v p f n l,p+1 -2f n 1,l,p + f n l,p-1 = - p C n ei,l,p v p ∆v.

Discrete model

The following scheme for the Maxwell-Ampere equation is used

E n+1 l -E n l ∆t = - j n+1 l α 2 .
(5.16)

Contrarily to the classical scheme (5.11) the current j in (5. [START_REF] Yu | Conservative nite-dierence schemes for the Fokker-Planck equation not violating the law of increasing entropy[END_REF]) is chosen implicit.

By using (5.16), we get

-α 2 E n+1 l -2E n l + E n-1 l ∆t 2 - ∆v 2∆x p v 2 p f n l+1,p -f n l-1,p -v p |v p | f n l+1,p -2f n l,p + f n l-1,p + 1 2 p v p E n+1 l f n l,p+1 -f n l,p-1 -|E n+1 l |v p f n l,p+1 -2f n 1,l,p + f n l,p-1 = - p C n ei,l,p v p ∆v. Remark 3. It is important to notice that |E n+1 l | p v p f n l,p+1 -2f n l,p + f n l,p-1 = 0.
Indeed a discrete integration by part gives

|E n+1 l | p v p f n l,p+1 -2f n l,p + f n l,p-1 = |E n+1 l | p v p (f n l,p+1 -f n l,p ) - p v p+1 (f n l,p+1 -f n l,p ) , = |E n+1 l | p (v p -v p+1 )(f n l,p+1 -f n l,p ) , = -|E n+1 l |∆v p (f n l,p+1 -f n l,p ) , = 0,
because of boundary condition f n l,pf = f n l,-pf = 0. Therefore, no linearisation nor approximation is required to compute E n+1 l .

5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime Finally, the Asymptotic-Preserving scheme for the second equation of (5.7) writes

-α 2 E n+1 l -2E n l + E n-1 l ∆t 2 - ∆v 2∆x p v 2 p f n l+1,p -f n l-1,p -v p |v p | f n l+1,p -2f n l,p + f n l-1,p + E n+1 l 2 p v p f n l,p+1 -f n l,p-1 = - p C n ei,l,p v p ∆v,
which is the numerical scheme for the reformulated Maxwell-Ampere equation in the case of a inhomogeneous collisional plasma. In the limit case when α tends to zero, the scheme becomes

E n+1 l = ∆v ∆x p v 2 p f n l+1,p -f n l-1,p -v p |v p | f n l+1,p -2f n l,p + f n l-1,p -2 p C n ei,l,p v p ∆v p v p f n l,p+1 -f n l,p-1 .
In the case the expression obtained is well consistent with the limit equation (5.6), this is a key point to obtain the asymptotic preserving property.

Generalisation to a non-homogeneous collisional plasma with electromagnetic elds.

In this part we derive the numerical scheme for the reformulated Maxwell-Ampere equation in the simplied case of 1 dimension in space and 3 dimensions in velocity. The scheme can be extended to the case of 3 dimensions in space. We consider a cartesian case with an electric and a magnetic eld of the form E = (E x (t, x, y), E y (t, x, y), 0), B = (0, 0, B z (t, x, y)).

Following the same method as for the electrostatic case, we derive the following numerical scheme for the reformulated Maxwell-Ampere equation

-α 2 E n+1 x,l -2E n x,l + E n-1 x,l ∆t 2 -∆v x ∆v y ∆v z i,j,k E n+1 x,l + j∆v y B n+1 z,l f n l,i,j,p -∆v 2 x ∆v z i,j,k v x,i E n+1 y,l -i∆v x B n+1 z,l f n l,i,j,p = - i,j,k C n ei,l,i,j,k v x,i ∆v x ∆v y ∆v z + ∆v x ∆v y ∆v z 2∆x i,j,k v 2 x,i f n l+1,i,j,k -f n l-1,i,j,k -v x,i |v x,i | f n l+1,j,k,p -2f n l,i,j,k + f n l-1,i,j,k , 5.3. Discrete model -α 2 E n+1 y,l -2E n y,l + E n-1 y,l ∆t 2 -∆v 2 y ∆v z i,j,k E n+1 x,l + j∆v y B n+1 z,l f n l,i,j,p -∆v x ∆v y ∆v z i,j,k v x,i E n+1 y,l -i∆v x B n+1 z,l f n l,i,j,p = - i,j,k C n ei,l,i,j,k v y,i ∆v x ∆v y ∆v z + α 2 β 2 ∆t B n+1 z,l+1 -B n+1 z,l-1 2∆x - B n z,l+1 -B n z,l-1 2∆x + ∆v x ∆v y ∆v z 2∆x i,j,k v y,i v x,i f n l+1,i,j,k -f n l-1,i,j,k -v y,i |v x,i | f n l+1,i,j,k -2f n l,i,j,k + f n l-1,i,j,k
, where l is the index for space, i the index for the rst coordinate in speed, j for the second and k for the third. Also ∆t, ∆x, ∆v x , ∆v y , ∆v z are respectively the time step, the space step, the velocity step in the rst, second and third dimension. In this case there are two equations, we notice they are coupled.

Stability property

The asymptotic-preserving property also requires that the scheme is uniformly stable with respect to the parameter α. The rigorous proof of the asymptotic stability property is challenging and in general, the few results presented describe simplied linearised models where a linear stability study is conducted [START_REF] Degond | Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit[END_REF][START_REF] Degond | Numerical approximation of the Euler-Maxwell model in the quasineutral limit[END_REF]. In the present case, because of the dependence of the space and velocity variables in addition to the collisional operators such a property cannot be easily derived and a rigorous stability analysis of the method seems beyond the scope of this work. However, we can give some elements of the proof in a simplied linearised collisionless homogeneous case with one velocity dimension (v ∈ R) without magnetic eld. The model reads

     ∂f ∂t -E ∂f ∂v = 0, α 2 ∂E ∂t = -j.
We consider the following linearisation around the equilibrium state given by a Maxwellian distribution function with no electric eld

f = f m + f 1 , E = 0 + E 1 ,
5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime with f m a Maxwellian distribution function. The linearised system reads

       ∂f 1 ∂t + 2E 1 vf m = 0, α 2 ∂E 1 ∂t = -j 1 .
(5.17)

In the numerical method proposed, the electric eld is chosen implicit as well as the electronic current. Then omitting the index 1 for simplicity, the numerical scheme reads

         f n+1 p -f n p ∆t + 2E n+1 v p f m p = 0, α 2 E n+1 -E n ∆t = -j n+1 = p f p=-p f f n+1 p v p ∆v.
(5.18)

The previous system can also be written in the following linear system form

         E f -p f f -p f +1 . . . f p f -1 f p f          n+1 = M α          E f -p f f -p f +1 . . . f p f -1 f p f          n ,
where the matrix M α is given by

M α =        A α α 2 ∆t A α ∆vv -pf • • • A α ∆vv pf - B α -p f α 2 ∆t 1 -B α -p f ∆vv -p f • • • -B α -pf ∆vv pf . . . . . . . . . . . . - B α p f α 2 ∆t -B α p f ∆vv -p f • • • 1 -B α pf ∆vv pf        , with A α = ∆t α 2 + 2∆t 2 p f p=-p f f m p v 2 p ∆v , B α p = 2∆t 2 f m p v p α 2 + 2∆t 2 p f p=-p f f m p v 2 p ∆v
.

The eigenvalues of the matrix M α are given by 1, 1, ..., 1

2p f -1 , K + i √ K -K 2 , K -i √ K -K 2 ,

Asymptotic-Preserving scheme for the

M 1 -Maxwell moments model with K = α 2 (α 2 + 2∆t 2 p f p=-p f f m p v 2 p ∆v)
.

As α ∈ [0, 1] one remarks that K ∈ [0, 1]. It follows that the eigenvalues of M α are in modulus less or equal than 1. The numerical scheme (5.18) for the simplied model (5.17) is then stable for all α. One remarks that in spite of the simplicity of the model (5.17), the form of the matrix M α is not trivial and an extension to the general model seems challenging. However, in a more general case, the numerical tests for the wide range of input parameters, witness of the stability of the method.

Kinetic codes are usually numerically expensive and limited to short time scales.

Angular moments models can be seen as a compromise between kinetic and uid models.

Asymptotic-Preserving scheme for the M 1 -Maxwell moments model

This part presents an Asymptotic-Preserving scheme for the M 1 model associated to the system (5.1). The derivation of the M 1 model has been detailed in chapter 2 and chapter 3. Therefore, we directly give the system we use for the derivation of the scheme.

M 1 moment model

As detailed in the previous chapter, the M 1 moment model without electric eld reads

   ∂ t f 0 + ∇ x .(ζf 1 ) -∂ ζ (Ef 1 ) = Q 0 (f 0 ), ∂ t f 1 + ∇ x .(ζf 2 ) -∂ ζ (Ef 2 ) + E (f 0 -f 2 ) ζ = Q 1 (f 1 ) + Q 0 (f 1 ), (5.19)
where the collisional operators Q 0 and Q 1 are given by

Q 0 (f 0 ) = 2 3 ∂ ζ ζ 2 A(ζ)∂ ζ ( f 0 ζ 2 ) -ζB(ζ)f 0 , Q 1 (f 1 ) = - 2f 1 ζ 3 . The coecients A(ζ) and B(ζ) write A(ζ) = ∞ 0 min( 1 ζ 3 , 1 ω 3 )ω 2 f 0 (ω)dω, B(ζ) = ∞ 0 min( 1 ζ 3 , 1 ω 3 )ω 3 ∂ ω ( f 0 (ω) ω 2 )dω.
5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime

Numerical scheme for the M 1 model

In this part the reformulation of the Maxwell-Ampere equation for the M 1 model is detailed. Considering a conservative scheme for the system (5.19) we write

f n+1 0,l,p -f n 0,l,p ∆t + (ζf n 1 ) l+ 1 2 ,p -(ζf n 1 ) l-1 2 ,p ∆x - (E n+1 f n 1 ) l,p+ 1 2 -(E n+1 f n 1 ) l,p-1 2 ∆ζ = 0, (5.20) f n+1 1,l,p -f n 1,l,p ∆t + (ζf n 2 ) l+ 1 2 ,p -(ζf n 2 ) l-1 2 ,p ∆x - (E n+1 f n 2 ) l,p+ 1 2 -(E n+1 f n 2 ) l,p-1 2 ∆ζ (5.21) + E n+1 ζ p (f n 0,l,p -f n 2,l,p ) = Q n 1,l,p + Q n 0,l,p .
The discrete collision operators involved in (5.21) are respectively given by

Q n 1,l,p = - 2f n 1,l,p ζ 3 p , Q n 0,l,p = 2 3∆ζ p (ζ 2 p+ 1 2 A(ζ p+ 1 2 ) 1 ∆ζ p+ 1 2 f n 1,l,p+1 ζ 2 p+1 - f n 1,l,p ζ 2 p -ζ p+ 1 2 B(ζ p+ 1 2 )f n 1,l,p+ 1 2
)

-(ζ 2 p-1 2 A(ζ p-1 2 ) 1 ∆ζ p-1 2 f n 1,l,p ζ 2 p - f n 1,l,p-1 ζ 2 p-1 -ζ p-1 2 B(ζ p-1 2 )f n 1,l,p- 1 2
) .

Using HLL numerical uxes in (5.20) and (5.21), it holds that

f n+1 0,l,p -f n 0,l,p ∆t + ζ p f n 1,l+1,p -f n 1,l-1,p -|ζ p | f n 0,l+1,p -2f n 0,l,p + f n 0,l-1,p 2∆x - E n+1 l f n 1,l,p+1 -f n 1,l,p-1 -|E n+1 l | f n 0,l,p+1 -2f n 0,l,p + f n 0,l,p-1 2∆ζ = 0, and f n+1 1,l,p -f n 1,l,p ∆t + ζ p f n 2,l+1,p -f n 2,l-1,p -|ζ p | f n 1,l+1,p -2f n 1,l,p + f n 1,l-1,p 2∆x - E n+1 l f n 2,l,p+1 -f n 2,l,p-1 -|E n+1 l | f n 1,l,p+1 -2f n 1,l,p + f n 1,l,p-1 2∆ζ (5.22) + E n+1 l ζ p (f n 0,l,p -f n 2,l,p ) = Q n 1,l,p + Q n 0,l,p .

Asymptotic-Preserving scheme for the M 1 -Maxwell moments model

Multiplying the previous equation (5.22) by -ζ p ∆ζ and summing in p leads to

-p ζ p f n+1 1,l ∆ζ + p ζ p f n 1,l ∆ζ ∆t - ∆ζ 2∆x p ζ 2 p f n 2,l+1,p -f n 2,l-1,p -ζ 2 p f n 1,l+1,p -2f n 1,l,p + f n 1,l-1,p (5.23)
+ 1 2 p ζ p E n+1 l f n 2,l,p+1 -f n 2,l,p-1 -|E n+1 l |ζ p f n 1,l,p+1 -2f n 1,l,p + f n 1,l,p-1 - p E n+1 l (f n 0,l,p -f n 2,l,p )∆ζ = - p ζ p Q n 1,l,p ∆ζ.
Here again, the term containing the electron-electron collision operator cancels. We use the denition of the dimensionless current j j = -

R + f 1 ζdζ,
which can be written on the discrete form

j n l = - p f n 1,l,p ζ p ∆ζ.
Therefore the scheme (5.23) becomes

j n+1 l -j n l ∆t - ∆ζ 2∆x p ζ 2 p f n 2,l+1,p -f n 2,l-1,p -ζ 2 p f n 1,l+1,p -2f n 1,l,p + f n 1,l-1,p (5.24)
+ 1 2 p ζ p E n+1 l f n 2,l,p+1 -f n 2,l,p-1 -|E n+1 l |ζ p f n 1,l,p+1 -2f n 1,l,p + f n 1,l,p-1 - p E n+1 l (f n 0,l,p -f n 2,l,p )∆ζ = - p ζ p Q n 1,l,p ∆ζ.
Using the scheme (5.16), expression (5.24) becomes

-α 2 E n+1 l -2E n l + E n-1 l ∆t 2 - ∆ζ 2∆x p ζ 2 p f n 2,l+1,p -f n 2,l-1,p -ζ 2 p f n 1,l+1,p -2f n 1,l,p + f n 1,l-1,p + 1 2 p ζ p E n+1 l f n 2,l,p+1 -f n 2,l,p-1 -|E n+1 l |ζ p f n 1,l,p+1 -2f n 1,l,p + f n 1,l,p-1 -E n+1 l p (f n 0,l,p -f n 2,l,p )∆ζ = - p ζ p Q n 1,l,p ∆ζ.
5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime Like for the kinetic scheme in (5.3.2), it holds that

p |E n+1 l |ζ p f n 1,l,p+1 -2f n 1,l,p + f n 1,l,p-1 = 0.
Therefore the nal scheme obtained reads

E n+1 l = -α 2 (2E n l -E n-1 l ) ∆t 2 + β 1 (f n 0,l , f n 1,l ) - α 2 ∆t 2 + β 2 (f n 0,l , f n 1,l )
,

where the coecients β 1 and β 2 are given by

β 1 = ∆ζ 2∆x p ζ 2 p f n 2,l+1,p -f n 2,l-1,p -ζ 2 p f n 1,l+1,p -2f n 1,l,p +f n 1,l-1,p - p ζ p Q n 1,l,p ∆ζ, β 2 = 1 2 p ζ p f n 2,l,p+1 -f n 2,l,p-1 - p (f n 0,p,l -f n 2,p,l )∆ζ.
Remark 4. The stability of this new scheme does not depend on the parameter α. So, the electrostatic eld can be obtained even if α becomes equal to zero.

Remark 5. Following the same procedure as for the Fokker-Planck-Maxwell system, this reformulation can be generalised for multi-dimension problems with magnetic elds.

Numerical test cases

This section presents two physically relevant numerical experiments where opposite regimes are considered. The rst one consider two counter-propagating beams of electrons where the collective electrostatic eects are predominant. The second one deals with the relaxation of a localised temperature prole in the quasi-neutral regime. In this regime collisions between particles dominate.

Two electron beams interaction

In this part we study the interaction between two electron beams. This collisionless test case enables us to study the regime where electrostatic eects are predominant. Therefore for this test case we have C ee = C ei = 0. Consider two electron beams propagating at velocity v 0 and v 1 . The dispersion relation is given by

1 - 1 (ω -kv 0 ) 2 - 1 (ω -kv 1 ) 2 = 0,

Numerical test cases

where v 0 and v 1 denote the beams velocities. This conguration can lead to electrostatic instabilities. Indeed, the solutions of the form Ae -iωt+ikx are unstable when ω I the imaginary part of ω is strictly positive. In the case v 0 = -v 1 we can show that the solution is stable if kv 0 ≥ √ 2. This test is problematic for the M 1 model. Indeed, if we consider two electron beams propagating with equal but opposite velocities the distribution function is well dened. Nevertheless, the M 1 model considers only the angular moments f 0 and f 1 . For the calculation of f 1 the contribution of two populations cancel out and we get f 1 = 0. The M 1 model sees an isotropic conguration which is not the reality. To overcome this problem we use the superposition principle that is valid because the model is linear. Two particle populations (one per beam) are considered.

For each time step the M 1 problem is solved for the rst population then for the second one. Hence the Maxwell equations are solved taking into account the two distribution functions.

In the case of two streams propagating with opposite velocities v d and -v d , the initial conditions are

f (t = 0, x, v) = 0.5[(1 + Acos(kx))M v d (v) + (1 -Acos(kx))M -v d (v)], with M ±v d (v) = n e ( m e 2πk B T e ) 3/2 exp -m e (v ∓ v d ) 2 2k B T e .
The parameter A is introduced to perturb the initial condition in order to enable the development of the electrostatic instability. The velocity modulus goes from 0 to 12 v th and the space scale from 0 to 25 λ De . With 100 points for the space grid and 128 points for the velocity modulus grid the results are converged. In Figure 5.1 the distribution function is represented in the phase space for the initial time and the nal time t = 30 plasma periods. In this example v d = 4, A = 0.001 and periodical boundary conditions are used. In the second plot the interaction between the two streams is observed.

Our results have been compared with a kinetic code [START_REF] Duclous | High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications[END_REF]. In Figure 5.2, the evolution of the electrostatic energy is represented as a function of time for the (M 1 - AP) code in green and for a kinetic code in red. The rst plot shows the results for A=0.001 and the second one for A=0.1. In the case of small perturbations (A=0.001), the M 1 model and the kinetic code give analogous results. In the case of strong perturbations (A=0.1), the (M 1 -AP) code and the kinetic code show some dierences after a long time. In the case of a strong perturbation, a non-linear regime is obtained and it is well-known that the M 1 model is not accurate enough [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF].

This numerical experiment shows the good behaviour of the (M 1 -AP) scheme in a regime where electrostatic eects are predominant.

Hot spot relaxation

We now study the relaxation of a localised temperature perturbation generated, for example, by a short laser pulse. Suppose that the laser impulse duration 5.5. Numerical test cases is shorter than the relaxation time. This phenomenon investigated in references [START_REF] Batishchev | Heat transport and electron distribution function in laser produced with hot spots[END_REF], [START_REF] Brantov | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF] corresponds physically to the heating of a plasma during a short time and to the relaxation phenomenon which follows. Steep temperature gradients due to the localised heating induce a non-local heat transport. Here, we consider the collisional regime. This conguration is particularly interesting because it enables to study the coupling of the M 1 model with the Maxwell-Ampere Asymptotic-Preserving scheme.

Initially the distribution function for electrons is a Maxwellian with a Gaussian temperature prole

T e (x, t = 0) = T 0 + T 1 exp(- x 2 D 2 ),
where the hot spot size D is a characteristic scale of inhomogeneity. First we make a few remarks on the formulation of the problem related to the ambipolar electric eld. In the case of a smooth temperature gradient, the following formula for the

electric eld is obtained [179] eE m e = - ∇ x R 3 F 0 v 7 dv 6 R 3 F 0 v 5 dv
, where F 0 is the isotropic part of the electron distribution function. For a Maxwellian distribution function, this eld is expressed through the classical formula eE = -T e ∇ x n e n e + 5 2

∇ x T e T e .
The local heat ux is given by the Spitzer-Harm formula [START_REF] Struchtrup | Macroscopic Transport Equations for Rareed Gas Flows[END_REF] q SH = -κ SH ∇ x T e , with conductivity κ SH = 128 3π Z + 0.24 Z + 4.2 n e v th λ ei .

Note that already for D -1 λ ei > 0.06/ √ Z, the classical transport theory is not applicable.

In a rst simulation presented here, we choose typical parameters for ICF studies T 0 = 1KeV, T 1 = 4KeV and D = 8, 44λ ei . There is no electric eld at the initial time. We choose the specular reection as the boundary conditions. The space scale goes from -80λ ei to 80λ ei . The velocity modulus scale goes from 0 to 50 v th .

Figure 5.3 shows the evolution of the temperature and electric eld proles until 30 τ ei . Then at t = 2 τ ei , we observe that the temperature prole starts to relax to a colder temperature and the electric eld, which is proportional to the gradient of temperature, also decreases. The numerical scheme reproduces a good behaviour of the hot spot relaxation phenomenon.

5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime The results of our M 1 -Asymptotic-Preserving scheme (M 1 -AP) have been compared with the ones obtained by a kinetic code [START_REF] Marocchino | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF]. In Figure 5.4, the temperature and the electrostatic eld proles are represented as a function of space for dierent times. The (M 1 -AP) results are given in green while the kinetic results are in red.

Both results show a good agreement. Small dierences are observed concerning the amplitude of the temperature and the electric eld. The relaxation phenomenon observed with the (M 1 -AP) code is faster than the one with the kinetic code.

It is interesting to notice that there is a large dierence of calculation time. The simulation with the kinetic code requires the use of 50 processes during several days while the (M 1 -AP) code only needs few minutes with one processor. Moreover, thanks to the rapidity of the M 1 Asymptotic-Preserving code a mesh convergence study has been performed. With 500 points for the space grid and 80 points for the energy grid the results are converged. The time step used is ∆t = 10 -3 τ ei in order to respect classical stability conditions. Remark 6. In this case the parameter α which represents the ratio between the electron-ion collision frequency and the electron plasma frequency is equal to 4.10 -4 . In order to avoid a severe constraint on the time step we use the new M 1 -Asymptotic-Preserving scheme. With the same CFL conditions, the classic Maxwell-Ampere numerical scheme breaks down from the very rst iterations. Remark 7. It is important to notice that the Asymptotic-Preserving scheme is stable even in the case α = 0.

In a second stage, a new simulation was performed in order to compare the results obtained using the (M 1 -AP) scheme with the ones obtained using another kinetic code [START_REF] Bobylev | Monte Carlo methods and their analysis for Coulomb collisions in multicomponent plasmas[END_REF] and a hydrodynamic code based on the classical transport theory

[179]- [START_REF] Struchtrup | Macroscopic Transport Equations for Rareed Gas Flows[END_REF]. For the simulation presented, we choose the parameters T 0 = 1KeV, T 1 = 2KeV, Z = 80 and D = 100λ ei . The results are given at time t = 120τ ei . The space scale goes from -2500λ ei to 2500λ ei . In Figure 5.5, the temperature and the heat ux are represented for the three codes. Dimensionless quantities are used here.

It appears that the three temperature proles are very close. The hydrodynamic temperature is slightly smaller than the two others while the (M 1 -AP) scheme and the kinetic scheme are in very good agreement. Dierent heat ux proles are also compared in Figure 5.5. The (M 1 -AP) ux and the kinetic ux are close and it appears that the (M 1 -AP) ux is slightly more spread out. The hydrodynamic ux on the contrary is much larger than the two others and is also more localised. In this regime, one can again observe the good behaviour of the (M 1 -AP) scheme. This scheme gives close results with the kinetic code while the hydrodynamic approach overestimates the heat ux. 5. Asymptotic-Preserving scheme for the M 1 -Maxwell system in the quasi-neutral regime

Conclusion

In this chapter, we have constructed an Asymptotic-Preserving scheme for the full Fokker-Planck-Landau-Maxwell system, which handles the quasi-neutral limit without any contraction of time and space steps. We have rst established a reformulated Fokker-Planck-Landau-Maxwell system then used it to construct the Asymptotic-Preserving scheme. The method has been extended to the general case of collisional plasmas in electromagnetic elds for multi-dimensions problems. An M 1 -Asymptotic-Preserving scheme has been derived. Next, the M 1 -Asymptotic-Preserving scheme has been implemented and two numerical test cases have been performed. The rst one corresponds to a regime where electromagnetic eects are predominant.

The second one on the contrary shows the eciency of the Asymptotic-Preserving scheme in the quasi-neutral regime. The scheme, accurate and fast, works in both regimes. In this chapter, we have scaled the system studied with the collisional parameter to study the quasi-neutral regime. In the next chapter the electronic M 1 model is studied in the diusive limit.

Chapter 6

Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

The study introduced in this chapter has been submitted for publication.

Introduction

In inertial connement fusion, nanosecond laser pulses are used to ignite a deuterium-tritium target. An accurate description of this process is necessary for understanding of laser-matter interactions and for the target design. Numerous physical phenomena such as, parametric [START_REF] Epperlein | [END_REF]173] and hydrodynamic [START_REF] Drake | Parametric instabilities of electromagnetic waves in plasmas[END_REF][START_REF] Shkarofsky | The Particle Kinetics of Plasmas M[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF] instabilities, laser-plasma absorption [190], wave damping [START_REF] Landau | On the vibration of the electronic plasma[END_REF], energy redistribution [180] inside the plasma and hot spots formation [START_REF] Brantov | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF][START_REF] Marocchino | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF] from which the thermonuclear reactions propagates depend on the electron heat transport. The most popular electron heat transport theory was developed by Spitzer and Härm [START_REF] Struchtrup | Macroscopic Transport Equations for Rareed Gas Flows[END_REF] who rst solved the electron kinetic equation by using the expansion of the electron mean free path to the temperature scale length (denoted ε in this work). Considering the distribution function of particles close to equilibrium, its deviation from the Maxwellian distribution function can be computed and the electron transport coecients in a fully ionised plasma without magnetic eld are derived. However, even if the electron heat transport is essential, it is not correctly described in large inertial connement fusion tools. Indeed, when the electron mean free path exceeds In chapter 2, it has been seen that the angular moments models represent an alternative method situated between kinetic and uid models. The M 1 model is largely used in various applications such as the radiative transfer [START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF][START_REF] Charrier | Multigroup model for radiating ows during atmospheric hypersonic re-entry[END_REF][START_REF] Dubroca | Entropic moment closure hierarchy for the radiative transfert equation[END_REF][START_REF] Ripoll | An averaged formulation of the M1 radiation model with presumed probability density function for turbulent ows[END_REF][START_REF] Ripoll | A factored operator method for solving coupled radiation-hydrodynamics models[END_REF][START_REF] Turpault | Multigroup half space moment appproximations to the radiative heat transfer equations[END_REF]202] or electronic transport [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF][START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF]. The M 1 model is known to satisfy fundamental properties such as the positivity of the rst angular moment, the ux limitation and conservation of total energy. Also, it correctly recovers the asymptotic diusion equation in the limit of long time behaviour with important collisions [START_REF] Dubroca | Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif[END_REF]. One challenging issue is to derive numerical schemes satisfying fundamental properties. For example, the classical HLL scheme [START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] ensures the positivity of the rst angular moment and the ux limitation property. However, this scheme fails in recovering the correct limit diusion equation in the asymptotic regime [START_REF] Audit | A radiation hydrodynamics scheme valid from the transport to the diusion limit[END_REF]. As explained in chapter 4, overcoming this major drawback a class of numerical schemes has emerged over the years called asymptotic-preserving schemes (AP). Asymptoticpreserving schemes in the sense of Jin-Levermore [START_REF] Jin | Fully discrete numerical transfer in diusive regimes[END_REF][START_REF] Jin | The discrete-ordinate method in diusive regimes[END_REF] are designed to handle multi-scale situations and behave correctly in the asymptotic limit considered. In this context many works have been performed following dierent approaches in a one dimensional framework [START_REF] Aregba-Driollet | Asymptotic High-Order Schemes for 2x2 Dissipative Hyperbolic Systems[END_REF][START_REF] Boscarino | High-order asymptotic-preserving methods for fully nonlinear relaxation problems[END_REF][START_REF] Bouchut | Upwinding of the source term at interfaces for Euler equations with high friction[END_REF][START_REF] Dimarco | Asymptotic Preserving Implicit-Explicit Runge Kutta Methods for Nonlinear Kinetic Equations[END_REF][START_REF] Filbet | A class of asymptotic preserving schemes for kinetic equations and related problems with sti sources[END_REF][START_REF] Latte | Asymptotic-preserving projective integration schemes for kinetic equations in the diusion limit[END_REF][START_REF] Lemou | A new asymptotic preserving scheme based on Micro-Macro formulation for linear kinetic equations in the diusion limit[END_REF]. In particular, one of the most productive approach from the work of Gosse-Toscani [START_REF] Gosse | Space localizaion and well-balanced schemes for discrete kinetic models in diusive regimes[END_REF] and which has been largely extended [START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF][START_REF] Berthon | Asymptotic preserving HLL schemes[END_REF][START_REF] Buet | Conservative and entropy decaying numerical scheme for the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Buet | Asymptotic preserving and positive schemes for radiation hydrodynamics[END_REF][START_REF] Chalons | Godunovtype schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction[END_REF], is based on the modication of approximate Riemann solvers. Some works also deal with the two dimensional case [START_REF] Berthon | An Asymptotic-Preserving Scheme for Systems of Conservation Laws with Source Terms on 2D Unstructured Meshes[END_REF][START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF][START_REF] Buet | Design of asymptotic preserving nite volume schemes for hyperbolic heat equation on unstructured meshes[END_REF].

In [START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF], an HLLC scheme is proposed to solve the M 1 model of radiative transfer in two space dimensions. The HLLC approximate Riemann solver is considered and a relevant numerical approximations of the extreme wavespeeds give the asymptoticpreserving property. Close ideas were also developed in [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF], where a relaxation scheme is exhibited. In order to derive suitable schemes pertinent for transport and diusion regimes, it was proposed to use the modied Godunov-type schemes in order to include sources terms [START_REF] Greenberg | A well-balanced scheme for the numerical processing of source terms in hyperbolic equations[END_REF]. The numerical viscosity is modied in [START_REF] Buet | Asymptotic analysis of uid models for the coupling of radiation and hydrodynamics[END_REF][START_REF] Buet | Asymptotic preserving and positive schemes for radiation hydrodynamics[END_REF][START_REF] Gosse | An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations[END_REF][START_REF] Gosse | Space localizaion and well-balanced schemes for discrete kinetic models in diusive regimes[END_REF] to correctly recover the expected diusion regimes but extensions seem to be challenging. In [START_REF] Berthon | Asymptotic preserving HLL schemes[END_REF], the approximate HLL Riemann solver is modied to include collisional source term. The resulting numerical scheme satises all the fundamental properties and a clever correction enables to recover the good diusion equation in the asymptotic limit.

In this work, we consider the M 1 model for the electron transport [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF][START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF][START_REF] Mallet | General moment system for plasma physics based on minimum entropy principle[END_REF]] in a Lorentzian plasma where ions are xed. Omitting the x and t dependency, the rst three angular moments f 0 , f 1 and f 2 of the electron distribution function f are 6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit dened by

f 0 (ζ) = ζ 2 1 -1 f (µ, ζ)dµ, f 1 (ζ) = ζ 2 1 -1 f (µ, ζ)µdµ, f 2 (ζ) = ζ 2 -1 -1 f (µ, ζ)µ 2 dµ. (6.1)
The moment system studied writes

         ∂ t f 0 (t, x, ζ) + ζ∂ x f 1 (t, x, ζ) + E(x)∂ ζ f 1 (t, x, ζ) = 0, ∂ t f 1 (t, x, ζ) + ζ∂ x f 2 (t, x, ζ) + E(x)∂ ζ f 2 (t, x, ζ) - E(x) ζ (f 0 (t, x, ζ) -f 2 (t, x, ζ)) = - 2α ei (x)f 1 (t, x, ζ) ζ 3 , (6.2)
The coecient α ei is a positive function which may depend of x, E represents the electrostatic eld as function of x and ζ is the velocity modulus. The fundamental point of the moments models is the denition of the closure which writes the highest moment as a function of the lower ones. This closure relation corresponds to an approximation of the underlying distribution function, which the moments system is constructed from. In the M 1 problem we need to dene f 2 as a function of f 0 and f 1 . As explained in chapter 2, the closure relation originates from an entropy minimisation principle [START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF][START_REF] Minerbo | Maximum entropy Eddigton Factors[END_REF] and the moment f 2 can be computed [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF][START_REF] Dubroca | Entropic moment closure hierarchy for the radiative transfert equation[END_REF] as a function of f 0 and f 1

f 2 (t, x, ζ) = χ f 1 (t, x, ζ) f 0 (t, x, ζ) f 0 (t, x, ζ), with χ(α) ≈ 1 + α 2 + α 4 3 . (6.
3)

The set of admissible states [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF] is dened by

A = (f 0 , f 1 ) ∈ R 2 , f 0 ≥ 0, |f 1 | ≤ f 0 . (6.4)
A challenging issue is to derive a numerical scheme for the electron M 1 model (6.2) satisfying all the fundamental properties and which handles correctly the diusive limit recovering the good diusion equation. Such a scheme could then have a direct access to all the nonlocal regimes and their related physical eects described above while the others numerical schemes fail in such regimes. Complications arise when considering such an issue. Firstly, the electron M 1 model (6.2) is nonlinear. Because, of the entropic closure, the angular moment f 2 is a nonlinear function of f 0 and f 1 .

Secondly, the approach undertaken must be suciently general to correctly take into account the source term -E(x)(f 0 (t, x, ζ)-f 2 (t, x, ζ))/ζ. One must notice, that this term is closely related to the term E∂ ζ f 2 (t, x, ζ), it plays an important role for low energies and cannot be treated as a collisional source term. Thirdly, for the purpose of realistic physical applications, one may require to correctly capture stationary states. In the case of near-equilibrium congurations a well-balancing property is then desired. Also, the physical parameter α ei is a function of x and cannot be treated as a constant. Finally, the space and velocity modulus dependence of the 6.2. Case without electrostatic eld angular moments, leads to a very complex diusion equation in the asymptotic limit with mixed derivatives.

In the rst part of this chapter, the case without electric eld and the homogeneous case with electric eld are studied. The generalisation to the general problem requires a deep understanding of the two congurations studied here. An extension to the general model is proposed in a second part. The approach retained is noticeably dierent with [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF][START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF][START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF]. The derivation of the numerical scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modied to enable the derivation of a scheme which also satises the admissibility conditions (6.4) and is well-suited for capturing stationary states. Moreover, it enjoys asymptotic-preserving properties and correctly handles the diusive limit recovering the good diusion equation.

In the rst part of this work we introduce the M 1 model without electrostatic eld and a homogeneous case with electric eld. The classical HLL scheme [START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] in the diusive limit is briey recalled before introducing the new numerical scheme.

The asymptotic-preserving property is exhibited. In Section 6.3 for the homogeneous case with electric eld, we point out diculties encountered when using a relaxation approach in order to include the source term -E(x

)(f 0 (t, x, ζ)-f 2 (t, x, ζ))/ζ. Then,
the derivation of an asymptotic-preserving scheme following the method introduced in the previous section is detailed and the well-balanced and asymptotic-preserving properties are analysed. In Section 6.4, dierent numerical tests are presented to highlight the eciency of the present method.

The second part is extending these ideas and introduce a numerical scheme for the general electronic M 1 model. In section 6.6, the scheme is modied to ensure the admissibility conditions (6.4) and to capture the non isotropic diusion then the asymptotic-preserving property is exhibited. The term -E(x)(f 0 -f 2 )/ζ is nally included in the scheme. In Section 6.7, numerical examples are presented to demonstrate of the eciency of the method. A conclusion is given in Section 6.8.

Case without electrostatic eld

The rst simplied case we consider is given by system (6.2) without electrostatic eld E. In this case the M 1 model (6.2) writes

   ∂ t f 0 + ζ∂ x f 1 = 0, ∂ t f 1 + ζ∂ x f 2 = - 2α ei ζ 3 f 1 .
(6.5) A very similar system was considered in [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF] in the frame of radiative transfer and a relaxation scheme was proposed. The same procedure could be applied in this case, however we introduce a dierent approach based on approximate Riemann solvers.

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

Model and diusive limit

We consider the following diusion scaling

t = t/t * , x = x/x * , ζ = ζ/v th , Ẽ = Ex * /v th . (6.6)
The parameters t * and x * are chosen such that τ ei /t * = ε 2 , λ ei /x * = ε, where τ ei is the electron-ion collisional period , λ ei the electron-ion mean free path and v th the thermal velocity dened by v th = λ ei /τ ei . The positive parameter ε is devoted to tend to zero. In that case, omitting the tilde notation, system (6.5) rewrites

   ε∂ t f ε 0 + ζ∂ x f ε 1 = 0, ε∂ t f ε 1 + ζ∂ x f ε 2 = - 2σ ζ 3 f ε 1 ε , (6.7)
where the coecient σ represents a positive function of x dened as

σ(x) = τ ei α ei (x) v 3 th .
Inserting the following Hilbert expansion of f ε 0 and f ε 1

f ε 0 = f 0 0 + εf 1 0 + O(ε 2 ), f ε 1 = f 0 1 + εf 1 1 + O(ε 2 ), (6.8)
into the second equation of (6.7) leads to f 0 1 = 0.

(6.9)

Using the denition (6.3), it follows that

f 0 2 = f 0 0 /3.
So, the second equation of (6.7) gives

f 1 1 = - ζ 4 6σ ∂ x f 0 0 . (6.10)
Using the previous equation and the rst equation of (6.7) nally leads to the diusion equation for f 0 0

∂ t f 0 0 (t, x) -∂ x ζ 5 6σ(x) ∂ x f 0 0 (t, x) = 0. (6.11)
Here we have omitted the tilde notation, writing this diusion equation in nonrescaled (dimensional) variables we obtain

∂ t f 0 0 (t, x) -∂ x ζ 5 6α ei (x) ∂ x f 0 0 (t, x) = 0.
(6.12) 6.2. Case without electrostatic eld

Numerical method

In this part, we rst recall the limit of the HLL scheme, usually used for the electronic M 1 model, for the diusive limit.

Limit of the HLL scheme

Introduce a uniform mesh with constant space step ∆x = x i+1/2 -x i-1/2 , i ∈ Z and a time step ∆t. We consider a piecewise constant approximate solution

U h (x, t n ) ∈ R 2 at time t n U h (x, t n ) = U n i if x ∈ [x i-1/2 , x i+1/2 ], with U n i = t (f n 0i , f n 1i ).
The classical HLL scheme [START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] for the system (6.11), in the case where the minimum and maximum velocity waves involved in the approximate Riemann solver are chosen equal to -ζ and ζ, writes

       ε f n+1,ε 0i -f n,ε 0i ∆t + ζ f n,ε 1i+1 -f n,ε 1i-1 2∆x -ζ∆x f n,ε 0i+1 -2f n,ε 0i + f n,ε 0i-1 2∆x 2 = 0, ε f n+1,ε 1i -f n,ε 1i ∆t + ζ f n,ε 2i+1 -f n,ε 2i-1 2∆x -ζ∆x f n,ε 1i+1 -2f n,ε 1i + f n,ε 1i-1 2∆x 2 = - 2σ i ζ 3 f n,ε 1i ε . (6.13)
We introduce the discrete Hilbert expansions

f ε 0i = f n,0 0i + εf n,1 0i + O(ε 2 ), f n,ε 1i = f n,0 1i + εf n,1 1i + O(ε 2 ). (6.14)
At the order ε -1 , the second equation of (6.13) gives

f n,0 1i = 0, and using the denition (6.3), it follows that

f n,0 2i = f n,0 0i /3.
At the order ε 0 , the second equation of (6.13) gives

f n,1 1i = - ζ 3 3σ i f n,0 0i+1 -f n,0 0i-1 2∆x .
However, because of the diusive part of the HLL scheme, the rst equation of (6.13) also leads to

f n,0 0i+1 -2f n,0 0i + f n,0 0i-1 ∆x 2 = 0,
which is not the diusion equation expected for f 0 0 . The diusive part of the HLL scheme gives an unphysical numerical viscosity and leads to the wrong asymptotic behaviour.

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

Derivation of the scheme

The ideas introduced in [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF][START_REF] Berthon | Asymptotic preserving HLL schemes[END_REF][START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for sources[END_REF][START_REF] Greenberg | A well-balanced scheme for the numerical processing of source terms in hyperbolic equations[END_REF] in order to include the contribution of source terms, urge to consider approximate Riemann solvers which own a stationary discontinuity (0-contact discontinuity). Therefore, we introduce the following approximate Riemann solvers at each cell interface, denoted by U R (x/t, U L , U R ),

dened by U R (x/t, U L , U R ) =            U L if x/t < -a x , U L * if -a x < x/t < 0, U R * if 0 < x/t < a x , U R if a x < x/t, (6.15)
where

U L * = t (f L * 0 , f * 1 ), U R * = t (f R * 0 , f * 1 )
and the minimum and maximum velocity waves -a x and a x . Note, we choose the two velocity waves to be opposite.

The structure solution of the approximate Riemann problem is displayed in Figure 6.1. At the interface x i+ 1 2 , the quantities U L and U R stand for U i = t (f 0i , f 1i ) and U i+1 = t (f 0i+1 , f 1i+1 ). Contrarily to the classical HLL scheme [START_REF] Touati | [END_REF] t n + ∆t is chosen as

U h (x, t n + ∆t) = U R x -x i+1/2 t n + ∆t , U i , U i+1 if x ∈ [x i , x i+1 ].
As the following CF L condition is respected ∆t ≤ ∆x 2a x ,

Case without electrostatic eld

the piecewise constant approximate solution is then obtained

U n+1 i = 1 ∆x x i+1/2 x i-1/2
U h (x, t n+1 )dx.

(6.16)

The intermediate states f L * 0 , f R * 0 and f * 1 must be dened. Integrating the rst equa- tion of (6.5) on [-a x ∆t, a x ∆t] × [0, ∆t] and multiplying by 1 2ax∆t , gives the following consistency condition

f L * 0 + f R * 0 2 = f L 0 + f R 0 2 - 1 2a x [ζf R 1 -ζf L 1 ].
(6.17)

The unknowns f L * 0 and f R * 0 will be chosen in order to satisfy this consistency condition (6.17). The same procedure using the second equation of (6.5) gives

f * 1 = f L 1 + f R 1 2 - 1 2a x (ζf R 2 -ζf L 2 ) - 2 ζ 3 1 2a x ∆t ax∆t -ax∆t ∆t 0 α ei (x)f 1 (x, t)dtdx. (6.18)
The following approximation is made

1 2a x ∆t ax∆t -ax∆t ∆t 0 α ei (x)f 1 (x, t)dtdx = ᾱei ∆tf * 1 , (6.19)
with ᾱei = α(0). Using (6.19) in (6.18), it follows that

f * 1 = ζ 3 ζ 3 + 2ᾱ ei ∆t f L 1 + f R 1 2 - 1 2a x (ζf R 2 -ζf L 2 ) .
Finally the following denition of f *

1 is chosen f * 1 = 2a x ζ 3 2a x ζ 3 + 2ᾱ ei ∆x f L 1 + f R 1 2 - 1 2a x (ζf R 2 -ζf L 2 ) .
(6.20)

It will be shown in the next part, that this choice enables to obtain the good asymptotic-preserving property. Also, this denition recovers the formalism introduced in [START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF][START_REF] Berthon | Asymptotic preserving HLL schemes[END_REF]. In order to respect the consistency relation (6.17), the unknowns

f L * 0 and f R * 0 are dened by f L * 0 = f0 -Γ, f R * 0 = f0 + Γ, with f0 = f L 0 + f R 0 2 - 1 2a x (ζf R 1 -ζf L 1 ),
and the coecient Γ is calculated using the classical Rankine-Hugoniot conditions

       f L * 0 = f L 0 - ζ a x (f * 1 -f L 1 ), f R * 0 = f R 0 - ζ a x (f R 1 -f * 1 ).
(6.21)

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

It follows that Γ = 1 2 [f R 0 -f L 0 - ζ a x (f L 1 -2f * 1 + f R 1 )]. (6.22)
In order to satisfy the admissibility conditions (6.4), we propose to modify the states

f L * 0 and f R * 0 such that f L * 0 = f0 -Γθ, f R * 0 = f0 + Γθ, (6.23)
where θ ∈ [0, 1] is xed to ensure the admissibility conditions. Remark 6.1. In the case θ = 0, the admissibility requirements (6.4) are fullled.

Indeed, in this case system (6.23) gives f R * 0 = f L * 0 = f0 and f * 1 is given by (6.20).

Since

2a x ζ 3 /(2a x ζ 3 + σ∆x) ≤ 1 it follows that f * 1 ≤ f R * 0 = f L * 0 .
Then the parameter θ is computed as the largest possible such that Finally, θ is chosen as θ = min( θ, 1).

     f R * 0 -|f * 1 | ≥ 0, f L * 0 -|f * 1 | ≥ 0, f R * 0 ≥ 0 and f L * 0 ≥ 0.
Therefore the unknowns f n+1 0i and f n+1 1i are computed using (6.16)

     f n+1 0i = a x ∆t ∆x f R * 0i-1/2 + (1 - 2a x ∆t ∆x )f n 0i + a x ∆t ∆x f L * 0i+1/2 , f n+1 1i = a x ∆t ∆x f * 1i-1/2 + (1 - 2a x ∆t ∆x )f n 1i + a x ∆t ∆x f * 1i+1/2 . (6.26)
The wavespeed a x is xed using the ideas introduced in [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF]. It is known that the electronic M 1 model without electric eld is hyperbolic symmetrizable [START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF] and the eigenvalues of the Jacobian matrix always belong in the interval [-ζ, ζ]. Therefore, we set a x = ζ.

Asymptotic-preserving properties

In this part we prove the asymptotic-preserving property of the scheme (6.20)-(6.23)- (6.26). It is shown that when ε tends to zero, the scheme (6.20)-(6.23)-(6.26) is consistent with the limit diusion equation (6.11).

Case without electrostatic eld

Theorem 6.2. When ε tends to zero, the unknown f n+1,0 0i given by the numerical scheme (6.26)-(6.23)-(6.20) satises the following discrete equation

f n+1,0 0i -f n,0 0i ∆t - ζ ∆x ζ 3 6σ i+1/2 ∆x (ζf n,0 0i+1 -ζf n,0 0i ) - ζ 3 6σ i-1/2 ∆x (ζf n,0 0i -ζf n,0 0i-1 ) = 0. (6.27)
Proof. Following the same approach as in [START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF][START_REF] Berthon | Asymptotic preserving HLL schemes[END_REF], using the diusive scaling and equation (6.26) leads to

       ε f n+1 0i -f n 0i ∆t = a x ∆x f L * 0i+1/2 - 2a x ∆x f n 0i + a x ∆x f R * 0i-1/2 , ε f n+1 1i -f n 1i ∆t = a x ∆x f * 1i+1/2 - 2a x ∆x f n 1i + a x ∆x f * 1i-1/2 , (6.28)
where the intermediate states f L * 0 and f R * 0 are given by (6.23) and (6.20) rewrites

f * 1 = 2a x ζ 3 2a x ζ 3 + 2σ∆x/ε f L 1 + f R 1 2 - 1 2a x (ζf R 2 -ζf L 2 ) . (6.29)
As soon as ε tends to zero, we obtain f * 1 = 0. We now suppose that f n 1i = 0 in the limit ε tends to zero. In this case, the denition (6.25) leads to

θ = f L 0 + f R 0 |f L 0 -f R 0 | ≥ 1.
Then the parameter θ is equal to 1.

Remark 6.3. In the diusive regime when ε tends to zero, no limitation on the intermediates states (6.23) is required.

Using the denition (6.23), it follows that the intermediate states f L * 0 and f R * 0 are given by

       f L * 0 = f L 0 - ζ a x (f * 1 -f L 1 ), f R * 0 = f R 0 - ζ a x (f R 1 -f * 1 ). (6.30)
The discrete Hilbert expansions (6.14) are now used. Inserting the previous expressions in the rst equation of (6.28), considered at the order ε 0 , gives no information since the terms cancel each other out. However, at the order ε 1 , the expressions (6.30), (6.29) and the rst equation of (6.28) lead to

       f L * ,1 0 = f L,1 0 - ζ a x (f * ,1 1 -f L,1 1 ), f R * ,1 0 = f R,1 0 - ζ a x (f R,1 1 -f * ,1 1 ), (6.31)
6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit with

f * ,1 1 = - ζ 3 6σ∆x ζf R,n,0 0 -ζf L,n,0 0 , (6.32)
and

f n+1,0 0i -f n,0 0i ∆t = a x ∆x f * ,1 0i+1/2 - 2a x ∆x f n,1 0i + a x ∆x f * ,1 0i-1/2 . (6.33)
Inserting expressions (6.31) into (6.33) leads to equation (6.27) which is consistent with the limit diusion equation (6.11).

To complete the proof, it is necessary to show that f n 1 tends to zero, when ε tends to zero. Equation (6.16) gives

∆xU n+1 i = x i x i-1/2 U R dx + x i+1/2 x i U R dx,
where U R is computed with the approximate Riemann problem (6.15). Then

x i x i-1/2 f 1 (x, ∆t)dx = a x ∆tf * 1i-1/2 + ( ∆x 2 -a x ∆t)f n 1i , and x i+1/2 x i f 1 (x, ∆t)dx = ( ∆x 2 -a x ∆t)f n 1i + a x ∆tf * 1i+1/2 .
A long but classical calculation [START_REF] Berthon | Asymptotic preserving HLL schemes[END_REF] leads to

f n+1 1i -f n 1i ∆t + 1 ∆x 2a x 2a x + σi+1/2 ∆x F i+1/2 - 2a x 2a x + σi-1/2 ∆x F i-1/2 (6.34) + 1 ∆x ∆xσ i+1/2 2a x + σi+1/2 ∆x (-a x f n 1i -ζf n 2i ) + ∆xσ i-1/2 2a x + σi-1/2 ∆x (-a x f n 1i + ζf n 2i ) = 0, with F i+1/2 = 1 2 ζf n 2i+1 + ζf n 2i -a x (f n 1i+1 + f n 1i ) .
Using the diusive scaling we obtain that f n 1i tends to zero when ε tends to zero.

Stability property

The asymptotic-preserving property requires that the scheme should be uniformly stable with respect to the small parameter ε. In the case of an uniform stable scheme the CFL stability condition in diusive regime should be that of a diusion scheme ∆t ≤ 3α ei ∆x 2 /ζ 5 (see Eq. 6.12). Also, in the case of a small collisional parameter α ei , the time step should be chosen according to the hyperbolic CFL condition ∆t ≤ ∆x/ζ. An uniform stability property is proved in [START_REF] Klar | Uniform stability of a nite dierence scheme for transport equations in the diusion limit[END_REF] or [START_REF] Liu | Analysis of an asymptotic preserving scheme for linear kinetic equations in the diusion limit[END_REF] in the framework of linear scalar equations. However, the model considered in this 6.2. Case without electrostatic eld work is a nonlinear system and the derivation of such a property is very challenging. Therefore, for the numerical test cases we consider the CFL condition ∆t ≤ max(∆x/a x , 3α ei ∆x 2 /ζ 5 ). (6.35) In practice, it has been observed that in the case of a very large collisional parameter α ei the CFL stability condition is that of a diusion scheme ∆t ≤ 3α ei ∆x 2 /ζ 5 and the proposed AP scheme is not stable. More precisely, in a very diusive regime when considering a parabolic CFL condition, it is observed that the quantity f n 1i does not behave in O(ε) in the long time regime as expected (see condition Eq. 6.9).

To overcome this drawback, instead of using the second equation of (6.26), we propose to consider the classical following scheme to compute f n 1i at each time step

f n+1 1i -f n 1i ∆t + ζ f n 2i+1 -f n 2i-1 2∆x -a x f n 1i+1 -2f n 1i + f n 1i-1 2∆x = - 2α ei ζ 3 f n+1 1i .
This scheme rewrites

f n+1 1i = ζ 3 ζ 3 + 2α ei ∆t f n 1i -∆t ζ f n 2i+1 -f n 2i-1 2∆x -a x f n 1i+1 -2f n 1i + f n 1i-1 2∆x . (6.36)
Obviously this scheme is consistent with the second equation of (6.5) and captures the correct asymptotic limits (6.9) and (6.10). Here, it is important to notice that we still consider the rst equation of (6.26) with the denitions (6.23)-(6.22)-(6.25)

to compute f n 0i at each time step. This choice enables to correctly capture the asymptotic limit and the use of the parabolic CFL condition in the diusive regime. In addition, the numerical solution needs to satisfy the admissibility requirements (6.4). Indeed, the correction parameter θ dened in (6.25) was proposed considering the second equation of (6.26) which is now replaced by (6.36). However, it can be shown that the condition (6.25) also enables the admissibility of the numerical solution using (6.36). Proposition 6.4. The numerical scheme (6.22)-(6.23)-(6.25)-(6.26)-(6.36) preserves the admissibility of the numerical solutions. Proof. We remark that equation (6.36)

rewrites f n+1 1i = α a x ∆t ∆x f1i-1/2 + α(1 - 2a x ∆t ∆x )f n 1i + α a x ∆t ∆x f1i+1/2 , (6.37
)

with α = ζ 3 /(ζ 3 + 2α ei ∆t) ∈ [0, 1] and f1i+1/2 = f n 1i + f n 1i+1 2 - 1 2a x (ζf n 2i+1 -ζf n 2i ).
Using the rst equation of (6.26) and (6.37) a direct calculation shows that the condition (6.25) ensure the admissibility of the numerical solution. Also, it can be seen geometrically since the admissible set is a convex cone and α belongs to [0, 1].

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

Homogeneous case with electric eld

The second simplied model studied, is given by (6.2) which is homogeneous in space but considering an electric eld. In this section, the diculties encountered when using a relaxation-type method to include the source term -E ζ (f 0 -f 2 ) are highlighted. Following the same procedure as in the case without electric eld, a numerical scheme is proposed and the source term

-E ζ (f 0 -f 2 ) is taken into account.
The scheme presented, satises a well-balanced property and is asymptoticpreserving. The collision coecient α ei is a function of x and is then constant in the present case. However, the method proposed here, is able to handle the case where α ei depends on ζ. Without spatial dependence, the model (6.2) simplies into

   ∂ t f 0 + E∂ ζ f 1 = 0, ∂ t f 1 + E∂ ζ f 2 - E ζ (f 0 -f 2 ) = - 2α ei f 1 ζ 3 . (6.38)
Using the Hilbert expansions (6.8) as in the previous case, the following diusion equation is obtained

∂ t f 0 0 (t, ζ) -E∂ ζ Eζ 3 6α ei ∂ ζ f 0 0 (t, ζ) - Eζ 2 3α ei f 0 0 (t, ζ) = 0. (6.39)

Limit of the relaxation approach

Using the ideas introduced in [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF], one can think of deriving a relaxation scheme for system (6.38). Even if the approach is similar, the relaxation scheme involved would be signicantly dierent from the one proposed in [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF] since the source term

-E ζ (f 0 -f 2 )
should be added. To assess such an issue, we rst consider the colli-

sionless case    ∂ t f 0 + E∂ ζ f 1 = 0, ∂ t f 1 + E∂ ζ f 2 - E ζ (f 0 -f 2 ) = 0. (6.40) Setting ∂ ζ z(ζ) = 1/ζ, we propose the following relaxation model                  ∂ t f 0 + E∂ ζ φ -E(f 1 -φ)z (ζ) = 0, ∂ t φ + E∂ ζ f 0 -2Ef 0 z (ζ) = µ(f 1 -φ), ∂ t f 1 + E∂ ζ π -E(f 0 -π)z (ζ) = 0, ∂ t π + E∂ ζ f 1 -2Ef 1 z (ζ) = µ(f 2 -π), ∂ t z = 0, (6.41)
where φ and π are relaxation variables. In the case µ = 0, the previous system is hyperbolic, the eigenvalues are -E, 0, E and are associated with linearly degenerate elds. Hence, the Riemann problem can be solved.

Homogeneous case with electric eld

Eigenvalue Multiplicity Riemann Invariants Eigenvectors In order to be consistent with the notations [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF], we introduce

E 2 f 0 + φ, f 1 + π, z t (0, 0, 1, 1, 0), t (1, 1, 0, 0, 0) -E 2 -f 0 + φ, -f 1 + π, z t (0, 0, -1, 1, 0), t (-1, 1, 0, 0, 0) 0 1 f 1 ζ 2 , f 0 ζ 2 , ζ(π -f 0 /3), ζ(φ -f 1 /3) t (2f 0 , f 1 -φ, 2f 1 , f 0 -π, 1)
w = t (f 0 , φ, f 1 , π, z), U = t (f 0 , f 1 ), F(U) = t Ef 1 , Ef 2 (f 0 , f 1 ) , Lemme 6.5. Let w L,R be equilibrium constant states with φ L,R = f L,R 1 and π L,R = f L,R 2
. Dening the initial condition of (6.41) by w 0 (x) = w L if x < 0 and w 0 (x) = w R if x > 0 for µ = 0, the solution of (6.41) writes

w(x, t) =            w L if x/t < -E, w L * if -E < x/t < 0, w R * if 0 < x/t < E, w R if E < x/t, (6.42)
with

f L * ,R * 0 = 3(ζ L,R ) 2 4(2(ζ R ) 6 + 2(ζ L ) 6 + 5(ζ R ) 3 (ζ L ) 3 ) (-f R 2 -2f R 1 + 3f R 0 )(ζ R ) 4 +(-f L 2 + 2f L 1 + 3f L 0 )(ζ L ) 4 + (f L 2 + 4f L 1 + 3f L 0 )(ζ R ) 3 (ζ L ) + (f R 2 -4f R 1 + 3f R 0 )(ζ R )(ζ L ) 3 , f L * ,R * 1 = 3(ζ L,R ) 2 4(2(ζ R ) 6 + 2(ζ L ) 6 + 5(ζ R ) 3 (ζ L ) 3 ) (3f R 2 -2f R 1 -f R 0 )(ζ R ) 4 +(-3f L 2 -2f L 1 + f L 0 )(ζ L ) 4 + (-3f L 2 -4f L 1 -f L 0 )(ζ R ) 3 (ζ L ) + (3f R 2 -4f R 1 + f R 0 )(ζ R )(ζ L ) 3 , z L * ,R * = z L,R , φ L * = f L 0 + f L 1 -f L * 0 , φ R * = -f R 0 + f R 1 + f R * 0 , π L * = f L 1 + f L 2 -f L * 1 , π R * = -f R 1 + f R 2 + f R * 1 ,
and

U L * ,R * = t (f L * ,R * 0 , f L * ,R * 1
) satisfy the admissibility conditions (6.4).

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

The computation of the intermediate states U L * ,R * is straightforward using the Riemann invariants given in Table 6.1. A long but easy calculation, using the expressions gives the admissibility conditions (6.4).

The relaxation model (6.41) enables the computation of a numerical scheme [START_REF] Berthon | Numerical approximations of the 10-moment Gaussian closure[END_REF][START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for sources[END_REF][START_REF] Jin | The relaxation scheme for systems of conservation laws in arbitrary space dimension[END_REF] for the model (6.40). However, one notices the complexity of the intermediate states U L * ,R * and an extension including the collisional term -2α ei f 1 /ζ 3 is very challenging. Dierent relaxation models were tested in order to include the collisional source term, but, because of their complexity, they lead to congurations where a Riemann invariant is missing and the problem remains unclosed. In a recent work [START_REF] Desveaux | Well-balanced schemes to capture non-explicit steady states. Part 1: Ripa model[END_REF], the same issue is encountered and an additional relation is arbitrarily imposed.

In the present situation, this strategy leads to particularly inconvenient solutions and the admissibility conditions are lost.

Numerical method

The numerical approach presented in the case without electric eld is now considered. Contrarily to the relaxation-type procedure, this method enables to include the source term

-E ζ (f 0 -f 2 ) naturally.
Integrating the second equation of (

f * 1 = 2a ζ ζ 3 2a ζ ζ 3 + 2α ei ∆ζ f L 1 + f R 1 2 - 1 2a ζ (Ef R 2 -Ef L 2 ) + ∆ζ 2a ζ S L,R , (6.43)
with

S L,R = 1 2 E ζ R (f R 0 -f R 2 ) + E ζ L (f L 0 -f L 2 ) .
The unknowns f L * 0 , f R * 0 , f n+1 0 and f n+1 1 are computed following the same approach as in the rst part

       f n+1 0i = a ζ ∆t ∆ζ f R * 0i-1/2 + (1 - 2a ζ ∆t ∆ζ )f n 0i + a ζ ∆t ∆ζ f L * 0i+1/2 , f n+1 1i = a ζ ∆t ∆ζ f * 1i-1/2 + (1 - 2a ζ ∆t ∆ζ )f n 1i + a ζ ∆t ∆ζ f * 1i+1/2 , (6.44)
where the unknowns f R * 0 and f L * 0 are given by

f L * 0 = f0 -Γθ, f R * 0 = f0 + Γθ, (6.45)
with

Γ = 1 2 [f R 0 -f L 0 - ζ a ζ (f L 1 -2f * 1 + f R 1 )],
6.3. Homogeneous case with electric eld and

f0 = f L 0 + f R 0 2 - 1 2a ζ [ζf R 1 -ζf L 1 ].
Using, the same arguments as in the case without electric eld, we set a ζ = |E|.

Properties

In this part, we are interested in the equilibrium solution of system (6.38). It is shown that the scheme (6.43)-(6.44)-(6.45) preserves this solution. Then, the asymptotic-preserving feature of the scheme is exhibited.

A stationary solution of system (6.38) satises

       E ∂f 1 ∂ζ = 0, E ∂f 2 ∂ζ - E ζ (f 0 -f 2 ) = - 2α ei f 1 ζ 3 . (6.46)
The rst equation of (6.46) implies that f 1 is independent of ζ. Using the denitions of the angular moments (2.1) and the denition (6.3), it follows that f 1 = 0 and f 2 = f 0 /3. Indeed the denitions (2.1) imply f 1 = 0 in ζ = 0. The second equation of the previous system is solved and gives the equilibrium solution of the model (6.38)

f 0 = Kζ 2 , f 1 = 0, (6.47)
where K is a scalar constant.

Theorem 6.6. The numerical scheme given by (6.43)-(6.44)-(6.45) is well-balanced in the sense that the stationary states (6.47) are exactly preserved by the scheme.

Proof. Using the stationary states (6.47) into the denition (6.43) leads to

f * 1 = 2a ζ ζ 3 a ζ ζ 3 + 2α ei ∆ζ - 1 3a ζ (EKζ 2 R -EKζ 2 L ) + ∆ζEK 3a ζ (ζ R + ζ L ) . Since (ζ 2 R -ζ 2 L ) = (ζ R + ζ L )(ζ R -ζ L ) = (ζ R + ζ L )∆ζ, the calculation of the previous equation gives f * 1 = 0.
Using the second equation of (6.44) leads to

f n+1 1 = 0.
6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

With the denition (6.23) it follows that

     f R * 0 = 1 2 [f 0L -θf 0L + f 0R + θf 0R ], f L * 0 = 1 2 [f 0R -θf 0R + f 0L + θf 0L ]. (6.48)
The initial conditions (6.47) imply that θ = 1 and inserting (6.48) into the rst equation of (6.44) give

f n+1 0i = a∆t ∆ζ Kζ 2 i + (1 - 2a∆t ∆ζ )Kζ 2 i + a∆t ∆ζ Kζ 2 i .
Finally, the previous equation simplies to give

f n+1 0i = Kζ 2 i .
The stationary solution (6.47) is then preserved by the scheme.

Using the ideas introduced in the rst section, we obtain that the scheme (

-(6.45) is consistent with the limit diusion equation (6.39) in the diusive limit.

Theorem 6.7. When ε tends to zero, the unknown f n+1 0 given by the numerical scheme (6.43)-(6.44)-(6.45) satises the following discrete equation

f n+1,0 0i -f n,0 0i ∆t - E ∆ζ ζ 3 i+1/2 6σ∆ζ (Ef n,0 0i+1 -Ef n,0 0i ) - ζ 3 i-1/2 6σ∆ζ (Ef n,0 0i -Ef n,0 0i-1 ) + ζ 3 i+1/2 S n,0 i+1/2 2σ - ζ 3 i-1/2 S n,0 i-1/2 2σ = 0, with S n,0 i+1/2 = E 3 f n,0 0i+1 ζ i+1 + f n,0 0i ζ i .
Proof. The proof is the same as in the case without electric eld.

As in the inhomogeneous case without electric eld, in practice the following stability CFL condition is used

∆t ≤ max(∆ζ/a ζ , 3α ei ∆ζ 2 /E 2 ζ 3 max ). (6.49)
Similarly, using the ideas of the rst part we consider the following scheme to compute f n 1i at each time step

f n+1 1i = ζ 3 i ζ 3 i + 2α ei ∆t f n 1i -∆t E f n 2i+1 -f n 2i-1 2∆ζ -a ζ f n 1i+1 -2f n 1i + f n 1i-1 2∆ζ + S n i+1/2 + S n i-1/2 2 , (6.50)

Numerical examples

where

S n i+1/2 = 1 2 E ζ i+1 (f n 0i+1 -f n 2i+1 ) + E ζ i (f n 0i -f n 2i ) .
This scheme enables the use of the parabolic CFL condition (6.49) in the case of a large collisional parameter α ei . In addition, the well-balanced property is ensured since the stationary state (6.47) is still preserved by this scheme.

Numerical examples

In this section we compare the asymptotic-preserving scheme to the standard HLL scheme [START_REF] Harten | On upstream dierencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] and to an explicit discretisation of the diusion equation in dierent regimes. For all the numerical test cases the time step considered for the asymptotic-preserving scheme is taken as the maximum of the hyperbolic time step and the diusion time step (see CFL condition Eq. 6.35). The numerical scheme is able to work with the diusion time step when it becomes larger than the hyperbolic time step.

Free transport without electric eld

We rst consider system (6.5), without collisions, to validate the numerical scheme proposed in (6.20)-(6.23)-(6.26) on a simple advection of an initial prole.

The solution is compared with the exact solution. Consider the initial conditions

       f 0 (x, 0) = 2 π exp(- (x + 5) 2 2 ), f 1 (x, 0) = 2 π exp(- (x + 5) 2 2 ),
with periodical boundary conditions. In this case we have xed ζ = 5. In Figure 6.2, we compare the numerical solution obtained with the scheme (6.20)-(6.23)-(6.26) displayed in dashed blue with the exact solution in red at time t=6 using ∆x = 4 • 10 -3 . In Table 6.2 the results of a convergence study are given. The scheme is rst order accurate.

Temperature gradient with collisions without electric eld

We now consider the system equation (6.5) with collisions to validate the numerical scheme (6.20)-(6.23)-(6.26) taking into account the collisional part. The solution obtained with the scheme presented in this paper is compared with the classical HLL scheme.

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit ∆x

L 1 error L 1 order L 2 error L 2 order L ∞ error L ∞ order
     f 0 (x, ζ, 0) = 2 π ζ 2 T ini (x) 3/2 exp - ζ 2 2T ini (x) , f 1 (x, ζ, 0) = 0, with T ini (x) = 2 -arctan(x),
and α ei = 1. On the right and left boundaries, we use a Neumann boundary condition: the values of f 0 and f 1 in the boundary ghost cells are set to the values in the corresponding real boundary cells. The energy range chosen is [0, 12] with an energy step ∆ζ = 0.1 and the space range is [-40, 40] with a space step ∆x = 0.2.

In Figure 6.3, we compare the numerical solution obtained with the AP scheme (6.20)-(6.23)- (6.26). The solution obtained with the Asymptotic-preserving scheme is displayed in continuous lines with the solution given by HLL scheme in dashed

Temperature gradient in the diusive regime without electric eld

In this numerical test, the same initial and boundary conditions that in the test case 3.2 are chosen. However, we consider a large collisional parameter and take α ei = 10 4 . The scheme (6.20)-(6.23)-(6.26) is veried in the diusive regime. The results are compared with the diusion solution and with the one obtained with the HLL scheme.

In Figure 6.4, the numerical solution obtained with the scheme (6.20)-(6.23)-(6.26) is displayed. The results obtained with the asymptotic-preserving scheme are displayed in continuous green lines with the solution given by HLL scheme in continuous purple lines and the diusion solution in dashed blue lines at time t=50, t=100, 500 and 1000. The AP numerical scheme and the diusion solution match perfectly while we remark for time t = 50 and t = 100 that the HLL scheme gives very inaccurate results. The results obtained with the HLL scheme at time t = 500 and t = 1000 are completely wrong and are not displayed, however we notice that in the long time regime the AP numerical scheme and the diusion solution still match.

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit Consider the initial conditions

         f 0 (x, ζ, 0) =      1 if x ≤ L/3, 0 if L/3 ≤ x ≤ 2L/3, 1 if L/3 ≤ x, f 1 (x, ζ, 0) = 0,
with periodical boundary conditions and α ei = 10 4 . In Figure 6.5, we compare the numerical solution obtained with the Asymptotic-preserving scheme displayed in red with the diusion solution in dashed blue and the HLL scheme in green at time t=200. The AP and diusion solutions match perfectly while the HLL scheme is very inaccurate. In Figure 6.6, the long time behaviour of the numerical solutions is considered. The AP scheme and the diusion solution are compared at time t=500, the results match.

-10 -5 0 5 10

x 0 0, 2 0,4 0,6 0,8 1 f 0
Initial condition HLL AP diffusion Figure 6.5: Comparison of the f 0 prole for the asymptotic-preserving scheme (AP), for the HLL scheme (HLL) and the diusion solution at time t=200.

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

   f 0 (ζ, 0) = 2 π exp(- ζ 2 2 ), f 1 (ζ, 0) = 0.
In Figure 6.7, we compare the numerical solution obtained with the scheme (6.43)-(6.44)-(6.45) displayed in red with the diusion solution in dashed blue and the HLL scheme at time t=20. The asymptotic-preserving and diusion solutions match perfectly while the HLL scheme is very diusive. In Figure 6.8, the results obtained with the AP scheme and the diusion solution are compared in the long time regime at time t=80.

   f 0 (x, ζ, 0) = ζ 2 exp(- x 2
2 ),

f 1 (x, ζ, 0) = 0.
In Figure 6.9, we compare the numerical solution obtained with the asymptoticpreserving scheme displayed in red with the diusion solution in dashed blue at time t=150. In this case, the asymptotic-preserving and diusion solutions also match perfectly. The HLL scheme results are not given in Figure 6.9, since the nal time t=150 is important the HLL results are completely wrong.

-4 -2 0 2 4

x 0 0,2

Initial condition AP diffusion

Figure 6.9: Relaxation of a Gaussian prole in the case of a linear collisional parameter: comparison of the f 0 prole for the asymptotic-preserving scheme (AP) and the diusion solution at time t=150.

General model and diusive limit

In the rst part of this work, a numerical scheme was proposed for the electron M 1 model in a particular case without electric eld and in the homogeneous case.

The scheme derived using the consistency with the integral form of the approximate Riemann solver ensures the admissibility conditions (6.4) and correctly captures the limit diusion equation. The method proposed naturally takes into account the source term -E(x)(f 0 -f 2 )/ζ, the non linearity of the model which comes from the M 1 model closure and the spatial dependencies of the electric eld and the collisional parameter. However, the general model considering the x and ζ dependences has not 6.6. Numerical scheme been considered. In such a general case, mixed derivatives arise in the diusion limit leading to complex diusion equation. In addition, the source term -E(x)(f 0 -f 2 )/ζ also contributes in the limit equation. In this part, the general electronic M 1 model (6.2) is considered. The aim is to propose a numerical scheme, extending the ideas of the rst part, in order to take into account the mixed derivatives in the diusive limit. Such a scheme must ensure the admissibility conditions (6.4) and include the contribution of the source term in the diusion -E(x)(f 0 -f 2 )/ζ limit.

After using the diusive scaling (6.6), the general model (6.2) writes

         ε∂ t f 0 (t, x, ζ) + ζ∂ x f 1 (t, x, ζ) + E(x)∂ ζ f 1 (t, x, ζ) = 0, ε∂ t f 1 (t, x, ζ) + ζ∂ x f 2 (t, x, ζ) + E(x)∂ ζ f 2 (t, x, ζ) - E(x) ζ (f 0 (t, x, ζ) -f 2 (t, x, ζ)) = - 2σ(x) ζ 3 f 1 (t, x , ζ) ε .
(6.51)

Inserting the Hilbert expansion (6.8) into the second equation of (6.51) gives at

order ε 0 f 1 1 = - ζ 4 6σ ∂ x f 0 0 - Eζ 3 6σ ∂ ζ f 0 0 + Eζ 2 3σ f 0 0 . (6.52)
Finally, using the previous equation in the rst equation of (6.2) at order ε 1 , the following limit equation is obtained

∂ t f 0 0 +ζ∂ x - ζ 4 6σ ∂ x f 0 0 - Eζ 3 6σ ∂ ζ f 0 0 + Eζ 2 3σ f 0 0 +E∂ ζ - ζ 4 6σ ∂ x f 0 0 - Eζ 3 6σ ∂ ζ f 0 0 + Eζ 2 3σ f 0 0 = 0. (6.53)
In the case E = 0, one recognises a classical diusion equation involving a second order space derivative with a diusion coecient of -ζ 5 /6σ. However, in the general case this limit equation involves mixed x and ζ derivatives leading to a non isotropic diusion. In addition, the source term E(f 0 -f 2 )/ζ also contributes in the diusive limit adding the term (Eζ 2 /(3σ))f 0 0 in the right side of (6.52) and in the x and ζ derivatives of (6.53). Such an asymptotic limit is unusual compared to what has been studied in radiative transfer for example [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF][START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF]. The dierence lies in the fact that here charged particles are considered. Then, the contribution of the electric eld must be taken into account leading to these unexpected limit involving mixed derivatives.

Numerical scheme

The aim of this part is to propose a numerical scheme, generalising the ideas introduced in the rst part, for the general model (6.2) and consistent, in the limit ε tends to zero, with equation (6.53). The main diculty comes from the derivation of a numerical scheme consistent in the diusive limit with equation (6.53) and in particular with the mixed-derivatives. The numerical scheme proposed must also be able to deal with the contribution of the source term E(f 0 -f 2 )/ζ.

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit 6.6.1 Case without the source term

E ζ (f 0 -f 2 )
We rst consider the case without the source term

E ζ (f 0 -f 2 )
. With the present approach, it will be seen in part 6.6.1 that this term can be naturally taken into account, therefore, for clarity, we start without considering it. The electronic M 1 model then reads

   ∂ t f 0 (t, x, ζ) + ζ∂ x f 1 (t, x, ζ) + E(x)∂ ζ f 1 (t, x, ζ) = 0, ∂ t f 1 (t, x, ζ) + ζ∂ x f 2 (t, x, ζ) + E(x)∂ ζ f 2 (t, x, ζ) = - 2σ(x)f 1 (t, x, ζ) ζ 3 , (6.54)
and its diusive limit equation

∂ t f 0 0 + ζ∂ x - ζ 4 6σ ∂ x f 0 0 - Eζ 3 6σ ∂ ζ f 0 0 + E∂ ζ - ζ 4 6σ ∂ x f 0 0 - Eζ 3 6σ ∂ ζ f 0 0 = 0. (6.55)

Derivation of the scheme

In this part the derivation of an numerical scheme for the model (6.54) is detailed.

Let us consider an uniform mesh with a constant space step ∆x = x i+1/2 -x i-1/2 , a constant energy step ∆ζ = ζ i+1/2 -ζ i-1/2 and a time step ∆t. Extending the ideas introduced in the rst part, we propose to consider the following numerical scheme

U n+1 ij -U n ij ∆t = a x ∆x U R * i-1/2j + 2a x ∆x U n ij + a x ∆x U L * i+1/2j (6.56) + a ζ ∆ζ U R * ij-1/2 + 2a ζ ∆ζ U n ij + a ζ ∆ζ U L * ij+1/2 ,
where the intermediate states of the approximated Riemann solver (see Figure 6.10)

U L * i+1/2j , U R * i-1/2j , U L * ij+1/2 and U R * ij-1/2 are dened by U R * i-1/2j = f R * 0i-1/2j f * 1i-1/2j , U L * i+1/2j = f L * 0i+1/2j f * 1i+1/2j , U R * ij-1/2 = f R * 0ij-1/2 f * 1ij-1/2j , U L * ij+1/2 = f L * 0ij+1/2 f * 1ij+1/2
.

The second components of the intermediate states at each interface are chosen equal,

ie f L * 1i+1/2j = f R * 1i+1/2j = f * 1i+1/2j and f L * 1ij+1/2 = f R * 1ij+1/2 = f * 1ij+1/2 .
Following [START_REF] Berthon | An asymptotic preserving relaxation scheme for a moment model of radiative transfer[END_REF], the velocity waves a x and a ζ are xed such that

a x = ζ j , a ζ = |E i |.
For clarity, in the following, we omit the dependency of the speed a x in energy and a ζ in space. However, the results presented hold in the general case. If the intermediate states are dened following the rst part the numerical scheme (6.56) recovers only the second order space and energy derivatives in the diusive limit. Therefore, in 6.6. Numerical scheme order to take into account the mixed-derivative terms in the diusive limit leading to an anisotropic diusion, we propose to modify the numerical viscosity of the intermediate state f * 1 used in equation (6.20) in the following way

-a x a x t x U R U L U * R U * L
f * 1i+1/2j = α i+1/2j f 1i+1j + f 1ij 2 - 1 2a x (ζ j f 2i+1j -ζ j f 2ij )-c i+1/2j ( ∂f 0 ∂ζ ) i+1/2j (1-α i+1/2j ) , (6.57)
f * 1ij+1/2 = β ij+1/2 f 1ij+1 + f 1ij 2 - 1 2a ζ (E i f 2ij+1 -E i f 2ij )-c ij+1/2 ( ∂f 0 ∂x ) ij+1/2 (1-β ij+1/2 ) . (6.58) with α i+1/2j = 2a x ζ 3 j 2a x ζ 3 j + σ i+1/2 ∆x , β ij+1/2 = 2a ζ ζ 3 j+1/2 2a ζ ζ 3 j+1/2 + σ i ∆ζ . ( 6.59)
In this case, the numerical viscosity contributes in the x and ζ directions. The terms ( ∂f 0 ∂ζ ) i+1/2j , ( ∂f 0 ∂x ) ij+1/2 and the coecients c and c are xed in order to obtained the relevant limit equation (6.53) in the diusion regime. We set

c i+1/2j = E i+1/2 ∆x 3a x , cij+1/2 = ζ j+1/2 ∆ζ 3a ζ . (6.60)
We use an upwind scheme for the discretisation of the terms ( ∂f 0 ∂ζ ) i+1/2j and ( ∂f 0 ∂x ) ij+1/2 . The coecient c is always positive then

cij+1/2 ( ∂f 0 ∂x ) ij+1/2 ≈ cij+1/2 f 0i+1j+1 -f 0ij+1 + f 0i+1j -f 0ij 2∆x , c i+1/2j ( ∂f 0 ∂ζ ) i+1/2j ≈        c i+1/2j f 0i+1j -f 0i+1j-1 + f 0ij -f 0ij-1 2∆ζ if c i+1/2j < 0, c i+1/2j f 0i+1j+1 -f 0i+1j + f 0ij+1 -f 0ij 2∆ζ if c i+1/2j > 0.
6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

The previous two conditions rewrite

c i+1/2j ( ∂f 0 ∂ζ ) i+1/2j ≈ c - i+1/2j f 0i+1j -f 0i+1j-1 + f 0ij -f 0ij-1 2∆ζ + c + i+1/2j f 0i+1j+1 -f 0i+1j + f 0ij+1 -f 0ij 2∆ζ ,
with (c) + = max(c, 0) and (c) -= min(c, 0). We introduce the following notations

f0i+1/2j = f 1i+1j + f 1ij 2 - (ζ j f 2i+1j -ζ j f 2ij ) 2a x (2 -α i+1/2j ) , f1i+1/2j = f 1ij+1 + f 1ij 2 - (E i f 2ij+1 -E i f 2ij ) 2a ζ (2 -β ij+1/2
) .

(6.61)

In the rst part of this work, the intermediate states of the considered approximate Riemann solvers were dened using consistency relations and a corrective coecient to ensure the admissibility conditions. Extending these ideas, the intermediates states f R * 0i+1/2j and f L * 0i+1/2j are dened by

f L * 0i+1/2j = f0i+1/2j -Γ i+1/2j θ 1i+1/2j , f R * 0i+1/2j = f0i+1/2j + Γ i+1/2j θ 1i+1/2j , (6.62)
with

Γ i+1/2j = 1 2 [f 0i+1j -f 0ij - ζ j a x (f 1ij -2f * 1i+1/2j + f 1i+1j )],
and the coecient θ 1i+1/2j is xed in order to ensure the admissibility conditions (6.4). Similarly, the denitions of f R * 0ij+1/2 and f L *

0ij-1/2 read f L * 0ij+1/2 = f0ij+1/2 -Γ ij+1/2 θ 2ij+1/2 , f R * 0ij+1/2j = f0ij+1/2 + Γ ij+1/2 θ 2ij+1/2 , (6.63)
with

Γ ij+1/2 = 1 2 [f 0ij+1 -f 0ij - ζ j a ζ (f 1ij -2f * 1ij+1/2 + f 1ij+1 )].
In order to ensure the admissibility conditions (6.4), the denitions of the intermediate states f * 1i+1/2j and f * 1ij+1/2 given in (6.57) and (6.58) are modied such that

f * 1i+1/2j = α i+1/2j f1i+1/2j -θ 1i+1/2j c i+1/2j ( ∂f 0 ∂ζ ) i+1/2j (1 -α i+1/2j ) , (6.64) f * 1ij+1/2 = β ij+1/2 f1ij+1/2 -θ 2ij+1/2 cij+1/2 ( ∂f 0 ∂x ) ij+1/2 (1 -β ij+1/2
) .

(6.65) Remark 6.8. In the case θ 1i+1/2j = 0 and θ 2ij+1/2 = 0, the admissibility requirements (6.4) are fullled. Proof. The numerical scheme (6.56) also writes as a convex combination of vectors of A

U n+1 ij =(1 - 2a x ∆t ∆x - 2a ζ ∆t ∆ζ )U n ij + a x ∆t ∆x U R * i-1/2j + a x ∆t ∆x U L * i+1/2j + a ζ ∆t ∆ζ U R * ij-1/2 + a ζ ∆t ∆ζ U L * ij+1/2 ,
Using the denitions of θ 1 and θ 2 given in (6.66) and (6.67) the intermediate states

U R * i-1/2j , U L * i+1/2j , U R * ij-1/2 and U L * ij+1/2 belong to A.
Since A is a convex space it follows that the updated states U n+1 i belongs to A.

Asymptotic-preserving properties

In this part, the consistency in the classical regime and the asymptotic-preserving property of the scheme in the diusive regime are exhibited. Theorem 6.10. (Consistency in the classical regime) The numerical scheme (6.56) is consistent, when ∆t and ∆x tend to zero, with the set of equation (6.54).

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit Proof. Using the denitions (6.57) and (6.58), the second component of (6.56) reads

f n+1 1ij -f n 1ij ∆t = a x ∆x α i+1/2j f1i+1/2j -θ 1i+1/2j c i+1/2j ( ∂f 0 ∂ζ ) i+1/2j (1 -α i+1/2j ) - 2a x ∆x f n 1ij + a x ∆x α i-1/2j f1i-1/2j -θ 1i-1/2j c i-1/2j ( ∂f 0 ∂ζ ) i-1/2j (1 -α i-1/2j ) (6.68) + a ζ ∆ζ β ij+1/2 f1ij+1/2 -θ 2ij+1/2 cij+1/2 ( ∂f 0 ∂x ) ij+1/2 (1 -β ij+1/2 ) - 2a ζ ∆ζ f n 1ij + a ζ ∆ζ β ij-1/2 f1ij-1/2 -θ 2ij-1/2 cij-1/2 ( ∂f 0 ∂x ) ij-1/2 (1 -β ij-1/2 ) .
Inserting the denitions (6.61) into (6.68) and using the following expressions for α i+1/2j and β ij+1/2

α i+1/2j = 2a x ζ 3 j 2a x ζ 3 j + σ i+1/2 ∆x = 1 - σ i+1/2 ∆x 2a x ζ 3 j + σ i+1/2 ∆x
, and

β ij+1/2 = 2a ζ ζ 3 j+1/2 2a ζ ζ 3 j+1/2 + σ i ∆ζ = 1 - σ i ∆ζ 2a ζ ζ 3 j+1/2 + σ i ∆ζ
, lead to the consistency with the second equation of (6.54) as ∆x and ∆t tend to zero.

A similar calculation gives the consistency with the rst equation of (6.54).

Theorem 6.11. (Consistency in the diusive regime)

In the diusive limit, the numerical scheme (6.56) degenerates into

f n+1,0 0ij -f n,0 0ij ∆t = ζ j ∆x ζ 4 j 6σ i+1/2 ∆x (f n,0 0i+1j -f n,0 0ij ) - ζ 4 j 6σ i-1/2 ∆x (f n,0 0i1j -f n,0 0i-1j ) + ζ 3 j E i+1/2 6σ i+1/2 ( ∂f n,0 0 ∂ζ ) i+1/2j - ζ 3 j E i-1/2 6σ i-1/2 ( ∂f n,0 0 ∂ζ ) i-1/2j + E i ∆ζ E i ζ 3 j+1/2 6σ i ∆ζ (f n,0 0ij+1 -f n,0 0ij ) - E i ζ 3 j-1/2 6σ i ∆ζ (f n,0 0i1j -f n,0 0ij-1 ) + ζ 4 j+1/2 6σ i ( ∂f n,0 0 ∂x ) ij+1/2 - ζ 4 j-1/2 6σ i ( ∂f n,0 0 ∂x ) ij-1/2 .
(6.69) 6.6. Numerical scheme Proof. Following the same approach as in [START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF][START_REF] Berthon | Asymptotic preserving HLL schemes[END_REF][START_REF] Guisset | Asymptotic-preserving well-balanced scheme for the electronic M 1 model in the diusive limit: particular cases[END_REF], using the diusive scaling and equation (6.56) leads to

ε U n+1,ε ij -U n,ε ij ∆t = a x ∆x U R * ,ε i-1/2j - 2a x ∆x U n,ε ij + a x ∆x U L * ,ε i+1/2j (6.70) + a ζ ∆ζ U R * ,ε ij-1/2 - 2a ζ ∆ζ U n,ε ij + a ζ ∆ζ U L * ,ε ij+1/2 ,
and equations (6.64) and (6.65) gives

f * ,ε 1i+1/2j = α ε i+1/2j f ε 1i+1/2j -θ 1i+1/2j c i+1/2j ( ∂f ε 0 ∂ζ ) i+1/2j (1 -α ε i+1/2j ) , f * ,ε 1ij+1/2 = β ε ij+1/2 f ε 1ij+1/2 -θ 2ij+1/2 cij+1/2 ( ∂f ε 0 ∂x ) ij+1/2 (1 -β ε ij+1/2 ) . (6.71) with α ε i+1/2j = 2a x ζ 3 j 2a x ζ 3 j + σ i+1/2 ∆x/ε , β ε ij+1/2 = 2a ζ ζ 3 j+1/2 2a ζ ζ 3 j+1/2 + σ i ∆ζ/ε . (6.72)
Then it follows that f * ,0 1i+1/2j = 0 and f * ,0 1ij+1/2 = 0.

(

The second component of (6.70) reads

ε f n+1,ε 1ij -f n,ε 1ij ∆t = a x ∆x f * ,ε 1i-1/2j - 2a x ∆x f n,ε 1ij + a x ∆x f * ,ε 1i+1/2j + a ζ ∆ζ f * ,ε 1ij-1/2 - 2a ζ ∆ζ f n,ε 1ij + a ζ ∆ζ f * ,ε 1ij+1/2 .
At order ε 0 the previous equation leads to f n,0 1ij = 0.

(6.74)

In the limit ε tends to zero, the results (6.73) and (6.74) give θ 1i+1/2j = 1, θ 2ij+1/2 = 1.

(6.75)

Indeed, when ε tends to zero, the denitions (6.66) and (6.67) lead to

θ1i+1/2j = f n,0 0i+1j + f n,0 0ij |f n,0 0i+1j -f n,0 0ij | ≥ 1, θ2ij+1/2 = f n,0 0ij+1 + f n,0 0ij |f n,0 0ij+1 -f n,0 0ij | ≥ 1.
The rst component of (6.70) reads

ε f n+1,ε 0ij -f n,ε 0ij ∆t = a x ∆x f R * ,ε 0i-1/2j - 2a x ∆x f n,ε 0ij + a x ∆x f L * ,ε 0i+1/2j + a ζ ∆ζ f R * ,ε 0ij-1/2 - 2a ζ ∆ζ f n,ε 0ij + a ζ ∆ζ f L * ,ε 0ij+1/2 .
Using the denitions (6.62) and (6.63), the result (6.75) and the previous equation considered at order ε 1 gives the numerical scheme (6.69).

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

f * 1ij+1/2 = β ij+1/2 f1ij+1/2 + ∆ζ 2a ζ S ij+1/2 -θ 2ij+1/2 cij+1/2 ( ∂f 0 ∂x ) ij+1/2 (1 -β ij+1/2 ) , with Si+1/2j = ζ 2 j 3σ i f 0i+1j + f 0ij 2 and S ij+1/2 = E i 2 f 0ij+1 -f 2ij+1 ζ j+1 + f 0ij -f 2ij ζ j .
In this case, as in the previous part the coecients θ 1 and θ 2 are also xed to ensure the admissibility requirements.

Theorem 6.12. In the diusive limit, the numerical scheme given by (6.56)-(6.62)-(6.63)-(6.76) is consistent with the limit equation (6.53).

Proof. The proof is the same than for Theorem 3, considering the intermediate states f * 1i+1/2j and f * 1ij+1/2 given in (6.76). A direct calculation using the Hilbert expansions leads to the result. The terms S ij+1/2 are consistent with the term E ζ (f 0 -f 2 ) while the terms Si+1/2j enable to correctly recover the contribution of the two terms Eζ 2 3σ f 0 in the x and ζ derivatives of the limit equation.

Numerical examples

In this section, the asymptotic-preserving scheme (6.56) is compared with the HLL scheme and an explicit discretisation of the diusion equation (6.53) in the diusive regime.

Relaxation of a Gaussian prole in the diusive regime

In this example, the numerical scheme (6.56)-(6.62)-(6.63)-(6.76) is validated in the diusive regime considering a inhomogeneous plasma with electric eld. In this case, the initial conditions are the following

f 0 (t = 0, x, ζ) = ζ 2 exp(-x 2 ) exp(2(ζ -3) 2 ), f 1 (t = 0, x, ζ) = 0.

Numerical examples

The prole of f 0 at initial time as a function of x and ζ is displayed in Figure 6.11. For this test we have set E = 1, α ei = 10 4 , the space range chosen is [-10,10] and the energy range [0,6]. In Figure 6.12, the solution obtained with the numerical scheme (6.56)-(6.62)-(6.63)-(6.76) is compared with the solution obtained with the HLL scheme and with an explicit discretisation of the limit diusion equation (6.53) at dierent times. At time t = 1, one remark that the f 0 prole obtained with the HLL scheme is already seriously spread out while the proles obtained with the AP scheme and the diusion equation do not have changed. At time t = 50, the AP scheme and diusion equation discretisation f 0 proles are spread out while the prole obtained with the HLL scheme has vanished. As observed at time t=100, in the long time regime, the AP scheme and the discretisation of the diusion equation behave identically.

     f 0 (t = 0, x, ζ) = 2 π ζ 2 T ini (x) 3/2 exp(- ζ 2 2T ini (x)
),

f 1 = 0, (6.77)
with T ini (x) = 2 -arctan(x).

In this case the electric eld is self-consistent meaning that at each time step it is calculated from the plasma prole. In this case we consider a Spitzer type model 6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

, t=100 (bottom), for the HLL scheme (left), AP scheme (middle) and the diusion equation. [START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF][START_REF] Struchtrup | Macroscopic Transport Equations for Rareed Gas Flows[END_REF], to evaluate the electric eld In Figure 6.13, the temperature prole is displayed at the initial time and at time t=80. The temperature proles obtained with the HLL scheme, the AP scheme 6.7. Numerical examples and a discretisation of the diusion equation (6.53) are compared at time t=80.

E(x) = - dT (x) dx , (6.78
On one hand, one remark that the HLL temperature prole is excessively spread out compared to the AP and diusion proles while on the other hand the AP and diusion proles match exactly at time t=80. This example demonstrates the inability of the HLL scheme in capturing the correct temperature prole while the AP scheme presented handle perfectly the diusive limit regime. In this example the interaction between two electron beams is considered. This collisionless test case enables us to validate the AP scheme (6.56)-(6.62)-(6.63)- (6.76) in a regime where electrostatic eects are predominant compared to the collisional eects, therefore we set α ei = 0. In the case of two streams propagating with opposite velocities v d and -v d , the initial electron distribution function is the following

f (t = 0, x, v) = 0.5[(1 + A cos(kx))M v d (v) + (1 -A cos(kx))M -v d (v)], with M ±v d (v) = exp - (v ∓ v d ) 2 2 .
The rst corresponding angular moments f 1 0 and f 2 0 of the rst and second popula- tion read

       f 1 0 (t = 0, x, ζ) = 0.5(1 + A cos(kx)) ζ v d exp(- (ζ -v d ) 2 2 ) -exp(- (ζ + v d ) 2 2 ) , f 2 0 (t = 0, x, ζ) = 0.5(1 -A cos(kx)) ζ v d exp(- (ζ -v d ) 2 2 ) -exp(- (ζ + v d ) 2 2
) .

6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit

The second angular moments f 1 1 and f 2 1 of the rst and second population read

       f 1 1 (t = 0, x, ζ) = 0.5(1 + A cos(kx)) 1 -ζv d v 2 d exp(- (ζ -v d ) 2 2 ) -exp(- (ζ + v d ) 2 2 ) , f 2 1 (t = 0, x, ζ) = -0.5(1 -A cos(kx)) 1 -ζv d v 2 d exp(- (ζ -v d ) 2 2 ) -exp(- (ζ + v d ) 2 2
) .

At each time step, the electrostatic eld is computed using the Maxwell-Ampere equation considering the contribution of the two population of particles

dE dt = +∞ 0 f 1 1 ζdζ + +∞ 0 f 2 1 ζdζ.
The parameter A is introduced to perturb the initial condition in order to enable the development of the electrostatic instability. The energy range chosen is [0,12] and the space range is [0,25]. In this example we set v d = 4, A = 0.001 and periodical boundary conditions are used. The results have been compared with a kinetic code [START_REF] Duclous | High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications[END_REF]. In Figure 6.14, the evolution of the electrostatic energy is displayed as a function of time using the AP scheme in red and the kinetic code in dashed blue. The AP scheme and the kinetic code give analogous results. This numerical experiment

shows the good behaviour of the AP scheme in a regime where electrostatic eects are predominant. 6.7.4 Relaxation of a temperature prole in the diusive regime with a self-consistent electric eld and non-constant collisional parameter

In this example, the initial conditions are the same than for the previous example where the initial temperature prole is given by (6.77) and the electric eld is

Case variable self-consistent collisional parameter

In the congurations occurring in plasma physics, the collisional parameter depends of the state of the plasma. The knowledge of the ion and electron distribution function is required to compute the collisional parameter. Therefore in this test case, 6. Asymptotic-preserving scheme for the electronic M 1 model in the diusive limit we choose to consider a nonlinear collisional parameter which depends of the solution itself

α ei (t, x, ζ) = exp(f 0 (t, x, ζ) + f 1 (t, x, ζ)).
In this case, E = 1, the space range chosen is [-10,10] and the energy range [0,[START_REF] Aregba-Driollet | Asymptotic High-Order Schemes for 2x2 Dissipative Hyperbolic Systems[END_REF].

The initial condition is given by

f 0 (t = 0, x, ζ) = ζ 2 exp(-(ζ -3) 2 )) exp(-x 2 /10), f 1 = 0.
We consider periodical boundary conditions. In Figure 6.16, the initial prole of f 0 is displayed at the initial time and at time t=3.

Conclusion

In the rst part of this work, a numerical scheme has been proposed for the electron M 1 model in the case without electric eld and in the homogeneous case with electric eld. We have exhibited an approximate Riemann solver that satises the admissibility conditions. Contrarily to the HLL scheme, the proposed numerical scheme is asymptotic-preserving and recovers the correct diusion equation in the diusive limit. It has been shown, in the homogeneous case, that the method presented, enables to include the source term -E(f 0 -f 2 )/ζ, while a relaxation type method seems inconvenient. In addition, the scheme is well-balanced, capturing the steady state considered. Several numerical tests have been performed, it has been shown that the presented scheme behaves correctly in the classical regime and in the diusive limit. Indeed, while, the HLL scheme is very inaccurate in the diusive regime, the asymptotic-preserving scheme matches perfectly with the expected 6.8. Conclusion diusion solution. Also, the method correctly handles the case where the collisional parameter is not constant.

In the second part of this work, the approach has been extended to the general electronic M 1 model (6.2). In order to deal with the mixed derivatives which arise in the diusive limit an anisotropic numerical viscosity has been considered. The numerical scheme preserves the realisibility domain and captures the correct limit equation. The contribution of the source term E(f 0 -f 2 )/ζ is integrated and the cases of non constant electric eld and collisional parameter are naturally included.

Numerical examples have been performed in non-collisional and diusive regimes.

It has been observed that the present scheme behaves correctly in both regimes. A possible perspective could be to consider an electron-electron collisional operator or the study of the coupling with the Maxwell's equations.

First step towards multi-species modelling: the angular M 1 model in a moving frame Chapter 7

Angular M 1 model in a moving reference frame

Introduction

This chapter is a rst step towards the modelling and simulation of the multispecies particles transport. Previously, the electron transport studies were performed considering immobile ions [START_REF] Duclous | Modélisation et Simulation Numérique multi-échelle du transport cinétique électronique[END_REF][START_REF] Mallet | Contribution à la modélisation et à la simulation numérique multi-échelle du transport cinétique électronique dans un plasma chaud[END_REF]. Indeed, because of their large mass compared to electrons, the ion motion could be neglected when considering small time intervals. However, if long time studies are considered, the ions motion needs to be taken into account. This would give access to a general and interesting physics relevant to the inertial connement fusion applications. A signicant work is then required in order to consider angular moments models for the multi-species particles transport studies. The long time perspective is to be able to work with charged particles of dierent species such as electrons and ions. In order to simplify the form of the electron-ion collisional operator and in order to ensure the galilean invariance property of angular moments models we will work in the framework of ions. This is signicantly dierent with the previous studies, since the M 1 model [START_REF] Dubroca | Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif[END_REF][START_REF] Guisset | Classical transport theory for the collisional electronic M1 model[END_REF][START_REF] Guisset | Limits of the M1 and M2 angular moments models for kinetic plasma physics studies[END_REF] has always been considered in the framework of immobile ions.

However, before considering complex congurations dealing with charged particles interactions, in the present chapter, we only consider the case on species of non-charged particles and work in the frame of the mean velocity. Here, the velocity framework is centred on the particle mean velocity. Therefore, this study can be seen as a rst step towards multi-species modelling since the problem investigated here contains some of the main diculties encountered in the case of multi-species charged particles. Also, even if the long time perspective is the study of multi-species charged particles, the present approach is already relevant when considering neutral gas dynamics applications. Indeed, at our knowledge it is the rst time that the M 1 angular moments model is used for the rareed gas dynamic.

In order to derive the M 1 angular moments model in the mean velocity frame, a velocity change is considered to derived the kinetic equation in a moving frame.

Derivation of the model

Note, that rescaled velocity approaches are largely used in dierent context see [START_REF] Bobylev | On some properties of kinetic and hydrodynamic equations for inelastic interactions[END_REF][START_REF] Filbet | A Rescaling Velocity Method for Kinetic Equations: the Homogeneous Case[END_REF][START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior[END_REF] for example. However, the numerical treatment of the additional terms which appear when considering such a procedure on the kinetic equation can be challenging. In [START_REF] Filbet | A Rescaling Velocity Method for Dissipative Kinetic Equations -Applications to Granular Media[END_REF], in the context of granular ows, a numerical algorithm based on a relative energy scaling is proposed. Then, a clever de-coupling with the hydrodynamics equation is used to avoid the problems related to the change of scales in velocity variables. In this work the velocity framework is chosen centred on the mean velocity of the particles and enables the reduction of the velocity modulus grid. The Galilean invariance property for maximum entropy moments systems is discussed in details in [START_REF] Junk | Maximum entropy moment systems and Galilean invariance[END_REF]. It is shown that the choice of non-polynomial weight function leads to moments systems incompatible with the Galilean invariance property. In the present case, angular moments models are investigated, it is pointed out that working in the mean velocity enables to ensure the Galilean invariance property of the model.

The plan of this chapter is the following. First of all, the derivation of the angular M 1 moments model in the mean velocity frame is introduced. The choice of the mean velocity framework in order to enforce the Galilean invariance property of the model is highlighted. In addition, it is shown that the model rewritten in terms of the entropic variables is Friedrichs-symmetric. Also, the derivation of the associated conservation laws and the zero mean velocity condition are detailed.

Secondly, a suitable numerical scheme, preserving the realisability requirement of the numerical solution for the angular M 1 moments model in the mean velocity frame is proposed. In order to enforce the correct discrete energy conservation and the zero mean velocity condition, a correction of the numerical solution is also presented. Thirdly, some numerical results obtained considering several test cases in dierent collisional regimes are displayed. Finally, some conclusion and perspectives are given.

Derivation of the model

Velocity change of variables procedures are used in various contexts (see [START_REF] Bobylev | On some properties of kinetic and hydrodynamic equations for inelastic interactions[END_REF][START_REF] Filbet | A Rescaling Velocity Method for Dissipative Kinetic Equations -Applications to Granular Media[END_REF][START_REF] Filbet | A Rescaling Velocity Method for Kinetic Equations: the Homogeneous Case[END_REF][START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior[END_REF] for example) and can enable the simplication of a collisional operator form or the reduction of the velocity grid size used for numerical applications. In the context of angular moments models [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF], it will be seen in the next section that moving frame formulations play an important role in enforcing the Galilean invariance property. In order to explain in details this point, in this section we introduce the kinetic formulation in a moving frame from which the M 1 angular moments model studied is derived.

where f represents the particle distribution function and α = (t, x, v) ∈ R + t ×R 3

x ×R 3

v .

The form of the collisional operator C is not detailed here but only the properties used in this study will be detailed.

We consider the vector

F (α) ∈ R + t × R 3 x × R 3 v dened as F (α) =   f (α) vf (α) 0   ,
then the kinetic equation (7.1) rewrites under the following form

div α (F (α)) = C(f (α)). Introducing φ ∈ C ∞ c (R + t × R 3 x × R 3 v ; R)
, we consider the associated weak formulation

α∈R + t ×R 3 x ×R 3 v div α (F (α)) -C(f (α)) φ(α)dα = 0, which rewrites α∈R + t ×R 3 x ×R 3 v ∂ α φ(α)F (α) + C(f (α))φ(α)dα = 0. (7.2)
In order to derive the kinetic equation in a moving velocity frame the following set of coordinates is considered

t β = (τ, y, c) = (t, x, v -u(t, x)),
where u is a relative velocity which depends of time and space. In particular, we dene the C 1 -dieomorphism Φ as β = Φ(α).

We remark here that only the velocity coordinates are transformed while time and space coordinates are kept unchanged. Eq. (7.2) considered in the new set of variables rewrites

β∈R + t ×R 3 y ×R 3 c ∂ α φ(Φ -1 (β))F (Φ -1 (β)) + C(f (Φ -1 (β)))φ(Φ -1 (β)) | det J Φ -1 |dβ = 0, (7.3)

Derivation of the model

where det J Φ -1 is the determinant of the Jacobian matrix of the transformation. For this transformation det J Φ -1 = 1, indeed the Jacobian matrix J Φ reads

J Φ =           1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 -∂ t u 1 -∂ x 1 u 1 -∂ x 2 u 1 -∂ x 3 u 1 1 0 0 -∂ t u 2 -∂ x 1 u 2 -∂ x 2 u 2 -∂ x 3 u 2 0 1 0 -∂ t u 3 -∂ x 1 u 3 -∂ x 2 u 3 -∂ x 3 u 3 0 0 1           .
The following quantities expressed in the mobile framework are introduced

G(β) = F (Φ -1 (β)), g(β) = f (Φ -1 (β)) and Ψ(β) = φ(Φ -1 (β)).
In order to express ∂ α φ(Φ -1 (β)), one remarks that

∂ β Ψ(β) = ∂ α φ(Φ -1 (β))∂ β (Φ -1 )(β).
Then by derivation of a reciprocal function

∂ β Φ -1 (β) = (∂ α Φ(Φ -1 (β))) -1 , it follows that ∂ α φ(Φ -1 (β)) = ∂ β Ψ(β)(∂ α Φ(Φ -1 (β))).
Using the two previous equations, equation (7.3) rewrites

β∈R + t ×R 3 y ×R 3 c ∂ β Ψ(β)∂ α Φ(Φ -1 (β))G(β)) + C(g(β))Ψ(β)dβ. Finally for all Ψ(β) ∈ C ∞ c (R + τ × R 3 y × R 3 c ; R) β∈R + t ×R 3 y ×R 3 c div β (∂ α Φ(Φ -1 (β))G(β)) -C(g(β)) Ψ(β)dβ = 0, it follows that div β (∂ β Φ(Φ -1 (β))G(β)) = C(g(β)).
In the case of the present change of variables

∂ β Φ(Φ -1 (β))G(β) =    g(β) vg(β) -( ∂u ∂t + ∂u ∂x v)g(β)    .
Finally, using the fact that v = c + u(x, t), one obtains the kinetic equation in a moving velocity frame

∂ τ g(β) + div y ((c + u)g(β)) -div c (∂ τ u + ∂ y u(c + u))g(β) = C(g)(β).
7. Angular M 1 model in a moving reference frame Since time and space are unchanged by the change of variable

∂ t g(t, x, c) + div x ((c + u)g(t, x, c)) -div c (∂ t u + ∂ x u(c + u))g(t, x, c) (7.4)
= C(g(t, x, c)).

This equation is used in the next sections to derive the angular M 1 moments model in a moving reference frame. Of course, an additional evolution equation is required to compute the velocity u. In this work, the velocity u is chosen as the particles mean velocity in the xed frame (laboratory frame). In order to derive the evolution equation for u, the kinetic equation (7.1) is integrated in velocity. This leads to the following conservation laws ∂n ∂t + div x (nu) = 0,

(7.5) ∂(nu) ∂t + div x ( v f v ⊗ vdv) = 0.
Injecting the following expansion into (7.5)

v ⊗ v = (v -u) ⊗ (v -u) + (v -u) ⊗ u + u ⊗ (v -u) + u ⊗ u,
and by using the following identities

v u ⊗ uf dv = nu ⊗ u, v u ⊗ (v -u)f dv = u ⊗ v (v -u)f dv = 0, v (v -u) ⊗ uf dv = v (v -u)f dv ⊗ u = 0,
one obtains the evolution equation for u expressed in the new frame quantities

∂(nu) ∂t + div x (nu ⊗ u) + div x ( v g(c)c ⊗ cdc) = 0, (7.6)
where n = c g(c)dc.

M 1 angular moments model in a moving frame

The M 1 angular moments model in a moving frame is derived by performing an angular moments extraction of the kinetic equation (7.4). One denes the following three rst angular moments of the distribution function g

g 0 (ζ) = ζ 2 S 2 g(Ω, ζ)dΩ, g 1 (ζ) = ζ 2 S 2 g(Ω, ζ)ΩdΩ, g 2 (ζ) = ζ 2 S 2 g(Ω, ζ)Ω⊗ΩdΩ,
               ∂ t g 0 + div x (ζg 1 + ug 0 ) -∂ ζ du dt .g 1 + ζ∂ x u : g 2 = 0, ∂ t g 1 + div x (ζg 2 + u ⊗ g 1 ) -∂ ζ g 2 du dt + ζg 3 ∂u ∂x + g 0 Id -g 2 ζ du dt + ( ∂u ∂x g 1 -g 3 ∂u ∂x = 0, (7.7)
where du dt is dened as

du dt = ∂u ∂t + ∂u ∂x u,
and the third order moments g 3 as

g 3 (ζ) = ζ 2 S 2 g(Ω, ζ)Ω ⊗ Ω ⊗ ΩdΩ. (7.8)
The evolution law (7.6), expressed in terms of the angular moments rewrites

∂(nu) ∂t + div x (nu ⊗ u) + div x ( +∞ 0 g 2 (ζ)ζ 2 dζ) = 0. (7.9)
As explained in Chapter 2, one needs to close the set (7.7) by expressing the higher order moments g 2 and g 3 as function of g 0 and g 1 . Using the M 1 minimisation problem introduced in Chapter 2, we recall that the distribution function from which the angular moments are derived writes

g(t, x, ζ, Ω) = exp(a 0 (t, x, ζ) + a 1 (t, x, ζ).Ω), (7.10)
where a 0 is a scalar function and a 1 a vector valued function. Then extending the ideas of [START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF][START_REF] Dubroca | Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif[END_REF][START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF] one can show that the closure relation for g 2 is given by

g 2 = g 0 3χ(α) -1 2 g 1 |g 1 | ⊗ g 1 |g 1 | + 1 -χ(α) 2 Id , (7.11)
where

χ(α) = 1 + |α| 2 + |α| 4 3 , α = g 1 /g 0 .
(7.12)

Similarly the higher order moment g 3 reads

g 3 = 3|g 1 | -χ 2 g 0 2 g 1 |g 1 | ⊗ g 1 |g 1 | ⊗ g 1 |g 1 | + χ 2 g 0 -|g 1 | 2 g 1 |g 1 | ∨ Id , (7.13)
7. Angular M 1 model in a moving reference frame with χ 2 (α) = 3|α| -|α| 3 + 3|α| 5 5 ,

and

g 1 |g 1 | ∨ Id = g 1 |g 1 | ⊗ e 1 ⊗ e 1 + e 1 ⊗ g 1 |g 1 | ⊗ e 1 + e 1 ⊗ e 1 ⊗ g 1 |g 1 | + g 1 |g 1 | ⊗ e 2 ⊗ e 2 + e 2 ⊗ g 1 |g 1 | ⊗ e 2 + e 2 ⊗ e 2 ⊗ g 1 |g 1 | + g 1 |g 1 | ⊗ e 3 ⊗ e 3 + e 3 ⊗ g 1 |g 1 | ⊗ e 3 + e 3 ⊗ e 3 ⊗ g 1 |g 1 | .
Before studying the models properties, the realisability conditions associated to the model (7.7) are introduced

A = (g 0 , g 1 ) ∈ R 2 , g 0 ≥ 0, |g 1 | ≤ g 0 . (7.14)
Since the distribution function g is a nonnegative quantity the realisability conditions (7.14) naturally needs to be satised. In addition, these conditions are related to the existence of a nonnegative distribution function from which the angular moments can be derived [START_REF] Pomraning | Maximum entropy Eddington factors and ux limited diffusion theory[END_REF].

Model properties

In this section the main properties of the angular M 1 model in a moving frame (7.7-7.9) are presented. It is rst proved that the choice of working in the mean velocity frame enables to ensure the Galilean invariance property of the model.

Secondly it is shown that this model, rewritten in terms of entropic variables, is Friedrichs-symmetric. Finally, the derivation of the conservation laws is detailed.

Galilean invariance property

Galilean invariance is a fundamental feature of the Boltzmann equation. Following [START_REF] Junk | Maximum entropy moment systems and Galilean invariance[END_REF], we start dening translational and rotational transformations. For any vector s ∈ R d and any rotation matrix R ∈ SO(d)

(T s f )(v) = f (v -s), (T R f )(v) = f (Rv), v ∈ R d .
The following translational and rotational invariance properties of the collisional operator C are considered

T s C(f ) = C(T s (f )), T R C(f ) = C(T R (f )). ( 7
f (t, x, ṽ) = f (t, x, v).
Consequently the following relations can be derived

∂ t f (t, x, v) = ∂ t f (t, x, ṽ) -s.∂ x f (t, x, ṽ), ∂ x f (t, x, v) = R∂ x f (t,
x, ṽ).

Therefore using (7.1), it follows that f satises

∂ t f (t, x, c) + div x(ṽ f (t, x, c)) = C( f (t, x, c)), (7.17)
which shows the Galilean invariance of (7.1).

The same property cannot be directly obtained when considering angular moments models. Indeed, when integrating (7.1) on the unit sphere and applying the change of variables (7.16) on the resulting M 1 angular moments model gives nonlinear terms and one observes that the form of the M 1 model is not invariant. In order to overcome this drawback, in this study, we propose to not derive the M 1 angular moments model from the kinetic equation (7.1) but from the kinetic equation (7.4) which is expressed in a mobile reference frame. In particular, in this work the velocity u used in (7.4) is chosen as the particles mean velocity dened by

u = 1 n R 3 f (v)vdv. (7.18)
In order to show the advantage in deriving the M 1 angular moments model from the kinetic equation (7.4), the kinetic equation (7.17) is rewritten in its mean velocity frame. This second kinetic equation expressed in a mobile frame reads

∂ t g(t, x, c) + div x((c + ũ)g(t, x, c)) -div c (∂ t ũ + ∂ x ũ(c + ũ))g(t, x, x) (7.19) = C(g(t, x, c)), where ũ = 1 n R 3 f (
ṽ)ṽdṽ.

(7.20)

The key point which will be useful when considering angular moments models is the relation between the two kinetic equations (7.4) and (7.19). Indeed, the two relative velocities u and ũ are linked through the following relation ũ = Ru -s.

∂ t u = t R(∂ t ũ -(∂ x ũ)s), ∂ x u = t R(∂ x ũ)R, (7.23)
and

∂ t g = ∂ t g -(∂ x g)s, ∂ x g = (∂ x g)R, ∂ c g = (∂ c g)R, (7.24)
a direct calculation enables to recover equation (7.19). The relationships between the dierent studied framework are summarised on Fig 7 .1. The starting point is the kinetic equation (7.1) expressed in the xed frame, denoted A 0 . Since this kinetic equation is Galilean invariant, one obtains (7.17) denoted B 0 , by using (7.16).

Secondly, the kinetic equation in a mobile frame (7.4) denoted A has been derived.

In the present case u is the particles mean velocity dened in (7.18). The same procedure can be applied on (7.17 The change of variables (7.22) makes the link between equations (7.4) and (7.19) x = Rx -st ṽ = Rv -s

x = Rx -st c = Rc ũ = Ru -s A 0 : xed frame ∂ t f + div x (vf ) = C(f ) A : Mobile frame ∂ t g + div x ((c + u)g) -div c (∂ t u + ∂ x u(c + u))g = C(g) B 0 : Uniform translation frame ∂ t f + div x(ṽ f ) = C( f ) B : Uniform translation mobile frame ∂ t g + div x((c + ũ)g) -div c (∂ t ũ + ∂ x ũ(c + ũ))g = C(g) c = v -u(t, x) nu(t, x) = v f vdv c = ṽ -ũ(t, x)
nũ(t, x) = ṽ f ṽdṽ In order to deal with the second equation of (7.7), we remark that using (7.25) the ijk th component of the higher order moments g 3 dened in (7.8) rewrites

g 3ijk = l,m,n ζ2 S2 t R il t R jm t R kn ( Ω ⊗ Ω ⊗ Ω) lmn d Ω. ( 7
       ∂ t g1 + div x( ζ g2 + ũ ⊗ g1 ) -∂ ζ g2 dũ dt + ζ g3 ∂ ũ ∂ x + g0 Id -g2 ζ dũ dt + ( ∂ ũ ∂ x g1 -g3 ∂ ũ ∂ x = 0.

Symmetrization property

In this section it is shown that the M 1 model in a moving frame (7.7-7.9) written in terms of the entropic variables is Friedrichs symmetric. Following [START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF], the M 1 model in a moving frame (7.7) can be rewritten in terms of the entropic variables a 0 and a 1 . This procedure is sometimes called a Godunov's symmetrisation [START_REF] Goudon | Analysis of the M1 model: well-posedness and diusion asymptotics[END_REF].

Theorem 7.2. The M 1 model in a moving frame (7.7-7.9) written in terms of the variables a 0 and a 1 is Friedrichs symmetric.

Model properties

where the notation . refers to the angular integration on the unit sphere. Consequently, after a direct calculation, the M 1 angular moments model in a moving frame (7.7) rewrites

A 0 (α)∂ t α 0 α 1 + j A j (α)∂ x j α 0 α 1 + B(α)∂ ζ α 0 α 1 + S(x, ζ, α) = 0 0 , (7.32)
where

A 0 (α) =< exp(α.m) 1 t Ω Ω Ω ⊗ Ω >, A j (α) =< (ζΩ j + u j ) exp(α.m) 1 t Ω Ω Ω ⊗ Ω >, B(α) =< -(ζ 2 du dt .Ω + ζ 3 ∂u ∂x : Ω ⊗ Ω) exp(α.m) 1 t Ω Ω Ω ⊗ Ω >, and S(x, ζ, α) =    (div x u)g 0 - 2 ζ du dt .g 1 -3 ∂u ∂x : g 2 ∂u ∂x g 1 - 2 ζ g 2 du dt -3g 3 ∂u ∂x + g 0 Id -g 2 ζ du dt + ( ∂u ∂x g 1 -g 3 ∂u ∂x )    .
Since A 0 (α) is a positive-denite symmetric matrix and A j (α) and B(α) are symmetric matrices, one obtains that the system (7.32) is Friedrichs-symmetric [START_REF] Benzonie-Gavage | Multi-dimensonal Hyperbolic Partial Differential Equations[END_REF][START_REF] Friedrichs | Systems of Conservation Equations with a Convex Extension[END_REF].

Conservation laws

In this section the derivation of the conservation laws derived from the angular M 1 model in a moving frame (7.7) is detailed.

Before deriving the mass and energy conservation equations, we point out that in this work the velocity u is chosen as the particles mean velocity. Therefore, in the considered framework the mean velocity is equal to zero. This point is expressed by the following condition +∞ 0 g 1 (t, x, ζ)ζdζ = 0. Green's formula that all the terms vanish two by two and that condition (7.33) is preserved over times.

The derivation of the mass conservation equation can be directly obtained by direct integration in ζ. Indeed, integrating the rst equation of (7.7) in ζ, one obtains

∂ t n + div x (nu) = 0, (7.34)
where condition (7.33) has been used.

In order to derive the energy conservation equation, one starts multiplying the rst

+ (∂ x u : +∞ 0 g 2 ζ 2 dζ) = 0.
One notices that since the mean velocity frame is considered, only an equation on the internal energy is obtained. The kinetic energy equation is derived from the evolution equation (7.9) and writes

∂ t (nu 2 ) + div x ( nu 2 2 u) + u.div x ( +∞ 0 g 2 ζ 2 dζ) = 0. (7.36)
The energy conservation equation is directly obtained by summing equation (7.35) with equation (7.36).

Numerical scheme

In this part an appropriate numerical scheme is proposed for the M 1 model in a moving framework in an one dimensional spatial geometry considering a standard BGK collision operator [START_REF] Gross | A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems[END_REF]. In this case, the collisional operator C(f ) used in (7.1) is specied

C(f ) = 1 τ (M f -f ), with M f (v) = n (2πT ) 3/2 exp(- (v -u) 2 2T ),
and τ is a collisional parameter which is xed depending of the collisional regime studied. In this case the M 1 model in a moving framework (7.7) writes

               ∂ t g 0 + ∂ x (ζg 1 + ug 0 ) -∂ ζ du dt g 1 + ζ ∂u ∂x g 2 = 1 τ (M g 0 -g 0 ), ∂ t g 1 + ∂ x (ζg 2 + ug 1 ) -∂ ζ du dt g 2 + ζ ∂u ∂x g 3 + du dt g 0 -g 2 ζ + ∂u ∂x (g 1 -g 3 ) = - 1 τ g 1 , (7.37)
where

M g 0 = 4πζ 2 n (2πT ) 3/2 exp(- ζ 2 2T
).

7.4. Numerical scheme

Derivation of the numerical scheme

In order to derive a suitable numerical scheme for the model (7.37) which preserves the admissibility of the solution, the dierent terms of (7.37) are studied separately. Then the admissibility requirement of the complete scheme is shown under a reduced CFL condition.

Step 1: the rst intermediate state is the following

∂ t g 0 + ∂ x (ζg 1 + ug 0 ) = 0, ∂ t g 1 + ∂ x (ζg 2 + ug 1 ) = 0. (7.38)
In order to derive a numerical scheme preserving the realisability of the numerical solution, we consider an underlying kinetic model from which the system (7.38) can be derived by direct angular moments extraction

∂ t F (t, x) + ∂ x (a(x)F (t, x)) = 0, (7.39)
with

F = ζ 2 f , a(x) = ζµ + u(x) and µ ∈ [-1, 1]
. Note that µ is the angular variable in the case of one space dimension.

A natural conservative numerical scheme is proposed for the kinetic equation (7.39)

F n+1 i -F n i ∆t + h n i+1/2 -h n i-1/2 ∆x = 0, (7.40)
with

h n i+1/2 = a - i+1/2 F n i+1 + a + i+1/2 F n i ,
and a ± = 1 2 (a ± |a|).

Rewriting equation (7.40) as a convex combination

F n+1 i = F n i 1 - ∆t 2∆x (a i+1/2 -a i-1/2 ) -∆t |a i+1/2 | + |a i-1/2 | 2∆x + F n i+1 2∆t ∆x |a i+1/2 | -a i+1/2 (7.41) + F n i-1 2∆t ∆x |a i-1/2 | -a i-1/2 ,
it follows that the positivity of the numerical distribution function is ensured under the following CFL condition

∆t 1 ≤ ∆x 2||u|| ∞ + ζ . (7.42)
7. Angular M 1 model in a moving reference frame

The numerical scheme (7.41) rewrites on the following viscous form

F n+1 i -F n i ∆t + a i+1/2 F n i+1 + (a i+1/2 -a i-1/2 )F n i -a i-1/2 F n i-1 2∆x (7.43) - |a i+1/2 |F n i+1 -(|a i+1/2 | + |a i-1/2 |)F n i + |a i-1/2 |F n i-1 2∆x = 0.
The angular integration can not be directly performed on the scheme (7.43) because of the angular variable µ which appears in the term |a| in the numerical viscosity.

Therefore we modify (7.43) and consider the following scheme which is suitable for the angular integration.

F n+1 i -F n i ∆t + a i+1/2 F n i+1 + (a i+1/2 -a i-1/2 )F n i -a i-1/2 F n i-1 2∆x (7.44) -||a|| ∞ F n i+1 -2F n i + F n i-1 2∆x = 0.
Remark 7.3. Considering (7.44), one observes that the numerical viscosity of the scheme is increased in order to enable the angular integration.Therefore the numerical scheme still preserves the nonnegativity of the numerical solution under CFL condition (7.42).

The angular integration of the scheme (7.44)

+ (ζg n 1i+1 + u i+1/2 g n 0i+1 ) + ((ζg n 1i + u i+1/2 g n 0i ) -(ζg n 1i + u i-1/2 g n 0i )) -(ζg n 1i + u i-1/2 g n 0i-1 ) 2∆x -(ζ + ||u|| ∞ ) g n 0i+1 -2g n 0i + g n 0i-1 2∆x = 0, g n+1 1i -g n 1i ∆t (7.46) + (ζg n 2i+1 + u i+1/2 g n 1i+1 ) + ((ζg n 2i + u i+1/2 g n 1i ) -(ζg n 2i + u i-1/2 g n 1i )) -(ζg n 2i + u i-1/2 g n 1i-1 ) 2∆x -(ζ + ||u|| ∞ ) g n 1i+1 -2g n 1i + g n 1i-1 2∆x = 0. Remark 7.4. Computing g n+1 0 + g n+1 1 and g n+1 0 -g n+1 1
, one can show the scheme (7. 45-7.46) preserves the realisability requirement of the numerical solution under the CFL condition (7.42).

Step 2: the second intermediate step we consider writes

     ∂ t g 0 -∂ ζ ( du dt g 1 + ζ∂ x ug 2 ) = 0, ∂ t g 1 -∂ ζ (g 2 du dt + ζg 3 ∂ x u) = 0. ( 7
∂ t F (ζ) -∂ ζ (( du dt µ + ζ∂ x uµ 2 )F (ζ)) = 0,
with the following corresponding scheme

F n+1 j -F n j ∆t + b j+1/2 F n j+1 + (b j+1/2 -b j-1/2 )F n j -b j-1/2 F n j-1 2∆ζ (7.48) -||b|| ∞ F n j+1 -2F n j + F n j-1 2∆ζ = 0, with b = du dt µ + ζ∂ x uµ 2 . The CFL condition associated reads ∆t 2 ≤ ∆ζ 2(|| du dt || ∞ + ζ||∂ x u|| ∞ ) . (7.49)
The angular integration of (7.48) leads to the following discretisation for the intermediate state (7.47)

g n+1 0j -g n 0j ∆t (7.50) + ( du dt g n 1i+1 -ζ j+1/2 ∂ x ug n 2j+1 ) + (( du dt g n 1j -ζ j+1/2 ∂ x ug n 2j ) -( du dt g n 1j -ζ j-1/2 ∂ x ug n 2j )) 2∆ζ - ( du dt g n 1j-1 -ζ j-1/2 ∂ x ug n 2j-1 ) 2∆ζ -(| du dt | + ||ζ|| ∞ |∂ x u|) g n 0j+1 -2g n 0j + g n 0j-1 2∆ζ = 0, g n+1 1j -g n 1j ∆t (7.51) + ( du dt g n 2j+1 -ζ j+1/2 ∂ x ug n 3j+1 ) + (( du dt g n 2j -ζ j+1/2 ∂ x ug n 3j ) -( du dt g n 2j -ζ j-1/2 ∂ x ug n 3j )) 2∆ζ - ( du dt g n 2j-1 -ζ j-1/2 ∂ x ug n 3j-1 ) 2∆ζ -(| du dt | + ||ζ|| ∞ |∂ x u|) g n 1j+1 -2g n 1j + g n 1j-1 2∆ζ = 0.
Remark 7.5. The scheme (7.50-7.51) preserves the realisability domain under the CFL condition (7.49).

Step 3: the third state we consider is the following

   ∂ t g 0 = 0, ∂ t g 1 + g 0 -g 2 ζ du dt = 0.

Angular M 1 model in a moving reference frame

We choose the following classical scheme for this rst model

   g n+1 0ij = g n 0ij , g n+1 1ij = g n 1ij -∆t g 0ij -g 2ij ζ j ( du dt ) i .
Remark 7.6. This scheme preserves the realisability conditions under CFL conditions

∆t 3 ≤ ζ | du dt | 1 + α 1 -χ(α) ,
where α is dened by (7.12).

Proof. This result is directly obtained by computing g n+1 0i ± g n+1 1i .

Remark 7.7. One remarks that the term

1 + α 1 -χ(α)
does not tend to zero as α tends to -1. Indeed, using the denition of χ given in (7.12), one can show that

1 + α 1 -χ(α)
tends to 1/2 as α tends to -1.

Step 4: the fourth intermediate step we consider writes

∂ t g 0 = 0, ∂ t g 1 + ∂ x u(g 1 -g 3 ) = 0.
Following the third step we propose

g n+1 0i = g n 0i , g n+1 1i = g n 1i + ∆t(∂ x u) i (g 1i -g 3i ).
Remark 7.8. This scheme preserves the realisability conditions under CFL condi-

tions ∆t 4 ≤ 1 |∂ x u| 1 + α α -χ 2 (α)
.

Using the denition of χ 2 , we remark that 1 + α α -χ 2 (α) tends to -1/2 as α tends to -1.

In order to derive a admissible numerical scheme for the complete model (7.37), we propose to consider the following time semi-discretisation

U n+1 = U n + ∆t N k=1 F k (U n ), (7.52)
where

U n+1 = g n+1 0 g n+1 1 , 7
U n+1 = N k=1 1 N [U n + (N ∆t)F k (U n )]. (7.53)
Setting ∆t = N ∆t, one shows that if each intermediate step

Ũ n+1 = U n + ∆tF k (U n ),
preserves the realisability conditions of the numerical solution under CFL condition ∆t ≤ C k .

Therefore the general scheme (7.52) preserves the realisability conditions of the numerical solution under the following CFL condition

∆t ≤ min k ( C k N ).
The following result is then obtained Theorem 7.9. The general scheme (7.52) preserves the realisability conditions under the following CFL condition ∆t ≤ 1 4 min(∆t 1 , ∆t 2 , ∆t 3 , ∆t 4 ). Proof. Each step preserves the realisability conditions under CFL condition. Therefore, by convexity of the admissible set, considering the convex combination (7.53) and using the condition (7.4.1), we directly obtain that the general scheme (7.52) preserves the realisability conditions under the CFL condition (7.54).

The discretisation of the collision operator is performed by using a standard implicit scheme. For the numerical test presented in the next section, an usual second order Van Leer's slope limiter [START_REF] Van Leer | Towards the ultimate conservative dierence scheme III. Upstream-centered nite-dierence schemes for ideal compressible ow[END_REF] is used.

Enforcement of the discrete energy conservation and zero mean velocity condition

In this section, the enforcement of the discrete energy conservation and zero mean velocity condition is discussed. In a recent work [START_REF] Rey | An Exact Rescaling Velocity Method for some Kinetic Flocking Models[END_REF], a numerical scheme has been proposed to enforce the discrete zero mean velocity condition considering a kinetic equation. However, this strategy does not directly apply in the present case [START_REF] Audit | A radiation hydrodynamics scheme valid from the transport to the diusion limit[END_REF]. Angular M 1 model in a moving reference frame since a nonlinear set of equations (7.37) is considered associated to the realisability conditions (7.14). The enforcement of the discrete energy conservation and the zero mean velocity condition while preserving realisability conditions (7.14) of the numerical solution is particularly challenging and beyond the scope of the present study. However, in order to be able to present numerical results, in this section a correction of the numerical solution is proposed.

In order to enforce the correct energy conservation, we start considering the following conservation laws associated to (7.37)

     ∂ t ρ + div x (ρu) = 0, ∂ t (ρu) + div x (ρu ⊗ u + p -s) = 0, ∂ t E + div x ((E + p -s)u + q) = 0, (7.55)
where E is the total energy. The pressure tensor p, the stress tensor s and the heat ux q expressed in terms of the angular moments read

p -s = m 2 +∞ 0 g 2 ζ 2 dζ, q = m 2 +∞ 0 g 1 ζ 3 dζ.
At each time time step, the set of conservation laws (7.55) is numerically solved.

Then the numerical solution is corrected by using g 0p = α exp(βζ p 2 )ḡ 0p , ∀p ∈ {1; ...; pf }, where g 0 is the corrected solution and ḡ0 the solution which requires a correction computed with the scheme (7.52). The coecients α and β are numerically computed such that m pf p=1

g 0p ∆ζ = ρ, m 2 pf p=1 g 0p ζ 2 p ∆ζ = E - ρu 2 2 ,
where the quantities E, ρu 2 2 and ρ are known at each time step since the set (7.55) has been numerically solved. This procedure enables the enforcement of the correct energy conservation. As it will be shown in the next section this correction is important for the numerical results, in particular in order to numerically capture shock waves.

In order to enforce the zero mean velocity condition (7.33) at the discrete level, one could think in proposing an adapted discretisation for the source terms which appears in the second equation of (7.37). However, this procedure leads to an unsuitable CFL condition when considering the realisability requirements (7.14) for the numerical solution. Therefore the following correction is proposed based on the resolution of the convex optimisation problem min where g 1 is the corrected solution and ḡ1 the solution before correction given by the scheme (7.52). One observes that this procedure does not enforce the realisable conditions of the numerical solutions. In such unfortunate case, g 1 is simply projected on the realisable set.

g 1 ∈R pf 1 2 ||g 1 -ḡ1 || 2 L 2 = 0,

Numerical results

In this section, several test cases are presented. Depending on the regime considered, the numerical results obtained with the scheme introduced in the previous part for the angular M 1 moments model in a moving frame, denoted M 1 mobile, are compared either with an exact solution or with a kinetic reference solution. The results are given with and without the correction procedure. In the following, the kinetic solution has been obtained considering a standard kinetic 1D3V BGK model using an usual Lax-Friedrichs scheme with the second order Van Leer's slope limiter [START_REF] Van Leer | Towards the ultimate conservative dierence scheme III. Upstream-centered nite-dierence schemes for ideal compressible ow[END_REF]. The results obtained with this scheme are denoted BGK 1D3V. In addition, the results obtained considering a second order HLL scheme for the Euler equations using the second order Van Leer's slope limiter are also given. These results obtained using this scheme are denoted Euler.

Test 1: Temperature gradient test case in dierent collisional regimes.

The rst test case we study consists in considering a strong temperature gradient at initial time and studying the temporal evolution of density, velocity and temperature. The initial distribution function is supposed to be a Maxwellian distribution function dened by

f ini (x, v) = n ini (x) (2πT ini (x)) 3/2 exp - (v -u ini (x)) 2
2T ini (x) , with n ini (x) = 1, u ini (x) = 0, T ini (x) = 2 -arctan(x).

The space range chosen is [-40, 40], and the velocity range [-15, 15] 3 . For the present test case, 400 cells in space and 200 3 cells in velocity have been considered for the 1D3V BGK kinetic approach. Also, 400 cells in space and 200 cells in velocity modulus have been considered for the M 1 mobile scheme. The rst regime we consider is the uid regime, then the collisional parameter τ is set equal to zero. In Figure 7.2, the density, velocity and temperature proles are displayed at time t = 10 for the kinetic BGK 1D3V scheme in continuous blue, the M 1 mobile scheme in dashed green, the M 1 mobile scheme with correction in dashed blue and for the Euler scheme in dashed-point pink. It is observed that all the schemes converge towards the same solution. This behaviour is expected since working in uid regime the distribution remains a Maxwellian distribution function and the three descriptions give the same solution.

1.b Rareed regime.

The second regime we consider is a rareed regime where the collisional parameter τ is set equal to 1. In Figure 7.3, the density, velocity, temperature and heat ux proles are displayed at time t = 10 for the kinetic BGK 1D3V scheme in continuous blue, the M 1 mobile scheme in dashed green, the M 1 mobile scheme with correction in dashed blue and for the Euler scheme in dashed-point pink. The Euler scheme gives the same results than in the previous case 1a. This is expected since the description is not able to see the dierent regimes. In this case the heat ux is equal to zero.

One observes that M 1 mobile scheme gives close results to the ones obtained with the kinetic BGK 1D3V scheme. When looking at the heat ux proles, one observes 7.5. Numerical results that the general trends are qualitatively similar with some notable dierences in the amplitude reached. Since the heat ux is a high order velocity moment, the dierences between the models are particularly visible. The M 1 model is accurate in collisional regimes, however as seen in Chapter 1 or in [START_REF] Guisset | Limits of the M1 and M2 angular moments models for kinetic plasma physics studies[END_REF], it can be inaccurate in collisionless regimes. The dierences observed here, are due to the inaccuracy to the M 1 model in rareed regime.

1.c Non-homogeneous collisional parameter.

When considering realistic physical applications, the collisional parameter varies according to the gas conditions. Therefore, in the third case we consider the collisional parameter τ is variable in space and is dened by τ (x) = 1 2 (arctan(1 + 0.1x) + arctan(1 -0.1x)).

In Figure 7.4, the density, velocity, temperature and heat ux proles are displayed at time t = 10 for the kinetic BGK 1D3V scheme in continuous blue, the M 1 mobile scheme in dashed green and for the Euler scheme in dashed-point pink. It is observed that the proles obtained using the M 1 mobile sheme and the BGK 1D3V scheme are very close. One also observes that even the heat ux proles are very similar. These results show the interest in using an angular moment model. exact solution is known, a rarefaction wave, a contact discontinuity and a shock wave appear. In Figure 7.5, the mass density, velocity and temperature solution proles are displayed at time t = 7.34 • 10 -2 . It is observed that the rarefaction wave (left side) is correctly captured by the M 1 mobile scheme (solution displayed in dashed green). However, one remarks that the shock amplitude is not correctly captured. It has been observed that this incorrect behaviour is due to the wrong discrete energy conservation. Indeed, by using the correction procedure introduced in the previous part the results in dashed blue are obtained, in this case the correct amplitude is recovered. We notice, the importance of the correct discrete energy conservation for capturing shock waves. This point is highlighted in the next test case.

Test 3: Double shock wave test case

The third test case we study is the double shock wave test case in uid regime.

The initial distribution function is supposed to be a Maxwellian distribution function dened by

f ini (x, v) = n ini (x) (2πT ini (x)) 3/2 exp - (v -u ini (x)) 2
2T ini (x) ,

Conclusion

In this work, the M 1 angular moments model in a moving frame has been derived.

Several fundamental properties of the model have been presented. In particular, the importance of working in the mean velocity frame has been highlighted. Indeed, this choice of framework is relevant when considering the Galilean invariance property of angular moments models. The derivation of the associated conservation laws has been detailed in addition to the zero mean velocity condition. A numerical scheme preserving the realisable sets has been proposed and validated with numerical test cases in dierent collisional regimes. Also, the importance of the correct discrete energy conservation has been emphasized.

As a short term perspective, one needs to derive a numerical scheme enforcing the discrete energy conservation and the zero mean velocity condition. Such an issue is challenging since it should be done preserving the realisable property of the numerical solution. As long term perspective, it would be interesting to study the electronic particle transport working in the ion mean velocity framework. This choice would enable a great simplication of the electron-ion collisional operator and an important step toward the multispecies particle transport for plasma physics applications.

particle species is required. In addition, the study of the transport of electrons and ions would give access to a large variety of physical phenomena.

Our contribution

When investigating the validity limits of angular moments models, it has been shown

that the M 1 model can be very inaccurate in collisionless regimes. Also, it has been seen that the M 2 angular moments model is much more accurate but remains unadapted when dealing with complex phenomena such as the laser-plasma absorption for example. However, the numerical experiments presented in this manuscript allow us to demonstrate the interest of using angular moments models in collisional regimes in order to capture kinetic eects.

In general it is then dicult to clearly dene a validity domain for the use of angular moments models. It is observed that the results depend on the physical phenomenon and the collisionnality level considered.

In [START_REF] Mallet | An entropic scheme for an angular moment model for the classical Fokker-Planck-Landeau equation of electrons[END_REF], collisional operators have been proposed for angular moments models.

In this thesis, a work has been performed in order to validate them. After the study of the fundamental properties of the collisional operators proposed for the M 1 model, it has been shown that accurate transport plasma coecients are recovered. This is an important result since it provides at our disposal a reliable model for the study of the collisional electronic transport. Therefore, this model represents a competitive tool which can be used for practical physical applications.

In this thesis a large amount of work has been devoted to the design and implementation of numerical schemes for the M 1 angular moments model. It has been shown that numerical schemes commonly used for angular moments models are not able to correctly capture some asymptotic regimes. Consequently, asymptoticpreserving schemes have been designed for the study of long time behaviours. The rst regime studied corresponds to a quasi-neutral plasma. At this scale the characteristic quantities are large compared to the plasma parameters. In order to work without any restriction on the time step an appropriate reformulation of the Maxwell-Ampere equation has been considered. A natural discretisation of the scheme has been proposed and several test cases have been performed showing the interest of the method.

In order to be able to perform numerical simulations on larger time scales the diusive limit has been studied. At this scale, the charateristic quantities are large compared to the collisional parameters. An asymptotic-preserving scheme has been proposed for the M 1 angular moments models in the diusive regime. It has been shown that, since charged particles are considered, mixed derivatives arise in that limit leading to an anisotropic diusion. The properties of the scheme have been studied and several test cases have been performed to show the interest of the method. Even if further details still need to be considered, this work represents an important step in the development of numerical methods for the study of long time regimes at an aordable numerical cost.

In the last part of this manuscript, the M 1 angular moments model in a mov-

Conclusion / Perspectives

ing non-inertial reference frame has been introduced. Before considering complex electron-ion interactions, the case of a single neutral particles specie has been considered. The equations are written in the mean velocity frame. This case applies to neutral gas dynamics studies. It represents an intermediate indispensable step before investigating multispecies congurations. This case allows to readily evaluate the mean diculties that appear in the general case. The model properties have been studied and an appropriate numerical scheme proposed. Then, several test cases in dierent collisional regimes have been presented to show the major properties of the method. This work represents a rst signicant step towards the study of the multispecies particle transport and numerous perspectives can be considered.

Short and long time perspectives

Several perspectives can be considered following the results obtained in this thesis.

First of all, other plasma instabilities can be studied within the formalism of angular moments models. In addition to the work performed in this thesis, collisional regimes may be considered since it has been shown that the M 1 and M 2 angular moments models are better adapted for collisionnal and weakly collisionnal regimes. This would further clarify the validity domain of angular moments models.

Secondly, following [START_REF] Braginskii | Reviews of Plasma Physics[END_REF] the transport plasma coecients need to be derived by taking into account the external and self-consistent electric and a magnetic elds. This study would be in the direct continuity of the work performed in this manuscript. It has been shown that, without an external electric eld and in regimes close to the equilibrium regimes, accurate transport plasma coecients are recovered by the collisional M 1 moments model. This provides a basis for extension of the present study to regimes with magnetic elds.

Concerning the numerical part of this manuscript, several directions can be considered. In the continuity of the work achieved in the diusive regime, one can account for the contribution of the electron-electron collisional operator which was removed for simplicity. This is a challenging issue because of the complex form of this operator. Dierent approaches can be pursued by adapting the ideas introduced in [START_REF] Filbet | A class of asymptotic preserving schemes for kinetic equations and related problems with sti sources[END_REF] for example.

Also, a numerical study of the complete system M 1 -Maxwell in the quasi-neutral and diusive regime may be conducted. This work would be a direct extension of the studies presented in this thesis. This would lead to a complete and computationally cheap numerical tool for the electronic transport studies in long time regimes.

The last part of this manuscript opens interesting perspectives for physical applications levels. On a short time scale the design of appropriate numerical methods is needed for solving the M 1 angular moments model in a moving frame. As a long term perspective one could include the electromagnetic coupling terms in order to study the electron transport within the M 1 model while describing the ion motion with an usual hydrodynamic description. This would provide a performant hybrid tool for the multispecies transport studies in a plasma.

(∂ t f 1 + ∇ x • (v(ζ)f 2 )) ϕ(ζ)dζ - p∈R 3 ϕ (ζ)Ω • (qEf ) Ωdp - ∞ 0 q ζ ϕ(ζ) ζ 2 S 2 f dΩ E + v(ζ)ζ 2 S 2
Ωf dΩ ∧ B dζ

+ ∞ 0 q ζ ϕ(ζ) ζ 2 S 2 Ω ⊗ Ωf dΩ E dζ , because Ω ⊗ Ω (v ∧ B) = 0, = ∞ 0 (∂ t f 1 + ∇ x • (v(ζ)f 2 )) ϕ(ζ)dζ - ∞ 0 qϕ (ζ) ζ 2 S 2 Ω ⊗ Ωf dΩ Edζ - ∞ 0 q ζ ϕ(ζ) (f 0 E + v(ζ)f 1 ∧ B) dζ + ∞ 0 q ζ ϕ(ζ) (f 2 E) dζ, = ∞ 0 (∂ t f 1 + ∇ x • (v(ζ)f 2 )) ϕ(ζ)dζ + ∞ 0 ∂ ζ (qf 2 E) ϕ(ζ)dζ - ∞ 0 q ζ ϕ(ζ) (f 0 E + v(ζ)f 1 ∧ B) dζ + ∞ 0 q ζ ϕ(ζ) (f 2 E) dζ, = ∞ 0 ∂ t f 1 + ∇ x • (v(ζ)f 2 ) + q ∂ ζ (f 2 E) - 1 ζ (f 0 E + v(ζ)f 1 ∧ B -f 2 E) ϕ(ζ)dζ,
which is veried for all test functions ϕ, hence the second equation of (A.1) is veried.

A.2 Angular moment extraction for the collision operators C ee and C ei

In this section the angular integration of the collision operators is performed.

A.2. Angular moment extraction for the collision operators C ee and C ei

A.2.1 Angular integration for the electron-ion collision operator

Firstly, we start computing the electron-ion collision C ei which is much simpler to compute. To establish this derivation we consider only the homogeneous case with the collision operator C ei ,

∂ t f = C ei [f ] = α ei ∇ v • |v| 2 I -v ⊗ v |v| 3 ∇ v f (v) = α ei ∇ v • [S(v)∇ v f (v)] ,
(A.4)

where we have introduced S(v) as,

S(v) = |v| 2 I -v ⊗ v |v| 3 = 1 ζ (I -Ω ⊗ Ω)) • (A.5)
The operator S(v) is the projection operator on the orhogonal plane to the vector v and we have S(v)v = 0. Now if we want to consider the inuence of the C ei operator for the equation about f 0 , we can use duality approach. If we set ζ = |v|, and we denote by ϕ some test function, which is again veried for all test functions ϕ, hence,

0 = v∈R 3 (∂ t f -α ei ∇ v • [S(v)∇ v f ]) ϕ(ζ)dv, = ∞ 0 ∂ t ζ 2
∂ t f 1 = - 2α ei ζ 3 f 1 ,
and in fact this gives exactly the operator Q 1 . The electron-ion collisions are only relevant for the f 1 behavior, not the f 0 .

A.2.2 Moment closure for the electron-electron collisions

To establish this closure we consider only the homogeneous case with the collision operator C ee ,

∂ t f = C ee [f, f ] = α ee ∇ v • v ∈R 3 S(u) [f (v )∇ v f (v) -f (v)∇ v f (v )] dv , (A.6)
where S(u) is given by (A.5) and u = v -v. The moments of this operator is rather complicated, we will assume that the main contribution from this operator comes

A.2. Angular moment extraction for the collision operators C ee and C ei

In this case, as we have assumed that f is isotropic, we have f 0 (ζ) = ζ 2 S 2 f (ζ) dΩ = 4πζ 2 f (ζ), we can conclude from (A.7) and (A.12) that,

∂ t f 0 = α ee ∂ ζ ζ ∞ 0 2 3 inf 1 ζ 3 , 1 ζ 3 ζ 2 ζ 2 f 0 (ζ ) ζ 2 1 ζ ∂ ζ f 0 (ζ) ζ 2 - f 0 (ζ) ζ 2 1 ζ ∂ ζ f 0 (ζ ) ζ 2 ζ 2 dζ , = 2α ee 3 ∂ ζ ζ 2 ∞ 0 inf 1 ζ 3 , 1 ζ 3 ζ 2 f 0 (ζ )dζ ∂ ζ f 0 (ζ) ζ 2 -ζ ∞ 0 inf 1 ζ 3 , 1 ζ 3 ζ 3 ∂ ζ f 0 (ζ ) ζ 2 dζ f 0 (ζ) , = 2α ee 3 ∂ ζ ζ 2 A(ζ)∂ ζ f 0 (ζ)

  Laboratoire d'accueil Laboratoire Celia, 351 Cours de la Libération F-33405 Talence cedex France / Institut de Mathématiques de Bordeaux, 351 cours de la Libération F-33405 Talence cedex France. Study of particle transport in plasmas suggestions suite à la lecture de ce manuscrit. Je remercie également Christophe Chalons, pour les discussions lors du congrès SMAI puis lors de sa venue à Bordeaux, Philippe Helluy pour le Cemracs 2014 à Luminy et Mohammed Lemou pour les échanges lors de l'été 2015 à Beijing. Je les remercie tous les trois d'avoir accepté le rôle d'examinateur dans ce jury. Je remercie Alain Bachelot d'avoir accepté de présider ce jury et pour sa sympathie. J'adresse un grand "Merci!" à Stéphane Brull et Bruno Dubroca pour leur encadrement et pour tout ce qu'ils m'ont apporté lors de cette thèse. Je les remercie pour leur patience malgré mes interminables questions et mon entêtement, pour leur pédagogie, leurs critiques et leurs conseils. Au-delà du travail de thèse, je les remercie pour leur bonne humeur et leur humour communicatif qui était toujours présent au quotidien. Malgré les dicultés rencontrées cela m'a permis de relativiser certaines situations et de rester optimiste. Je remercie mon co-directeur de thèse Emmanuel d'Humières pour ses conseils avisés et sa gentillesse. Échanger avec lui m'a appris à être plus diplômate et à garder une ouverture d'esprit vis à vis de thèmes de recherches diérents ou de nouvelles idées. Je tiens à remercier Vladimir Tikhonchuk pour sa disponibilité et pour le partage de son expérience. J'ai souvent été impressionné par la précision et la profondeur de ses remarques. Ceci m'a amené à me poser de nombreuses questions et à eectuer de nombreux calculs relatifs à la compréhension de la physique des plasmas. Je remercie Denise Aregba pour toutes les explications concernant les schémas de relaxation, les problèmes de Riemann et bien d'autres choses. Je la remercie pour les séances de travail et pour ses conseils clairs et avisés sur la thèse et bien au-delà. Merci à Rodolphe Turpault d'avoir pris du temps pour discuter des limites de diusion et plus généralement des schémas asymptotic-preserving. Je le remercie aussi pour son écoute et ses conseils lors de mon activité d'enseignement. Je remercie Rachel Nuter pour toutes les discussions portant sur la physique des plasmas et pour avoir eu le courage de vérier mes calculs des taux d'absorption laser-plasma! Je remercie Thanh-Ha Nguyen-Bui pour les discussions sur la physique en général

Figure 2 :

 2 Figure 2: International Thermonuclear Experiment Reactor project (ITER -Cadarache) for the magnetic connement fusion (left) and the laser installation Laser MegaJoule (LMJ -Bordeaux) for Inertial Connement Fusion (right).

  tion and was proposed by Anatoly Vlasov in 1945. This kinetic model describes particles interacting by long range forces: electromagnetic and gravitational. The Vlasov equation describes the distribution function on large scales, greater than the Debye length and applies to low density plasmas. This equation does not take into account the uctuations which become important when particles are close to each other therefore the Vlasov equation applies to non-collisional plasmas. Considering the contribution of the correlation term δf micro in the microscopic distribution function (1.7) it comes that

(1. 20 )

 20 This equation is valid only considering elastic collisions. In the case of ionisation or recombination, the right side of the equation can become non-zero. The transport equation on the momentum is derived multiplying the kinetic equation by the momentum p = m α v and integrating on the momentum. This equation reads

  ) and (1.15) by E and B. The sum of the two obtained equations leads to 1 2

  is derived performing an angular moment extraction from the Fokker-Planck-Landau equation. For the sake of clarity, we omit in the following, the x and t dependence of the distribution function. If S 2 is the unit sphere Ω = v/| v| represents the direction of propagation of the particle. By setting ζ = | v|, the distribution function f writes in the spherical coordinates in the phase space f ( Ω, ζ). The rst three angular moments of the distribution function are dened by

Figure 2 . 1 :ff

 21 Figure 2.1: The coordinates system used for the calculation of angular moments of the electron distribution function.

Figure 2 . 2 :

 22 Figure 2.2: Representation of the dissipation coecient as a function of k for the Vlasov equation and for the M 1 , two populations M 1 and M 2 models.

2 model

 2 [START_REF] Chalons | Godunovtype schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction[END_REF]. Ignoring the constant values, the real part of the conductivity varies in ω 2 pe ω/v 2 th k 2 exactly like for the Vlasov equation and the imaginary part varies in ω 2 pe ω 4 /v 5 th k 5 instead of ω 2 pe /v th k for the Vlasov equation. The expression for the M

Figure 2 . 4 :

 24 Figure 2.4: Representation of the modulus integrand of the impedance coecient as a function of wave number in the limit ω/k >> v th in the case ω/ω pe = 0.3 and v th /c = 0.1.

Chapter 3 Classical

 3 transport theory for the collisional electronic M 1 model The study introduced in this chapter has been published. The reference is: S. Guisset, S. Brull, E. d'Humières, B. Dubroca, V. Tikhonchuk. Classical transport theory for the collisional electronic M1 model. Physica A: Statistical Mechanics and its Applications, Volume 446, Pages 182-194 (2016).

  2)-(3.4) satisfy mass and energy conservation properties. Here, we complete these results characterising the equilibrium state of the collisional operators (3.2)-(3.4) which corresponds to an isotropic Maxwellian function.

  From (3.7) and (2.1), we get the relation

+∞ 0 F 0 F 0 3 +∞ 0 F 1 ε 3 +∞ 0 F 1 ζ

 000301301 (t, x, ζ)ζ 2 dζ = 0 and +∞ 0 (t, x, ζ)ζ 4 dζ = 0. Equations for the density and temperature are following from the integration over ζ of the electronic M 1 model (3.1) and denitions (3.23-3.24) + ∇ x .(n e u e ) = 0, ∂T e ∂t + u e .∇ x (T e ) + 2 3 T e ∇ x .( u e ) + 2 3n e ∇ x .( q) = 2 3n e j. E (3.25)where we retained only linear terms in the Knudsen number ε. The temporal evolution of n e and T e in these equations is driven by the uxes of the particles and energy that are expressed through the electric current density and the electron heat ux dened by j = -en e u e = -4πeε ζ 3 dζ, q = 2πm e 5 dζ.

(3. 26 )

 26 In order to close the hydrodynamic system(3.25), one needs to express the electric current and the heat ux(3.26) in terms of the macroscopic variables n e , T e . More precisely, the term F 1 should be derived explicitly in terms of the gradients of n e and T e , then denitions(3.26) give the electric current and the heat ux. In the quasi-stationary case (∂/∂t << ν ei ) the second equation of the electronic M 1 model (3.1) reads

(3. 33 )

 33 Equation (3.27) has been established from the collisional electronic M 1 model (3.1).This equation is identical to the one obtained using the full Fokker-Planck-Landau equation (1.26), (see[START_REF] Balescu | Transport Processes in Plasma[END_REF]) with the exception of the electron-electron collisional operator. Therefore, the possible dierences in the plasma transport coecients between the collisional electronic M 1 model (3.1) and the Fokker-Planck-Landau equation (1.10)-(1.26) are due to the electron-electron collisional operator. More precisely, the approximations made to derive the electron-electron collisional operator (3.2)-(3.3) for the electronic M 1 model (3.1) may lead to dierent plasma transport coecients. The aim of the following subsections, is to derive the plasma transport coecients using the collisional electronic M 1 model (3.1) and to compare them to the ones obtained using the Fokker-Planck-Landau equation (1.10)-(1.26).

  [START_REF] Brull | Degenerate anisotropic elliptic problems and magnetised plasma simulations[END_REF] into expressions(3.31) and(3.32) gives the transport coecients for a high Z plasma[START_REF] Balescu | Transport Processes in Plasma[END_REF] σ 0 = 32 3π e 2 n e m e ν ei , α 0 = 16 π en e m e ν ei , χ 0 = 200 3π n e v Te λ ei .

Figure 3 . 1 :

 31 Figure 3.1: Representation of the velocity-dependent (y = v/v Te ) particle ux, j V = -ζ 3 f 1 in red and the electron energy ux q V = m e f 1 ζ 5 -5T e f 1 ζ 3 in green in the case Z >> 1 (Lorentzian approximation).

  [START_REF] Boscarino | High-order asymptotic-preserving methods for fully nonlinear relaxation problems[END_REF]) by ζ 3 one obtains an equation more suitable for numerical integration. Indeed, the term 1/ζ 3 in the electron-ion collision operator makes the equation (3.27) very sti when ζ becomes close to zero.

  and 3.4 as a function of Z for the electron-electron Landau collision operator C ee given in (1.27) and for the electron-electron M 1 collision operator (3.2)-(3.3) using six Laguerre polynomials.

Figure 3 . 2 :

 32 Figure 3.2: Representation of the velocity-dependent particle ux, j V = -ζ 3 f 1 , in the case Z = 1 (blue), Z = 4 (yellow), Z = 16 (green) and Z >> 1 (Lorentzian approximation) in red.

Figure 3 . 3 :

 33 Figure 3.3: Representation of γ σ (left) and γ α (right) as a function of Z for the Landau (red) and the M 1 (green) collision operators using six Laguerre polynomials.

Figure 3 . 4 :

 34 Figure 3.4: Representation of γ χ (left) and γ κ (right) as a function of Z for the Landau (red) and the M 1 (green) collision operators using six Laguerre polynomials.

(4. 7 )

 7 Using (4.4) and (4.7) by Green's formula we obtain

Figure 4 . 1 :

 41 Figure 4.1: Structure of the HLL approximate Riemann solver.

Remark 1 .

 1 Notice that the fourth equation of system (5.1), called Maxwell-Gauss equation (or Poisson equation) is not used. Indeed, the second equation of (5.1), called Maxwell-Ampere equation and Poisson equation are equivalent if the

  ) where the Maxwell-Poisson and Maxwell-Thomson equations have to be satised at the initial time. Remark 2. The fth equation of the system (5.1) called the Maxwell-Thomson equation is not used. Indeed the third and fth equations (5.1) called the Maxwell-Faraday and Maxwell-Thomson equations are equivalent if the Maxwell-Thomson

2 = 2 , v p+ 1 2 [ where v p-1 2 = (p - 1

 22221 ]v p , v p+1 [ ∀p ∈ {-p f ; p f } where v p = p∆v and p ∈ N represents the number of points which discretize the velocity domain. ∆v represents the energy discretisation step, which is xed. Denote by D its associated dual mesh consisting of cells D p =]v p-1

5 .Figure 5 . 1 :

 551 Figure 5.1: Distribution function as a function of space and velocity at initial time (left) and after 30 plasma periods (right).

  regime (A = 0.1)

Figure 5 . 2 :

 52 Figure 5.2: Temporal evolution of the electrostatic energy (dimensionless units) in the linear regime (top) and in the non-linear regime (bottom).

Figure 5 . 3 :

 53 Figure 5.3: Representation of the temperature and electric eld as function of space for dierent times.

Figure 5 . 4 :

 54 Figure 5.4: Comparison of the temperature (left) and electric eld (right) for akinetic code[START_REF] Marocchino | Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses[END_REF] and the M 1 Asymptotic-Preserving scheme.

Figure 5 . 5 :

 55 Figure 5.5: Comparison of the temperature (left) and heat ux (right) for a kinetic code [24], a hydrodynamic code and the M 1 Asymptotic-Preserving scheme.

about 2 . 10 - 3

 2103 times the temperature gradient length, the local electron transport model of Spitzer and Härm fails. The transport coecients were derived in the case where the isotropic part of the electron distribution function remains close to the Maxwellian function. The results of Spitzer and Härm have been reproduced in several approaches[START_REF] Balescu | Transport Processes in Plasma[END_REF][START_REF] Braginskii | Transport phenomena in a completely ionized two-temperature plasma[END_REF][START_REF] Shrimpton | Statistical treatment of turbulent polydisperse particle systems[END_REF] which develop another technique of solution to the integral equation for the electron distribution function introduced many years before by Chapman and Enskog[START_REF] Chapman | The Mathematical Theory of Non-Uniform Gases[END_REF] for neutral gases. Therefore, kinetic approaches seem 6.1. Introduction necessary in the context of inertial connement fusion. In such multiscale issues, kinetic solvers are often very computationally expensive and usually limited to time and length much shorter than those studied with hydrodynamic simulations. It is then a challenge to describe kinetic eects using reduced kinetic code on uid time scales.

1 = f R * 1 = f * 1 .

 111 two intermediate states U L * and U R * are introduced. The second components of the two intermediate states are chosen equal, ie f L * The approximate solution at time

Figure 6 . 1 :

 61 Figure 6.1: Structure solution of the approximate Riemann problem.

Figure 6 . 2 :

 62 Figure 6.2: Free transport: comparison of the numerical solution for ∆x = 4 • 10 -3 and the exact solution (red) at time t=6.

6. 4 .Figure 6 . 3 :

 463 Figure 6.3: Temperature gradient: comparison of the temperature prole for the numerical solution (AP) and for the HLL scheme (HLL) at time 0.25 and 0.5.

Figure 6 . 4 :

 64 Figure 6.4: Temperature gradient in the diusive limit: comparison of the temperature prole of the asymptotic-preserving scheme (AP), the HLL scheme (HLL) and the diusion solution at time t=50, 100, 500 and 1000.

Figure 6 . 6 :

 66 Figure 6.6: Comparison of the f 0 prole for the Asymptotic-preserving scheme (AP), and the diusion solution at time t=500.

Figure 6 . 7 :Figure 6 . 8 :

 6768 Figure 6.7: Relaxation of a Gaussian prole: comparison of the f 0 prole for the asymptotic-preserving scheme (AP), for the HLL scheme (HLL) and the diusion solution at time t=20.

Figure 6 . 10 :

 610 Figure 6.10: Structure of the approximate Riemann solver.

Figure 6 . 11 :

 611 Figure 6.11: f 0 prole at the initial time.

Figure 6 . 12 :

 612 Figure 6.12: Representation of the f 0 prole as function of x and ζ at time t=1 (top), t=50 (middle), t=100 (bottom), for the HLL scheme (left), AP scheme (middle) and

ζ 2 f

 2 0 dζ -u 2 n e ,

Figure 6 . 13 :

 613 Figure 6.13: Temperature prole at time t=80.

Figure 6 . 14 :

 614 Figure 6.14: Temporal evolution of the electrostatic energy.

6. 7 .

 7 Numerical examples computed using(6.78). In this case the collisional parameter α ei is not constant and follows the linear prole σ(x) = ax + b, with α ei (x min = -40) = 5.10 3 and α ei (x max = 40) = 10 5 . It follows that the coecients a and b reads a = 10 5 -5.10 3 x max -x min , b = 5.10 3 -ax min . The space range is [-40,40] and the energy range [0,6]. In Figure 6.15, the temperature prole is displayed at the initial time and at time t=5000 for the AP scheme and an explicit discretisation of the diusion equation (6.53). After a long time (t=5000) and despite the strong spatial variation of the function α ei the AP and diusion proles give very close result. One remark on the space interval [-40,0] the AP curve in red is slightly dierent to the diusion curve in dashed blue while on the interval [0,[START_REF] Buet | Fast algorithm for numerical, conservative and entropy approximation of the Fokker-Planck-Landau equation[END_REF] the results match perfectly. This could be explained as the collisional parameter α ei becomes larger for important x, therefore, the limit diusive regime is fully reach for large x where the comparison with the diusion equation is valid.

Figure 6 . 15 :

 615 Figure 6.15: Temperature prole at time t=5000.

Figure 6 . 16 :

 616 Figure 6.16: Representation of the f 0 prole as function of x and ζ at the initial time (left) and t=3 (right).

7. Angular M 1 model in a moving reference frame 7 . 2 . 1

 721 Kinetic equation in a moving frame We start considering the following kinetic equation written in the laboratory framework ∂f (α) ∂t + div x (vf (α)) = C(f (α)),

(7. 21 ) 7 .

 217 Angular M 1 model in a moving reference frame Therefore, we propose to consider the following change of variable x = Rx -st, c = Rc, ũ = Ru -s.

(7. 22 )

 22 Indeed, by injecting the change of variable (7.21-7.22) into (7.4) and using the following relations

  ) to obtain(7.19), denoted B. Finally, one remarks that (7.4) and (7.19), denoted A and B are linked by the change of variable(7.22).

Figure 7 . 1 :

 71 Figure 7.1: Diagram relation between the dierent frames

  Proof. Settingt m = (1, Ω), t α = (α 0 , α 1 ), the distribution function (7.10) reads g(t, x, ζ, Ω) = exp(α.m), and the solution of (7.7)-(7.11)-(7.13) writes t (g 0 , g 1 ) = ζ 2 exp(α.m)m ,

(7. 33 )

 33 Multiplying the second equation of (7.7) by ζ and integrating in ζ, one shows using

g

  1p ζ p ∆ζ = 0,

  Finally, 400 cells in space have been considered for the Euler description. Neumann boundary conditions are considered, the values in the boundary ghost cells set to the values in the corresponding real boundary cells.

7. Angular M 1

 1 model in a moving reference frame 1.a Fluid regime.

Figure 7 . 2 :

 72 Figure 7.2: Test 1a -Solution proles obtained for the temperature gradient test case with τ = 0 at time t = 10.

Figure 7 . 3 :

 73 Figure 7.3: Test 1b -Solution proles obtained for the temperature gradient test case with τ = 1 at time t = 10.

7. Angular M 1

 1 model in a moving reference frame

Figure 7 . 4 :

 74 Figure 7.4: Test 1c -Solution proles obtained for the temperature gradient test case with variable collisional parameter at time t = 10.

Figure 7 . 5 :

 75 Figure 7.5: Test 2 -Sod tube test case with τ = 0 at time t = 7.34 • 10 -2 .

7. Angular M 1

 1 model in a moving reference frame

Figure 7 . 6 :

 76 Figure 7.6: Test 3 -Double shock wave test case with τ = 0 at time t = 0.15.

A. 1 . 1 0 = p∈R 3 ( 3 ( 3 ∇ 0 ∂ t ζ 2 S 2 f 0 ∇ x • v(ζ)ζ 2 S 2 3 ϕ 0 ( 3 ϕ 0 ( 2 S 2 0 ( 0 ( 3 = p∈R 3 ( 3 ( 3 ∇ 3 ϕ 0 ∂ t ζ 2 S 2 0 ∇ x • v(ζ)ζ 2 S 2 Ω 3 ϕ

 1133302202230302200333330220223 Derivation of the model for the angular moments f 0 and f∂ t f + v(p) • ∇ x f + q (E + v(p) ∧ B) • ∇ p f ) ϕ(ζ)dp, = p∈R ∂ t f + v(p) • ∇ x f ) ϕ(ζ)dp -p∈R p ϕ(ζ) • (q (E + v(p) ∧ B) f ) dp, from (A.2), = ∞ dΩ ϕ(ζ)dζ + ∞ Ωf dΩ ϕ(ζ)dζ -p∈R (ζ) p ζ • (q (E + v(p) ∧ B) f ) dp, as p = ζΩ and dp = ζ 2 dζdΩ, = ∞ ∂ t f 0 + ∇ x • (v(ζ)f 1 )) ϕ(ζ)dζ -p∈R (ζ)Ω • (qEf ) dp, as p • (v(p) ∧ B) = 0, = ∞ ∂ t f 0 + ∇ x • (v(ζ)f 1 )) ϕ(ζ)dζ -∞ 0 ϕ (ζ)qE • ζ Ωf dΩ dζ, = ∞ ∂ t f 0 + ∇ x • (v(ζ)f 1 )) ϕ(ζ)dζ + ∞ 0 ∂ ζ (qE • f 1 ) ϕ(ζ)dζ, = ∞ ∂ t f 0 + ∇ x • (v(ζ)f 1 ) + ∂ ζ (qE • f 1 )) ϕ(ζ)dζ,which is veried for all test functions ϕ, hence the rst equation of (A.1) is veried. Now if we consider the case of the vector f 1 , we need to compute ∇ p Ω,∇ p Ω = ∇ p |p| 2 I -p ⊗ p) = 1 ζ (I -Ω ⊗ Ω)) , (A.3)where I is the identity matrix. Now, we can consider the proof of the second equation of (A.1). Multiplying the kinetic equation by ϕ(ζ)Ω, and integrating leads to A. Derivation of the electron M 1 model0 ∂ t f + v(p) • ∇ x f + q (E + v(p) ∧ B) • ∇ p f ) Ωϕ(ζ)dp, = p∈R ∂ t f + v(p) • ∇ x f ) Ωϕ(ζ)dp -p∈R p ϕ(ζ) • (q (E + v(p) ∧ B) f ) Ωdp -p∈R (ζ)∇ p Ω (q (E + v(p) ∧ B) f ) dp, from (A.2), = ∞ ΩdΩ ϕ(ζ)dζ + ∞ ⊗ Ωf dΩ ϕ(ζ)dζ -p∈R (ζ) p ζ • (q (E + v(p) ∧ B) f ) Ωdp -p∈R 3 1 ζ ϕ(ζ)q (E + v(p) ∧ B) f dp + p∈R 3 1 ζ ϕ(ζ)Ω ⊗ Ω (q (E + v(p) ∧ B) f ) dp, from (A.3), = ∞ 0

S 2 f 3 ∇ 0 ∂ 3 ϕ 0 ∂ 3 ϕ 0 ∂ 3 ( 0 ∂ t ζ 2 S 2 3 ∇α ei v∈R 3 ϕ 0 ∂ 3 ϕ 0 ∂ 0 ∂ 3 ϕ(ζ) ζ 4 ∇ 0 ∂ 3 ϕ(ζ) ζ 4 0 ∂ 0 2 ζ 3 ζ 2 S 2 0 ∂ t f 1 + 2α ei ζ 3 f

 2303030302233030034034002203 dΩ ϕ(ζ)dζ + α ei v∈R v ϕ(ζ) • [S(v)∇ v f ] dv, = ∞ t f 0 ϕ(ζ)dζ + α ei v∈R (ζ) v ζ • [S(v)∇ v f ] dv, = ∞ t f 0 ϕ(ζ)dζ + α ei v∈R (ζ) S(v) v ζ • ∇ v f dv, as S(v) is symmetric, = ∞ t f 0 ϕ(ζ)dζ, as S(v)v = 0,which is veried for all test functions ϕ, hence,∂ t f 0 = 0,and the C ei operator has no inuence on the equation for f 0 .A. Derivation of the electron M 1 modelNow, we consider the inuence of the C ei operator for the equation about f1 , ∂ t f -α ei ∇ v • [S(v)∇ v f ]) Ωϕ(ζ)dv, = ∞ Ωf dΩ ϕ(ζ)dζ + α ei v∈R v ϕ(ζ) • [S(v)∇ v f ] Ωdv + (ζ)∇ v Ω [S(v)∇ v f ] dv, = ∞ t f 1 ϕ(ζ)dζ + α ei v∈R ) [S(v)∇ v f ] dv, from (A.3), = ∞ t f 1 ϕ(ζ)dζ -α ei v∈R 3 ϕ(ζ)∇ v • 1 ζ S(v) f dv, as S(v) is symmetric, S(v)S(v) = 1 ζ S(v) and S(v)v = 0, = ∞ t f 1 ϕ(ζ)dζ -α ei v∈R v • [ζ 2 I -v ⊗ v]f dv, as S(v)v = 0, = ∞ t f 1 ϕ(ζ)dζ + α ei v∈R 2vf dv, as ∇ v • [|v| 2 I -v ⊗ v] = -2v, = ∞ t f 1 ϕ(ζ)dζ + α ei ∞ Ωf dω ϕ(ζ)dζ, as dv = ζ 2 dζdΩ, = ∞ 1 ϕ(ζ)dζ,

ζ 2 -

 2 ζB(ζ)f 0 (ζ) , = 2α ee 3 ∂ ζ A(ζ)∂ ζ f 0 (ζ) -ζB(ζ) + 2A(ζ) ζ f 0 (ζ) , w)dw.We retrieve exactly the formula (3.2) for the collision operators involved by the equation on f 0 . We neglect the operator Q ee for the equation on f 1 because we retain only the isotropic part of this operator and f 1 represent the anisotropic part of the distribution function.This holds true for all test function φ then one obtains the rst equation for the M 1 model in a moving frame∂ t g 0 + div x (ζg 1 + ug 0 ) -Introduce a test function φΩ, we consider the integral c ∂ t g + div x ((c + u)g) -div c ( du dt g + ∂u ∂x cg) φ(ζ)Ωdc = 0.By using the Green formulaec ∂ t g + div x ((c + u)g) φ(ζ)Ωdc + c ∂c = φ (ζ)Ω ⊗ Ω + φ(ζ) Id -Ω ⊗ Ω ζ .Then the second term of the left side of the equation gives cFinally one obtains the second equation for the M 1 model in a moving frame

  Cette partie traite de la résolution numérique du système M 1 -Maxwell dans le régime quasi-neutre[START_REF] Crispel | An asymptotic preserving scheme for the two-uid Euler-Poisson model in the quasi-neutral limit[END_REF][START_REF] Degond | Numerical approximation of the Euler-Maxwell model in the quasineutral limit[END_REF]. Dans ce régime les équations permettant de calculer le champ électrique deviennent singulières. Une reformulation de

	étudiés [4, 160, 175, 177, 195]. La fonction de distribution sous-jacente est donnée travers les modèles aux moments angulaires. La deuxième partie de ce document
	par l'exponentielle d'une fonction polynomiale dépendant du module de vitesse et traite de la dérivation de méthodes numériques pour l'étude du transport de partic-Chapitre 4. Les modèles aux moments angulaires sont des systèmes hyper-
	est ainsi non négative. De plus les modèles aux moments basés sur des fermetures ules en temps long. Les schémas classiques utilisés pour les modèles aux moments boliques. Dans ce chapitre, quelques concepts pour la résolution numérique de
	entropiques satisfont des propriétés mathématiques fondamentales [112, 160, 172] angulaires préservent l'admissibilité de la solutions numérique. Cependant, de tels systèmes non linéaires sont rappelés en suivant les idées présentées dans [52, 101,
	comme l'hyperbolicité et la dissipation d'entropie. Cependant, d'un point de vue schémas ne sont, en général, pas capables de capturer la bonne solution numérique 102, 118, 159]. Ces méthodes s'appliquent directement aux modèles aux moments
	numérique, même si la fermeture est bien dénie, des dicultés numériques de-lorsque l'on considère des régimes asymptotiques. Le but est de proposer des sché-angulaires et sont utilisées dans les prochains chapitres.
	meurent. En particulier, la résolution de problèmes de minimisation d'entropie sous mas numériques qui capturent la bonne limite asymptotique sous des contraintes
	contraintes peuvent êtres particulièrement complexes et nous renvoyons à [4] pour raisonnables sur le pas de temps et d'espace. La troisième partie est une contri-Chapitre 5.
	un traitement spécique. bution à la modélisation multi-espèce. L'objectif nal est d'étudier les dynamiques
	électronique et ionique avec un modèle précis qui est aussi numériquement peu coû-
	doit être considéré. Ceci donnerait accès à une physique plus générale et plus riche
	pour des applications en fusion par connement inertiel. Un travail important est
	donc requis pour l'utilisation des modèles aux moments angulaires pour l'étude de
	transport de particules multi-espèces.
	Structure du manuscrit

Les modèles aux moments angulaires représentent une méthode alternative située entres les modèles cinétiques et les modèles uides. Ils nécessitent un coût informatique moins important que les modèles cinétiques et fournissent des résultats de plus grande précision que les modèles uides. Ils sont basés sur une moyenne angulaire

[START_REF] Minerbo | Maximum entropy Eddigton Factors[END_REF][START_REF] Pomraning | Maximum entropy Eddington factors and ux limited diffusion theory[END_REF]

des équations cinétiques tout en conservant le module de la vitesse (noté ζ dans ce travail) comme variable. Ceci permet de considérer des distributions énergétiques de particules éloignés de l'équilibre tout en travaillant avec une description réduite. De tels modèles sont obtenus par l'intégration angulaire de l'équation cinétique (intégration sur la sphère unité). Ainsi une hiérarchie d'équations aux moments peut être obtenue. Il existe plusieurs modèles aux moments dont les différences proviennent de choix de la relation de fermeture. Dans ce document nous Study of particle transport in plasmas considérons les modèles aux moments angulaires basés sur un principe de minimisation d'entropie

[START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]

. Les problèmes de minimisations d'entropie ont été largement Le modèle aux moments angulaires M 1 est largement utilisé dans le contexte du transfert radiatif

[START_REF] Berthon | An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions[END_REF][START_REF] Charrier | Multigroup model for radiating ows during atmospheric hypersonic re-entry[END_REF][START_REF] Dubroca | Entropic moment closure hierarchy for the radiative transfert equation[END_REF][START_REF] Ripoll | An averaged formulation of the M1 radiation model with presumed probability density function for turbulent ows[END_REF][START_REF] Ripoll | A factored operator method for solving coupled radiation-hydrodynamics models[END_REF][START_REF] Turpault | Multigroup half space moment appproximations to the radiative heat transfer equations[END_REF]202]

, cependant, il est relativement nouveau de l'utiliser pour des applications en physiques des plasmas

[START_REF] Dubroca | Angular moment model for the Fokker-Planck equation[END_REF]

. Ainsi, le premier objectif de ce travail est de fournir une meilleure compréhension des avantages et des limites de ce modèle réduit pour la modélisation en physique des plasmas.

Puisque les modèles aux moments angulaires peuvent êtres considérés comme un compromis entres les descriptions uides et cinétiques, ils représentent des candidats naturels pour capturer des eets cinétiques sur de longues échelles de temps.

Ainsi, un autre objectif important est de proposer des méthodes numériques appropriées pour calculer des solutions numériques de modèles aux moments angulaires pour l'étude de régimes en temps longs. En eet, dans le cas où les grandeurs caractéristiques du problème considéré deviennent grandes devant les paramètres plasmas ou les paramètres collisionnels, des raideurs apparaissent dans le système d'équation étudié et le modèle dégénère. En général, les schémas numériques classiques utilisés pour les modèles aux moments angulaires ne sont pas capables de correctement capturer les bonnes limites asymptotiques sous des conditions acceptables sur les pas de temps et d'espace. Aussi, en plus de se comporter correctement dans les régimes asymptotiques, les méthodes numériques doivent êtres capables de préserver les propriétés fondamentales des modèles aux moments angulaires comme la préservation des ensembles admissibles. Ce point est étudié en détail dans ce document. Un autre objectif concerne la simulation du transport multi-espèce. Les études portant sur le transport électronique sont souvent eectuées en considérant les ions immobiles

[START_REF] Duclous | Modélisation et Simulation Numérique multi-échelle du transport cinétique électronique[END_REF][START_REF] Mallet | Contribution à la modélisation et à la simulation numérique multi-échelle du transport cinétique électronique dans un plasma chaud[END_REF]

. En eet, en raison de leur masse importante comparée aux électrons, le mouvement des ions est souvent négligé car de petits intervalles de temps sont considérés. Cependant, lors des études sur de longues échelles, le mouvement ionique Ce travail se situe entre la modélisation en physique des plasmas, les mathématiques Introduction appliquées et l'analyse numérique. Ce manuscrit est organisé en trois parties. La première partie est une contribution à la modélisation en physique des plasmas à teux pour les applications. Dans ce travail un problème plus simple est étudié considérant des particules non-chargées. Chapitre 1. Dans ce premier chapitre nous introduisons les concepts de bases essentiels en modélisation pour la physique des plasmas. Ces éléments sont utilisés dans les chapitres suivants. Chapitre 2. Dans ce chapitre, nous commençons par rappeler le principe de construction des modèles aux moments basés sur une fermeture par minimisation d'entropie. Alors qu'en limite fortement collisionnelle les modèles aux moments angulaires fournissent une bonne approximation d'une équation cinétique complète, leur domaine de validité en limite non-collisionnelle est inconnu. Le travail ce chapitre est dévoué à l'étude du domaine de validité du modèle M 1 et de ces extensions, le modèle M 1 double population et le modèle M 2 pour des applications cinétiques non-collisionnelle. Trois eets sont étudiés, l'interaction de faisceaux de particules chargées, l'amortissement Landau et l'absorption d'une onde électromagnétique dans un plasma semi-inni très dense. Pour chaque phénomène une analyse perturbative est conduite et une relation de dispersion est établie en utilisant les modèles aux moments angulaires. Ces relations sont comparées avec celles obtenues en considérant l'équation de Vlasov. Les domaines de validité de chaque modèle sont étudiés.

Chapitre 3. Dans ce chapitre, le modèle M 1 électronique est utilisé pour l'étude du transport électronique dans un plasma collisionnel. La prise aux moments de l'opérateur de collision électron-électron de Landau est dicile et des approximations sont nécessaires. Premièrement, les opérateurs de collisions utilisés pour le modèle M 1 électronique sont détaillés [166, 167]. Les propriétés de ces opérateurs sont données puis suivant [32] les coecients de transport plasma sont dérivés. Il est montré que dans la limite Z élevé (degré d'ionisation élevé), le modèle M 1 électronique et l'équation de Fokker-Planck-Landau coïncide proche équilibre. Il est aussi montré que l'opérateur de collision électron-électron proposé pour le modèle Study of particle transport in plasmas M 1 électronique permet de retrouver des coecients de transport plasma proches de ceux obtenus avec l'équation Fokker-Planck-Landau. l'équation de Maxwell-Ampère est alors considérée puis un schéma numérique est proposé. Ce schéma est consistent avec le modèle intermédiaire et le modèle limite. Ce schéma se comporte correctement en régime quasi-neutre sans restrictions sur le pas de temps ni d'espace. Deux cas tests numériques sont enn présentés. Chapitre 6. Ce chapitre est dédié à la dérivation d'un schéma préservant l'asymptotique pour le modèle M 1 électronique en régime de diusion. Dans la première partie de cette section, le cas sans champ électrique puis le cas homogène avec champ électrique sont étudiés. La construction du schéma est basée sur un solveur de Riemann approché dont les états intermédiaires sont choisis consistent avec la forme intégrale du modèle M 1 . Le schéma proposé préserve l'admissibilité de la solution numérique et capture la bonne limite asymptotique. Diérents cas test numériques sont présentés. La deuxième partie de ce travail concerne l'extension de la méthode proposée au modèle général. Dans ce cas, des dérivés croisées apparaissent en limites de diusion et une modication du schéma est nécessaire. Diérents cas tests numériques sont enn présentés dans le cas général. Chapitre 7. Cette étude représente un premier important vers la modélisation de particules chargées multi-espèces. Avant de considérer des congurations complexes traitant de l'intéraction de particules chargées, dans ce chapitre, nous considérons seulement des particules non-chargées et présentons le modèle aux moments angulaires M 1 pour l'étude de la dynamique des gaz raréés. Dans ce travail, l'origine du repère en vitesse est centrée sur la vitesse moyenne des particules. Premièrement, la dérivation du modèle au moment M 1 dans le référentiel de vitesse moyenne est présentée. Il est montré que le choix de ce référentiel permet d'assurer la propriété d'invariance galiléenne du modèle. Il est aussi montré que le modèle écrit en fonctions des variables entropiques est symétrique au sens de Friedrichs. La dérivation de lois des conservations associées est aussi détaillée. Deuxièmement, un schéma numérique préservant l'admissibilité de la solution numérique est proposé. Finalement, plusieurs cas tests numériques menés en considérant diérents régimes collisionnels sont présentés. Introduction Conclusion. Nous présentons nos conclusions et plusieurs perspectives à ces travaux.

1 .

 1 Basic concepts for plasma physics modelling where ∂ t = ∂/∂t is the temporal partial derivative and d t = d/dt the temporal total derivative. The previous equation rewrites

  l for ζ = ζ l . Using the previous values in (2.26), we obtain that D = (ω -k l ζ l ) 2 = (ω -kv l ) 2 , for ζ = ζ l , and the value of the integral in (2.26) becomes -1

	n	n l=1 v 2 l /(ω -kv l ) 2 . We can rewrite the dispersion relation
	(2.26) as,

  Using the previous values in (2.26) we obtain that D = ω 2 -k 2 ζ 2 /3 and the dispersion relation (2.26) writes as follows,

  instead of 0.1398/k 3 and the coecient in the exponential is 1/2k 2 instead of 3/2k 2 . Figure2displays the Landau dissipation coecient as a function of k for the M 1 model (dotted curve) and the Vlasov equation (solid curve). The M 1 model clearly underestimates the Landau dissipation. This gure highlights the impossibility for the M 1 model to accurately model the Landau

						3,
	and	δω = -	3 √ 6π 4k 5 exp(-	5 2	) exp(-	3 2k 2 ) ≈ -	0.267 k 5 exp(-	3 2k 2 ).	(2.34)
	The dissipation found by the M 1 model (2.34) is signicantly dierent from the
	Landau dissipation term (2.28) computed with the Vlasov equation. Indeed the
	pre-exponential factor varies in 0.267/k 5 damping.
	2.5.2 Two populations M 1 model: plasma wave dispersion
	We propose here to study the possibility to model the Landau damping with the
	two populations M 1 model (2.2)-(2.18). The stationary solution for the two parts of
	the distribution function reads

  )where erf is the error function. Next, inserting the denition of Q 0 (3.2) and expressions(3.40) and (3.41) into (3.39) a long but straight calculation leads to the following expression for ce nm

	+∞
	ce nm =
	0

  with 50 Laguerre polynomials. In this work, 6 Laguerre polynomials were used to ensure a high accuracy of the transport coecients. The sixth polynomial expansion leads to the + 3306.34Z 3 + 851.07Z 4 + 90.44Z 5 + 3.39Z 6 173.69 + 2826.28Z + 3603.55Z 2 + 1604.84Z 3 + 320.28Z 4 + 29.31Z 5 + Z 6 , + 2263.58Z 3 + 702.46Z 4 + 83.77Z 5 + 3.39Z 6 173.69 + 2826.28Z + 3603.55Z 2 + 1604.84Z 3 + 320.28Z 4 + 29.31Z 5 + Z 6 .

	following approximations
	φ E 0 ≈ -ν -1 ei 670.42Z + 4467.79Z 2 φ Q 0 ≈ -5 2ν ei 29.38Z + 1611.93Z 2 + 1595.33Z 3 + 462.03Z 4 + 52.26Z 5 + 2.03Z 6 173.69 + 2826.28Z + 3603.55Z 2 + 1604.84Z 3 + 320.28Z 4 + 29.31Z 5 + Z 6 ,
	φ E 1 ≈ ν -1 ei	-86.09Z + 1177.61Z 2 + 1414.61Z 3 + 437.38Z 4 + 51.18Z 5 + 2.03Z 6 173.69 + 2826.28Z + 3603.55Z 2 + 1604.84Z 3 + 320.28Z 4 + 29.31Z 5 + Z 6 ,
	φ Q 1 ≈	2ν ei 5	163.98Z + 2155.57Z 2

Table 6 .

 6 1: Features of the Riemann problem

Table 6 .

 6 2: Convergence study of the method. The order of the method is given for the L 1 , L 2 and L ∞ norms.

	4 • 10 -3	0.08	1.09	0.04	0.95	0.03	0.93
	Consider the initial conditions

  6.6. Numerical schemeThen θ 1i+1/2j and θ 2ij+1/2 are xed in the interval [0, 1], the larger possible such that the admissibility requirements (6.4) are fullled. A simple calculation gives the Theorem 6.9. (Admissibility) If for all (i, j) ∈ N 2 , U n i,j ∈ A, then for all (i, j) ∈ N 2 ,

	following conditions
		θ1i+1/2j =	f0i+1/2j -α i+1/2j | f1i+1/2j | |Γ i+1/2j | + |α i+1/2j ( ∂f 0 ∂ζ ) i+1/2j c i+1/2j |	,	(6.66)
	and
		θ2ij+1/2 =	f0ij+1/2 -β ij+1/2 | f1ij+1/2 | |Γ ij+1/2 | + |β ij+1/2 ( ∂f 0 ∂ζ ) ij+1/2 cij+1/2 |	.	(6.67)
	Finally, θ 1i+1/2j = min( θ1i+1/2j , 1) and θ 2ij+1/2 = min( θ2ij+1/2 , 1).
	U n+1 i,j	∈ A as soon as the following CFL condition holds
			∆t ≤	∆ζ∆x (2a x ∆ζ + 2a ζ ∆x)	.

  6.6.2 General case with the termE ζ (f 0 -f 2 )

	As specied in part 3.1, in order to take into account the contribution of the
	source term	E ζ (f 0 -f 2 ), we simply propose to modify the intermediate states f * 1i+1/2j
	and f * 1ij+1/2 given in (6.64) and (6.65) such that
	f * 1i+1/2j = α i+1/2j f1i+1/2j -θ 1i+1/2j c i+1/2j (	∂f 0 ∂ζ	) i+1/2j -	Si+1/2j 2	(1 -α i+1/2j ) ,
						(6.76)

  7.2. Derivation of the modelwhere S 2 is the unit sphere. The complete derivation of the M 1 angular moments model in a moving frame is given in Appendix B. Removing the collisional operators contribution, the studied model reads

  (7.1) can be shown. Indeed, the reference coordinates system (t, x, v) and a new set of coordinates (t, x, ṽ) can be linked by the following relations

	x = Rx -st,	ṽ = Rv -s,	(7.16)
	for any constant vector s ∈ R d . Distribution function f in the moving frame is
	dened as

.15) Note that the Boltzmann collision operator or the BGK collision operator satisfy such properties. Using equation (7.15) the Galilean invariance property of the kinetic 7.3. Model properties equation

  7. Angular M 1 model in a moving reference frame

	equation of (7.7) by	m 2	ζ 2 and integrate in ζ to obtain the following internal energy
	equation
	∂ t (	1 2	0	+∞	g 0 ζ 2 dζ) + div x (	1 2	0	+∞	g 1 ζ 3 dζ + u	1 2	0	+∞	g 0 ζ 2 dζ)	(7.35)

  .4. Numerical scheme F k represents the discretisation proposed for the k th intermediate step and N is the number of intermediate step considered. Equation (7.52) rewrites under the form of

	a convex combination

  ∂ t g 1 + div x (ζg 2 + u ⊗ g 1 ) -∂ ζ g 2 .Derivation of the angular M 1 model in a moving frameThe angular M 1 model in a moving frame reads∂ t g 0 + div x (ζg 1 + ug 0 ) -∂ ζ du dt .g 1 + ζ∂ x u : g 2 = 0, ∂ t g 1 + div x (ζg 2 + u ⊗ g 1 ) -∂ ζ g 2

	
	  
	  	du dt	+ ζg 3	∂u ∂x	+	f 0 Id -f 2 ζ	du dt	+ (	∂u ∂x	g 1 -g 3	∂u ∂x	= 0.
	du dt	+ ζg 3	∂u ∂x	+	f 0 Id -f 2 ζ	du dt	+ (	∂u ∂x	g 1 -g 3	∂u ∂x	= 0.

B(B.1)

Sébastien GUISSET

Sébastien GUISSET

(3.7)Study of particle transport in plasmas

(3.15) Study of particle transport in plasmas

Study of particle transport in plasmas

5.6. Conclusion

Remerciements

Tout d'abord, je remercie Bruno Després et Francis Filbet d'avoir accepté de rapporter cette thèse. Je les remercie pour leurs remarques, commentaires, critiques,

Model properties

also enables to link the angular M 1 model derived from the kinetic equation (7.4) to the angular M 1 model derived from the kinetic equation (7.19). This point is detailed in the following result. The form of the M 1 angular moments model (7.7-7.9) expressed in the mean velocity frame is invariant by rotational and translational transformations. Using the fact that

and the equations (7.24), the following relations can be derived

and

Using the denition of g2 , we remark that

Then injecting (7.26-7.27) into the rst equation of (7.7) and using (7.28) and (7.23) a direct calculation gives

Conclusion / Perspectives

The present work aimed at the development of models and numerical methods for the study of the charged particles transport in a hot plasma. In this context angular moments models based on an entropy minimisation principle have been introduced.

Questions addressed

First of all, one needs to understand the limits of angular moments models when considering plasmas physics or neutral gas dynamics applications. Indeed, the angular integration performed leads to reduced models which can be inaccurate depending on the physical phenomenon studied. The objective is to obtain a better understanding of the validity regimes of angular moments models.

A second issue deals with the design and validation of collisional operators for angular moments models. Indeed, relevant kinetic physical collisional operators such as the Landau collisional operator or the Boltzmann collisional operator are nonlinear. Therefore the angular moments extraction is not straightforward and some approximations are required. Of course, the collisional operators proposed need to satisfy fundamental properties such as the entropy dissipation or the realisability preservation of the angular moments.

A third issue concerns the derivation of appropriate numerical methods for angular moments models. Since angular moments models can be seen as intermediate models between kinetic and hydrodynamics descriptions, one can hope to capture more accurately physical processes studied compared to hydrodynamics models while keeping a computational cost more aordable compared to the ones required for kinetic models. This makes these models naturally attractive to study the kinetic eects on long time scales. Therefore, suitable numerical schemes with correct asymptotic behaviour are required to access such regimes. The design of these adapted methods needs to be done in the framework of the asymptotic-preserving schemes and ensuring the fundamental properties of the model such as the preservation of the realisable sets.

The last question we address deals with the development of models and numerical methods for the study of the transport of multiple particle species. When studying the electrons transport, the motion of ions is generally neglected because of their large mass compared to the one of electrons. However, when considering long time regimes this approximation may not be valid and the study of the transport of both Appendix A

Derivation of the electron M 1 model

A.1 Derivation of the model for the angular moments f 0 and f 1

In this section we proove that in the case without collisional operators, the angular M 1 model writes under the following form

Before beginning the proof, we can compute the divergence of the magnetic part of the Lorentz force,

To proof these equalities, we can use duality approach.

If we set ζ = |p| = m|v(p)|γ(|p|) = mv(ζ)γ(ζ), and we denote by ϕ some test function. Here, we suppose that the test function ϕ depends only of ζ. Therefore, multiplying the kinetic equation by ϕ and integrating in p gives A.2. Angular moment extraction for the collision operators C ee and C ei from the isotropic part of f . In the remainder of this subsection, we will assume that f is isotropic, f = f (|v|) = f (ζ). This is a classical hypothesis useful in plasma physics.

We need introduce the notation Γ(v) which is the term into the divergence in v,

. Moreover if f is isotropic as we assume, Γ(v)v is isotropic. In fact, in this case Γ(v)v = g(|v|) doesnt depend on the direction of vector v. If we set ζ = |v|, and we denote by ϕ some test function, then,

and

•

Or Γ(v) • v involves the S(u)v matrix-vector product, from (A.9) we can obtain,

•

A. Derivation of the electron M 1 model

To achieve the computation of Γ(v) • v we need to know some scalar product, A v = S(u)v •v and A Ω = S(u)v •Ω . We introduce µ the cosine of that angle between of the vector Ω and v, such that Ω • v = ζµ, and we obtain,

, (A.10)

The comutation of the Γ(v) • v term can now be completed,