Stéphane Brull
email: stephane.brull@u-bordeaux.fr

Xavier Lhébrard
email: xavier.lhebrard@u-bordeaux.fr

Bruno Dubroca
email: bruno.dubroca@u-bordeaux.fr

Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic field

Keywords: BGK models, hydrodynamic limit, relaxation method, nonconservative hyperbolic system, discrete entropy inequalities, discrete entropy minimum principle Mathematics Subject Classification: 65M08, 35L60, 65M12

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Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic field

Introduction

The present paper is devoted to the formal derivation and the approximation of the nonconservative bitemperature Euler system with transverse magnetic field.

This fluid model consists of four conservation equations for mass, momentum (two components), magnetic field, and two nonconservative equations for energies. Physically, this model describes the interaction of a mixture of one species of ions and one species of electrons in thermal nonequilibrium, subjected to a transverse variable magnetic field. The pressure of each species is supposed to satisfy a gamma-law with its own γ constant. Moreover the system owns a 1 dissipative strictly convex entropy, closely related to the classical entropy for the Euler sytem [START_REF] Tadmor | A minimum entropy principle in the gas dynamics equations[END_REF].

Solving nonconservative hyperbolic systems is a delicate problem because the definition of weak solutions remains unclear. In order to define nonconservative products, Dal Maso, Lefloch and Murat proposed in [START_REF] Maso | Definition and weak stability of nonconservative products[END_REF] a new theory based on the definition of family of paths. In [START_REF] Pares | Numerical methods for nonconservative hyperbolic systems: a theoretical framework[END_REF], path-conservative schemes are defined by using the concepts developed in [START_REF] Maso | Definition and weak stability of nonconservative products[END_REF]. However, it is shown, in [START_REF] Abgrall | A comment on the computation of nonconservative products[END_REF], that even if the correct path is known, the numerical solution can be far from the expected solution. Let us mention some works dealing with nonconservative Euler type system, where the magnetic fields are neglected. In [START_REF] Berthon | Why many theories of shock waves are necessary : Kinetic relations for nonconservative systems[END_REF], the authors consider an hyperbolic system having n -p equations in a conservative form, the remaining p equations being nonconservative. In [START_REF] Pelanti | A mixture-energy-consistent six-equation two phase numerical model for fluids with interfaces, cavitation and evaporation waves[END_REF], the authors use a Roe solver, and an HLLC solver which neglects the nonconservative part of the system. They validate the approach by comparing their results to theoretical temperatures/pressures curves ( [START_REF] Saurel | Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures[END_REF], [START_REF] Saurel | Modelling phase transition in metastable liquids: application to cavitating and flashing flows[END_REF]). In [START_REF] Coquel | Numerical methods for weakly ionized gas[END_REF], the authors assume that the electronic entropy is conserved by all weak solutions including shocks and the system is approached by a system of conservation laws.

Despite these difficulties, the nonconservative formulation is physically relevant. Indeed we show that the bitemperature MHD system admits an underlying conservative kinetic model, which consists of a BGK model coupled with Maxwell equation in the quasi-neutral regime. This result generalizes the approach of [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF] in the case of a transverse magnetic field. This BGK model possesses different interspecies collision frequencies in order to take into account discrepancies in the masses of the particles. This point has been in particular mentioned in [START_REF] Greene | Improved Bhatnagar-Gross-Krook model of electron-ion collisions[END_REF], where a conservative formulation is suggested. Hence following this idea we perform an hydrodynamic limit and we formally get the nonconservative MHD system.

In another way, the present bitemperature MHD system is approached numerically. An important issue in multidimensional simulations is to minimize the numerical viscosity by using accurate solvers, in particular on contact discontinuities; while being robust, see for example [START_REF] Fuchs | Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations[END_REF][START_REF] Waagan | A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability, implementation and tests[END_REF][START_REF] Balsara | Multidimensional HLLC Riemann solver for unstructured meshes with application to Euler and MHD flows[END_REF].

If dependency is only one spatial variable x, the nonconservative two species Euler equations with transverse magnetic field are given by:

∂ t ρ + ∂ x ρu 1 = 0, (1.1)
∂ t (ρu 1 ) + ∂ x (ρu 2 1 + p e + p i + B 2 3 /2) = 0, (1.2)

∂ t (ρu 2 ) + ∂ x (ρu 2 u 1 ) = 0, (1.3)
∂ t E e + ∂ x u 1 E e + p e + c e B 2 3 /2 -u 1 ∂ x (c i p e -c e p i ) = S ei , (1.4)

∂ t E i + ∂ x u 1 E i + p i + c i B 2 3 /2 +u 1 ∂ x (c i p e -c e p i ) = S ie , (1.5)
∂ t B 3 + ∂ x (u 1 B 3 ) = 0, (1.6)
where ρ = ρ e + ρ e is the total density of the plasma, u = (u 1 ,u 2 ) is the average velocity of the plasma, B 3 is the vertical component of the magnetic field. We denote by T e , T i the temperatures of electrons and ions, ρ e = n e m e , ρ i = n i m i the densities of the electrons and ions, n e , n i the concentrations of the electrons and ions, m e , m i , the masses of the electrons and ions particles. The source terms for exchanges between electron and ion species S ei and S ie are defined by

S ei = ν 1,ei (T i -T e )+ ν 2,ei (∂ x B 3 ) 2 , S ie = -ν 1,ei (T i -T e )+ ν 2,ie (∂ x B 3 ) 2 , (1.7)
where ν 1,ei ≥ 0 is the frequency exchange between temperatures, ν 2,ei , ν 2,ie ≥ 0 are frequencies due to the drift velocity u 2 ± ∂ x B 3 . These quantities we will be defined later. The concentrations n e and n i are related by the average ionization number Z = n e /n i ≥ 1 which is considered here constant. This implies that the mass fractions c α = ρ α /ρ, α = e,i are also constant and c e and c i write

c e = Zm e m i + Zm e , c i = 1 -c e .
The electrons and ions pressures and temperatures are related by

p e = n e k B T e , p i = n i k B T i ,
where k B is the Boltzmann constant. The internal energies are given by e = k B T e m e (γ e -1)

, i = k B T i m i (γ i -1)
,

where γ e , γ i are constant numbers belonging to the interval [START_REF] Abgrall | A comment on the computation of nonconservative products[END_REF][START_REF] Bouchut | A new model for shallow viscoelastic fluids[END_REF]. The quantity

E α = ρ α ε α + 1 2 ρ α (u 2 1 + u 2 2 ) + c α B 2 3 /2
is the total energy associated to each species α = e, i. This model is closely related to the model derived in [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF], the novelty here is to take as an additional variable the vertical component of the magnetic field B 3 .

The homogeneous system associated to (1.1) -(1.6) is endowed with an entropy inequality: ∂ t (-ρ (s e + s i )) + ∂ x (-ρu (s e + s i )) ≤ 0, (1.8) where s α , α = e,i is the classical specific entropy

s α (ρ, ε α ) = c α m α (γ α -1) ln (γ α -1)ε α ρ (γα-1) + C. (1.9)
Here C is a nonnegative constant.

The homogeneous system associated to (1.1) -(1.6) is an hyperbolic system. The three eigenvalues of the system are u, u -(γ e p e + γ i p i + B 2

3 ) /ρ and u + (γ e p e + γ i p i + B 2

3 ) /ρ of multiplicity four, one and one, respectively. The eigenvalue u is linearly degenerate and the associated contact discontinuities is called a material contact. The jump relation associated is as follows. Across a material contact, the quantities u 1 and p e + p i + B 2 3 /2 are constant. We note here that unlike the 7-wave full MHD system [START_REF] Bouchut | A multiwave approximate Riemann solver for ideal MHD based on relaxation I -Theoretical framework[END_REF] or the 5-wave shallow water MHD system [START_REF] Bouchut | A 5-wave relaxation solver for the shallow water MHD system[END_REF], the transverse magnetic system has a 3-wave structure and does not admit Alfven waves.

A finite volume scheme for this homogeneous quasilinear system is classically built following Godunov's approach, by considering piecewise constant approximation of

U = ρ, ρu 1 , ρu 2 , E e + c e B 2 3 /2, E i + c i B 2 3 /2, B 3 ∈ R 6 (1.10)
and invoking an approximate Riemann solver at the interface between two cells, see for example [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] or [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources[END_REF]Section 2.3]. A difficulty is however that the system is not conservative. In this paper we apply the relaxation approach of [START_REF] Bouchut | Entropy satisfying flux vector splitting and kinetic BGK models[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] Bouchut | A multiwave approximate Riemann solver for ideal MHD based on relaxation I -Theoretical framework[END_REF][START_REF] Bouchut | A multiwave approximate Riemann solver for ideal MHD based on relaxation II -numerical implementation with 3 and 5 waves[END_REF][START_REF] Coquel | Relaxation of fluid systems[END_REF] to the bitemperature transverse MHD system, in order to get an approximate Riemann solver that is entropy preserving, ensuring robustness, while being exact on isolated material contacts. The relaxation system is of Suliciu type as introduced in [START_REF] Suliciu | Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations[END_REF], and the approximate Riemann solver belongs to the family of HLLC solvers, as in [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[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] Berthon | An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions[END_REF][START_REF] Sangam | An HLLC scheme for ten-moments approximation coupled with magnetic field[END_REF][START_REF] Berthon | A local entropy minimum principle for deriving entropy preserving schemes[END_REF][START_REF] Fuchs | Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations[END_REF][START_REF] Balsara | Multidimensional HLLC Riemann solver for unstructured meshes with application to Euler and MHD flows[END_REF].

Let us emphasize that the discrete entropy inequalities are established by arguing a suitable extension of technique introduced in [START_REF] Berthon | A local entropy minimum principle for deriving entropy preserving schemes[END_REF], also used in the context of solving the ten moment equations [START_REF] Berthon | An entropy preserving relaxation scheme for ten-moments equations with source terms[END_REF]. Put in other words, at the discrepancy with the previous works [START_REF] Bouchut | Entropy satisfying flux vector splitting and kinetic BGK models[END_REF][START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources[END_REF], an entropy extension is obtained implicitly and thus the entropy dissipation is not evaluated in the proposed discrete entropy inequalities.

This work is organized as follows. In the next section we present the kinetic model involved in this paper. Firstly we consider the Vlasov-BGK model from which the MHD system is derived. Starting from an ad-hoc scaling the construction of the MHD system is performed. In section 3 we derive a relaxation scheme. In section 4 we establish the stability of our scheme. Numerical tests are performed in section 5 to illustrate both accuracy and robustness of the proposed scheme in 1D.

Kinetic model

Kinetic models are described by the distribution function f α of each species depending on the time variable t ∈ R + , on the positions x ∈ R 3 and on the velocity v ∈ R 3 . The macroscopic quantities can be obtained by extracting moments on these distribution function w.r.t. the velocity variable. Indeed density, velocity and total energy of the species α can be defined as

n α = R 3 f α dv, u α = 1 n α R 3 vf α dv, E α = 3 2 ρ α k B m α T α + 1 2 ρ α u 2 α = R 3 m α |v| 2 2 f α dv,
where m α is the mass particle, ρ α = m α n α , and T α is the temperature of species α. The current in the plasma j and the total charge ρ are defined by

ρ = R 3
(q e f e + q i f i )dv = n e q e + n i q i , (2.1)

j = R 3
v(q e f e + q i f i )dv = n e q e u e + n i q i u i .

(2.2)

Kinetic model for a transverse magnetic field

In this section, we present the BGK model developed in this paper describing a plasma interacting with an electric field E = (E 1 , E 2 , 0) ∈ R 3 and a magnetic field B = (0, 0, B 3 ) ∈ R 3 . The model writes

∂ t f α + v 1 ∂ x f α + q α m α (E 1 + B 3 v 2 ) ∂f α ∂v 1 + q α m α (E 2 -B 3 v 1 ) ∂f α ∂v 2 = 1 τ α (M α (f α ) -f α ) + 1 τ αβ M α (f α , f β ) -f α , (2.3)
with τ α > 0, τ αβ > 0. We denote by 1 τα the collision frequency for the interaction between α particles and 1 τ αβ the collision frequency for the ion/electron interaction. The frequencies τ ei and τ ie being of order of the mass ratios, [START_REF] Greene | Improved Bhatnagar-Gross-Krook model of electron-ion collisions[END_REF] suggested to take τ ei = τ ie . The quantity q α is the charge of the species α. Moreover we define the velocity and the temperature of the mixture by

u = ρ e u e + ρ i u i ρ e + ρ i , T = 1 2 α ρ α (u 2 α -u 2 ) + 3 2 α n α k B T α 3 2 nk B . (2.4)
Let us highlight that one can check that the temperature of the mixture satisfies T ≥ 0. Here fictitious velocity and temperature u # , T # introduced in [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF] are defined by

u # = ρ e τ ei u e + ρ i τ ie u i ρ e τ ei + ρ i τ ie , T # = 1 2 α ρα τ αβ (u 2 α -(u # ) 2 ) + 3 2 k B α nα τ αβ T α 3 2 k B ne τei + ni τie . (2.5)
The two Maxwellian distribution functions M α and M α are defined by:

M α (f α ) = n α (2πk B T α /m α ) 3/2 exp - (v 1 -u 1,α ) 2 + (v 2 -u 2,α ) 2 + v 2 3 2k B T α /m α , (2.6)
and

M α (f α ) = n α (2πk B T # /m α ) 3/2 exp - (v 1 -u # 1 ) 2 + (v 2 -u # 2 ) 2 + v 2 3 2k B T # /m α . (2.7)
The two Maxwellian distributions satisfy the constraints

R 3 (M α -f α )   m α m α v m α v 2 2   dv = 0, R 3 (M α -f α )dv = 0 (2.8) and R 3 1 τ ei (M α -f α ) m α v m α v 2 2 + 1 τ ie (M α -f α ) m α v m α v 2 2 dv = 0. (2.9)
This model is coupled with the Maxwell equations which write in the transverse magnetic field

∂ x E 1 = ρ ε 0 , ∂ t E 1 = - j 1 ε 0 , ∂ t B 3 + ∂ x E 2 = 0, ∂ t E 2 -c 2 ∂ x B 3 = - j 2 ε 0 , (2.10)
where j = (j 1 ,j 2 ) has been defined in (2.2), c represents the speed of light and ε 0 is the vacuum permittivity.

Hydrodynamic limit

Scaling on the one dimensionnal BGK model

In order to use a Chapman-Enskog procedure, we introduce a small parameter κ and the BGK model (2.3) is rescaled as:

∂ t f α + v 1 ∂ x f α + q α m α (E 1 + B 3 v 2 ) ∂f α ∂v 1 + q α m α (E 2 -B 3 v 1 ) ∂f α ∂v 2 = 1 κ (M α (f α ) -f α ) + 1 τ αβ M α (f α , f β ) -f α . (2.11)
Moreover this model is coupled with Maxwell equations (2.10), which are rescaled as:

∂ x E 1 = ρ κ 2 , ∂ t E 1 = - j 1 κ 2 , ∂ t B 3 + ∂ x E 2 = 0, ∂ t E 2 + 1 ε 2 ∂ x B 3 = - j 2 κ 2 .
(2.12)

Derivation of Euler equations

In a previous work [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF], the nonconservative bitemperature Euler system has been established as the hydrodynamic limit of a conservative kinetic system. In particular the authors deal with the case B 3 = 0, which implies that the current j = (j 1 , j 2 ) at equilibrium satisfies j = 0 and enables to express the electric field as a combination of first order spatial derivatives of the variables ρ, T e , T i . The novelty in our case is that we have j 1 = 0 and j 2 = -∂ x B 3 = 0. Thus we have a slightly different approach and we proceed in two steps.

• First we perform an hydrodynamic limit of (2.11) -(2.12). The first component E 1 will behave as in [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF] and will be expressed as a combination of first order spatial derivatives of the variables ρ, u 2 , T e , T i , B 3 . However, the evolution of E 2 will be given by the second order PDE (2.21).

• Second we use the smallness of the mass ratio m e /m i which enables to simplify the equation satisfied by E 2 .

After those two steps we recover the bitemperature Euler system with transverse magnetic field (1.1)-(1.6).

Proposition 2.1. The kinetic conservative system (2.11) -(2.7) converges formally to the following system

∂ t ρ + ∂ x ρu 1 = 0, (2.13)
∂ t (ρu 1 ) + ∂ x (ρu 2 1 + p e + p i + B 2 3 /2) = 0, (2.14)
∂ t (ρu 2 ) + ∂ x (ρu 2 u 1 ) = 0, (2.15)
∂ t E e + c e B 2 3 /2 + ∂ x (u 1 (E e + p e ) + c e E 2 B 3 ) -u 1 ∂ x (c i p e -c e p i ) -(E 2 -u 1 B 3 ) (q e n e u 2 + (c e -c i )∂ x B 3 ) = S ei , (2.16
)

∂ t E i + c i B 2 3 /2 + ∂ x (u 1 (E i + p i ) + c i E 2 B 3 ) + u 1 ∂ x (c i p e -c e p i ) + (E 2 -u 1 B 3 ) (q e n e u 2 + (c e -c i )∂ x B 3 ) = -S ei , (2.17)
∂ t B 3 + ∂ x E 2 = 0, (2.18)
where

E α = ρ α ε α + 1 2 ρ α (u 2 1 + u 2 2,α ), α = e,i (2.19)
with u 2,α defined by (2.33). Moreover the first component of electric field E 1 is explicitly given by the Ohm's law

c i ∂ x p e -c e ∂ x p i + (c i -c e )∂ x (B 2 3 /2) = q e n e (E 1 + u 2 B 3 ) , (2.20)
and the second component of electric field E 2 satisfies the following equation:

∂ t (∂ x B 3 ) + ∂ x (u 1 ∂ x B 3 ) + (q e n e ) 2 ρ ρ e ρ i (E 2 -B 3 u 1 ) = - ∂ x B 3 τ ie c e + τ ei c i . (2.21)
In addition the source term S ei is defined by

S ei = ν 1,ei (T i -T e ) + ν 2,ei (∂ x B 3 ) 2 + ν 3,ei u 2 ∂ x B 3 , (2.22)
with the coefficients ν 1,ei , ν 2,ei , ν 3,ei given by

ν 1,ei = 3 2 k B n e n i τ ie n e + τ ei n i , (2.23)
ν 2,ei = c e c i ρ (c e -c i ) n i c i τ 2 ei + n e c e τ 2 ie -2c e c i (n i -n e )τ ei τ ie (τ ie n e + τ ei n i ) (q e n e ) 2 (τ ie c e + τ ei c i ) 2 , (2.24)
and ν 3,ei = c e c i ρ (q e n e ) (c e τ ie + c i τ ei ) .

(2.25)

Note that with Proposition 2.1, we do not recover yet (1.1) - (1.6). Using the following approximation we are able to simplify the previous system and recover the bitemperature Euler system with transverse magnetic field. Proposition 2.2. We consider the previous system (2.13)- (2.25). Using the approximation of a small mass ratio between electron and ion species m e /m i = κ << 1, the equation (2.21) formally converges to

E 2 = B 3 u 1 .
(

ν 2,ei = c e c i ρ (τ ie n e + τ ei n i ) (τ ie c e + τ ei c i ) + (c i n e -c e n i ) τ ei τ ie 2(q e n e ) 2 (τ ie n e + τ ei n i ) (τ ie c e + τ ei c i ) 2 , (2.27)
ν 2,ie = c e c i ρ (τ ie n e + τ ei n i ) (τ ie c e + τ ei c i ) -(c i n e -c e n i ) τ ei τ ie 2(q e n e ) 2 (τ ie n e + τ ei n i ) (τ ie c e + τ ei c i ) 2 .
(2.28)

Remark 2.3. In the case B 3 = 0 we recover all the results of [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF]. Concerning the Ohms law, compared to [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF], in (2.20) we have additional terms

(c e -c i )∂ x (B 2 3 /2) + q e n e u 2 B 3 .
Proof of Proposition 2.1. For this proof, for any g belonging in

L 1 2 = {g ∈ L 1 /(1 + v 2 )f ∈ L 1 }, we use the notation g = R 3
gdv, the equilibrium states of (2.11) write

f α = M α , α = e,i, ρ = 0, j 1 = 0, ∂ t B 3 + ∂ x E 2 = 0, j 2 = -∂ x B 3 .
This system implies that n e q e + n i q i = 0, n e q e u 1,e + n i q i u 1,i = 0,

∂ t B 3 + ∂ x E 2 = 0, n e q e u 2,e + n i q i u 2,i = -∂ x B 3 ,
which is equivalent to n e q e + n i q i = 0, (2.29)

u 1,e = u 1,i = u, (2.30
)

n e q e (u 2,e -u 2,i ) = -∂ x B 3 , (2.31)
∂ t B 3 + ∂ x E 2 = 0, (2.32)
with u defined by (2.4). Moreover combining (2.4) and (2.31) we compute

     u 2,e = u 2 - c i q e n e ∂ x B 3 , u 2,i = u 2 - c e q i n i ∂ x B 3 .
(2.33) Next we establish the hydrodynamic limit associated to the previous equilibrium. Hence f α writes

f α = M α + κg α , (2.34)
with

R 3 m α 1 |v| 2 g α = 0, R 3 m α v 1 v 2 g α = 0. (2.35)
Plugging (2.34) into (2.11), we compute g α as follows

∂ t M α + v 1 ∂ x M α + q α m α (E 1 + B 3 v 2 ) ∂ v1 M α + q α m α (E 2 -B 3 v 1 ) ∂ v2 M α = -g α + 1 τ αβ M α -M α + O(κ).
(2.36)

From here we will take the moments of the previous equation. The procedure we use here is very similar to the one detailed in [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF]. So we skip some details of computation.

Our goal is to obtain the system (2.13)-(2.18) with E 1 , E 2 defined by (2.20), (2.21), respectively. a) Firstly we proceed similarly to [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF]. Multiplying (2.36) by m α , integrating w.r.t. v and by summing the two equations for electrons and ions gives (2.13).

b) Then multiplying (2.36) by m α v 1 and integrating w.r.t. v leads to

∂ t (ρ α u 1 ) + ∂ x (ρ α u 2 1 + p α ) -q α n α (E 1 + B 3 u 2,α ) = 0. (2.37)
By summing for electrons and ions (2.37) and using (2.29), (2.30) we get (2.14). Next considering as in [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF] the nonconservative form of (2.37) with the additional term qαnα ρα B 3 u 2,α , we get the generalized Ohm's law (2.20) for E 1 .

c) Then we multiply by m α v 2 and we integrate w.r.t. v:

∂ t (ρ α u 2,α ) + ∂ x (ρ α u 2,α u 1 ) -q α n α (E 2 -B 3 u 1 ) = ρ α τ αβ u # 2 -u 2,α . (2.
1 τ αβ u # 2 -u 2,α = c β c α τ βα + c β τ αβ ∂ x B 3 q α n α . (2.39)
Then we multiply (2.38) by q α /m α and we use (2.39) to get

∂ t (q α n α u 2,α ) + ∂ x (q α n α u 2,α u 1 ) - (q α n α ) 2 ρ α (E 2 -B 3 u 1 ) = c β ∂ x B 3 c α τ βα + c β τ αβ .
(2.40) Finally using (2.29) and q e n e u 2,e + q i n i u 2,i = -∂ x B 3 , we sum for electrons and ions (2.40) and we obtain (2.21). d) Then we multiply by m α |v| 2 2 and we integrate w.r.t. v:

∂ t E α + ∂ x (u 1 (E α + p α )) -q α n α (E 1 + B 3 u 2,α ) u 1 -q α n α (E 2 -B 3 u 1 ) u 2,α = S, (2.41) whith E α defined by (2.19) and S = 1 τ αβ m α |v| 2 2 M α -M α . (2.42)
Using (2.20) and (2.33) we get

∂ t E α + ∂ x (u 1 (E α + p α )) + u 1 (c α ∂ x p β -c β ∂ x p α ) + u 1 c α B 3 ∂ x B 3 -(E 2 -B 3 u 1 ) (q α n α u 2 -c β ∂ x B 3 ) = S. (2.43)
Moreover, we multiply (2.32) by c α B 3 and we get

∂ t (c α B 2 3 /2) + ∂ x (c α B 3 E 2 ) -c α E 2 ∂ x B 3 = 0. (2.44)
Adding (2.43) and (2.44) we get, for α = β

∂ t E α + c α B 2 3 /2 + ∂ x (u 1 (E α + p α ) + c α E 2 B 3 ) + u 1 (c α ∂ x p β -c β ∂ x p α ) -(E 2 -B 3 u 1 ) (q α n α u 2 + (c α -c β )∂ x B 3 ) = S. (2.45)
At this point we deal with S defined by (2.42). Using (2.6) and (2.7) we write S = 3 2

k B n α τ αβ T # -T α + 1 2 ρ α τ αβ (u # 2 ) 2 -u 2 2,α . (2.46)
Next by using the definition of T # (2.5) in (2.46), we get

S = 3 2 k B n α n β τ βα n α + τ αβ n β (T β -T α ) + 1 2 n α n β τ βα n α + τ αβ n β m α (u # 2 ) 2 -u 2 2,α -m β (u # 2 ) 2 -u 2 2,β . (2.47) Now we deal with (u # 2 ) 2 -u 2 2,α . First using (2.33) in the definition of u # in (2.5) we write u # 2 = u 2 + c α c β ∂ x B 3 (τ αβ -τ βα ) (q α n α ) (c α τ βα + c β τ αβ ) .
Moreover using (2.33) we can compute u # 2 -u 2,α and u # 2 + u 2,α and after some simple computations we get that

m α (u # 2 ) 2 -u 2 2,α = 2m α c β τ αβ q α n α (c α τ βα + c β τ αβ ) u 2 ∂ x B 3 + m α c 2 β τ αβ ((c α -c β )τ αβ -2c α τ βα ) (q α n α ) 2 (c α τ βα + c β τ αβ ) 2 (∂ x B 3 ) 2 .
At this point we sum last equality for electron and ion species and we get 1 2 Proof of Proposition 2.2. We use Proposition 2.1 and we deal with the system (2.13) -(2.21). In addition we use the small mass ratio assumption κ = m e /m i . We also use the constant ionization number Z = n e /n i and we get

n e n i τ ie n e + τ ei n i m e (u # 2 ) 2 -u 2 2,e -m i (u # 2 ) 2 -u 2 2,i = ν 2,ei (u 2 ∂ x B 3 ) + ν 3,ei (∂ x B 3 )
ρ ρ e ρ i = n e m e + n i m i n e m e n i m i = (κZ + 1) (m i n i )κZ = O 1 κ . ( 2
∂ t E e + c e B 2 3 /2 + ∂ x u 1 E e + c e B 2 3 /2 + p e + c e B 2 3 /2 -u 1 ∂ x (c i p e -c e p i ) = S ei ,
(2.50)

∂ t E i + c i B 2 3 /2 + ∂ x u 1 E i + c i B 2 3 /2 + p i + c i B 2 3 /2 + u 1 ∂ x (c i p e -c e p i ) = -S ei ,
(2.51)

∂ t B 3 + ∂ x (u 1 B 3 ) = 0, ( 2
2 u 1 = - c i q e n e ν 3,ei (∂ x B 3 ) 2 + ν 3,ei u 2 ∂ x B 3 . (2.55)
On the other hand, we combine (2.13), (2.15) and we obtain

∂ t ρ e u 2 2 2 + ∂ x ρ e u 2 2 2 u 1 = 0. (2.56)
Next following (2.53), we add (2.50), (2.56) and we substract (2.55), it leads to (1.4). Moreover we obtain the following formulas for ν 1,ei , ν 2,ei , ν 2,ie :

ν 1,ei = ν 1,ei , ν 2,ei = ν 2,ei + c i q e n e
ν 3,ei , ν 2,ie = -ν 2,ei + c e q e n e ν 3,ei .

Finally using (2.24), (2.25) one recovers (2.27).

Numerical approximation

The numerical approximation we use has two steps:

• First step. We use an approximate Riemann solver for the homogeneous system. Let U n+1/2 be the obtained solution.

• Second step. We take the temperatures interaction into account implicitly: the approximate solution of system at time t n+1 is defined by

ρ n+1 = ρ n+1/2 , u n+1 1 = u 1 n+1/2 , u n+1 2 = u 2 n+1/2 , B n+1 3 = Bn+1/2 3 and    E n+1 e = E e n+1/2 + ∆t S n+1 ei , E n+1 i = E i n+1/2 -∆t S n+1 ei , where S n+1 ei = ν 1,ei (T n+1 i -T n+1 e ) + ν 2,ei (∂ x B 3 ) n+1 2 .
This system is linear and owns an explicit solution.

From here we will focus on the first step of the numerical approximation. We will explain how to derive an efficient approximate Riemann solvers for the homogeneous part of the system (1.1) - (1.6).

In order to get those approximate Riemann solvers, we use a standard relaxation approach, introduced in [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources[END_REF] for the gas dynamic equations. This approach has been developed in [START_REF] Bouchut | A multiwave approximate Riemann solver for ideal MHD based on relaxation I -Theoretical framework[END_REF] for the MHD equations, in [START_REF] Bouchut | A new model for shallow viscoelastic fluids[END_REF] for shallow elastic fluids, in [START_REF] Bouchut | A 5-wave relaxation solver for the shallow water MHD system[END_REF] for shallow water MHD equations, in [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF] for the bitemperature Euler system. An abstract general description can be found in [START_REF] Coquel | Relaxation of fluid systems[END_REF], and related works are [START_REF] Chalons | Well-balanced time implicit formulation of relaxation schemes for the Euler equations[END_REF][START_REF] Chalons | Modified Suliciu relaxation system and exact resolution of isolated shock waves[END_REF]. This technique enables to naturally handle the entropy inequality (1.8), and also preserves the positivity of density and internal energies.

Relaxation approach

Approximate Riemann solver

We introduce new variables π e ,π i , the relaxed pressures, and a intended to parametrize the speed. The form of the relaxation system is as follows,

∂ t ρ + ∂ x ρu 1 =0, (3.1)
∂ t (ρu 1 ) + ∂ x (ρu 2 1 + π e + π i + B 2 3 /2) =0, (3.2)
∂ t (ρu 2 ) + ∂ x (ρu 2 u 1 ) =0, (3.3)
∂ t E e + ∂ x u 1 E e + π e + c e B 2 3 /2 -u 1 ∂ x (c i π e -c e π i ) =0, (3.4)
∂ t E i + ∂ x u 1 E i + π i + c i B 2 3 /2 + u 1 ∂ x (c i π e -c e π i ) =0, (3.5)
∂ t B 3 + ∂ x (u 1 B 3 ) =0, (3.6)
∂ t (ρπ e ) + ∂ x (ρu 1 π e ) + c e a 2 -ρB 2 3 ∂ x u 1 =0, (3.7)
∂ t (ρπ i ) + ∂ x (ρu 1 π i ) + c i a 2 -ρB 2 3 ∂ x u 1 =0, (3.8)
∂ t (ρa) + ∂ x (ρau 1 ) =0.

(3.9)

The approximate Riemann solver can be defined as follows, starting from left and right values U l , U r at an interface.

• Solve the Riemann problem of the system (3.1)-(3.9) with initial data obtained by completing U l , U r by the equilibrium relations

π e,L = p e,L ≡ (γ e -1)ρ L ε e,L , π i,L = p i,L ≡ (γ i -1)ρ L ε i,L , π e,R = p e,R ≡ (γ e -1)ρ R ε e,R , π i,R = p i,R ≡ (γ i -1)ρ R ε i,R , (3.10)
and with suitable positive values of a l , a r that will be discussed further on, essentially in Section 3.2.1.

• Retain in the solution only the variables ρ, ρu 1 , ρu 2 , E e , E i , B 3 . The result is a vector called R(x/t, U l ,U r ).

Intuitively, the solver is consistent because of the equations (3.1)-(3.6), that are consistent with the equations (1.1)- (1.6). The specific values used for a do not play any role in this consistency.

The accuracy of the solver on isolated contacts is described by the following lemma.

Lemma 3.1. The approximate Riemann solver R(x/t, U l ,U r ) solves exactly the material contact discontinuities.

Proof. Material contacts are solutions to the bitemperature MHD system (1.1) -(1.6) with u 1 , p e + p i + B 2 3 /2 constant. These solutions are obviously solutions to the relaxation system (3.1) - (3.8) with π e = p e , π i = p i . Thus for these data, R coincides with the exact solver, which concludes the proof.

Godunov scheme

Following the Godunov approach, the numerical scheme can be defined by the approximate Riemann solver as follows. We consider a mesh of cells

(x i-1/2 , x i+1/2 ), i ∈ Z, of length ∆x i = x i+1/2 -x i-1/2
, discrete times t n with t n+1 -t n = ∆t, and cell values U n i approximating the average of U over the cell i at time t n . We can then define an approximate solution U appr (t,x) for t n ≤ t < t n+1 and x ∈ R by

U appr (t,x) = R( x -x i+1/2 t -t n , U n i , U n i+1 ) for x i < x < x i+1 , (3.11)
where

x i = (x i-1/2 + x i+1/2 )/2.
This definition is coherent under a half CFL condition, formulated as

x/t < - ∆x i 2∆t ⇒ R(x/t, U i , U i+1 ) = U i , x/t > ∆x i+1 2∆t ⇒ R(x/t, U i , U i+1 ) = U i+1 . (3.12)
The new values at time t n+1 are defined by

U n+1 i = 1 ∆x i x i+1/2 x i-1/2
U appr (t n+1 -0, x)dx.

Notice that it is only in this averaging procedure that the choice of the particular pseudo-conservative variable U as (1.10) is involved. We can follow the computations of [9, Section 2.3], the only difference being that the system is not conservative. We obtain the update formula

U n+1 i = U n i - ∆t ∆x i (F l (U n i , U n i+1 ) -F r (U n i-1 , U n i )), (3.13)
where

F l (U l ,U r ) = F (U l ) - 0 -∞ (R(ξ, U l , U r ) -U l )d ξ, F r (U l ,U r ) = F (U r ) + ∞ 0 (R(ξ, U l , U r ) -U r ) dξ. (3.14)
The variable ξ stands for x/t, and the pseudo-conservative flux is chosen as (3.15), the fourth and fifth components could be chosen differently since the two energy equations in our system are not conservative. We can remark that the choice of F has no influence on the update formula (3.13).

F (U ) ≡ ρu 1 , ρu 2 1 + p e + p i + B 2 3 /2, ρu 1 u 2 , u 1 E e + p e + c e B 2 3 /2 , u 1 E i + p i + c i B 2 3 /2 , u 1 B 3 . (3.15) In

Subcharacteristic condition

We are interested on necessary stability conditions for smooth solutions. In order to address such an issue, we consider (3.7), (3.8) in which we add righthand-sides:

∂ t (ρπ e ) + ∂ x (ρu 1 π e ) + c e a 2 -ρB 2 3 ∂ x u 1 = ρ κ (p e -π e ) , (3.16)
∂ t (ρπ i ) + ∂ x (ρu 1 π i ) + c i a 2 -ρB 2 3 ∂ x u 1 = ρ κ (p i -π i ) . ( 3
∂ t (ρp i ) + ∂ x (ρp i u 1 ) + γ i ρp i ∂ x u 1 = 0. (3.19)
Next when the relaxation parameter κ tends to zero one has

π α = p α + δκ + O(κ 2 ), α = e,i. (3
δ = c α ρB 2 3 + γ α ρp α -a 2 ∂ x u 1 + O(κ).
Hence the stability condition needed by the parameter a is a 2 > ρB 2 3 + γ e ρp e + γ i ρp i .

(3.21)

Intermediate states

By using the variable V = (ρ,u 1 ,u 2 ,B 3 , ε e , ε i , π e , π i , a), one can easily compute the eigenvalues of the system (3.1) -(3.9). They read as {u -a/ρ, u, u + a/ρ}, where u is an eigenvalue of order 7. All the fields are linearly degenerated. As a consequence, Rankine-Hugoniot conditions are well-defined (the weak Riemann invariants do not jump through the associated discontinuity), and are equivalent to any conservative formulation.

In the solution to the Riemann problem, the speeds corresponding to the previous eigenvalues will be denoted by

Σ 1 < Σ 2 < Σ 3 . (3.22)
Thus we get a 3-wave solver with two intermediate states. The variables take the values "L" for x/t < Σ 1 , "L*" for Σ 1 < x/t < Σ 2 , "R*" for Σ 2 < x/t < Σ 3 , "R" for Σ 3 < x/t, see Figure 1. There are 7 strong Riemann invariants for the central wave (i.e. quantities that lie in the kernel of

∂ t + u 1 ∂ x ), which are u 2 , a, B 3 ρ , w 1,e , w 1,i , w 2,e , w 2,i , (3.23)
with w 1,e , w 1,i , w 2,e , w 2,i defined by

w 1,α = π α + c α B 2 3 /2 + a 2 1 c α ρ , w 2,α = ε α + B 2 3 2ρ - π α + c α B 2 3 /2 2 2(c α a 1 ) 2 . [!htbp] x U R x/t = Σ 3 U * L U * R x/t = Σ 2 U L x/t = Σ 1 0
u 1 , π, (3.24)
with π = π e + π i + B 2 3 /2. We shall denote u * 1 the common value of u 1 on the left and on the right of the central wave. Computations using the Riemann invariants (3.23), (3.24) give the following intermediate states:

       1 ρ * L = 1 ρ L + a R (u 1,R -u 1,L ) + π L -π R a L (a L + a R ) ,
1 ρ * R = 1 ρ R + a L (u 1,R -u 1,L ) + π R -π L a R (a L + a R ) , u * 1 = a R u 1,R + a L u 1,L + π L -π R a L + a R , (3.25)
B * 3,L = ρ * L ρ L B 3,L , B * 3,R = ρ * R ρ R B 3,R , for α = e,i we have              π * α,L = π α,L + c α B 2 3,L 2 1 - ρ * L ρ L 2 + a 2 L c α 1 ρ L - 1 ρ * L , π * α,R = π α,R + c α B 2 3,R 2 1 - ρ * R ρ R 2 + a 2 R c α 1 ρ R - 1 ρ * R ,                                      ε * α,L = ε α,L + B 2 3,L 2ρ L 1 - ρ * L ρ L + π * α,L + c α B * 3,L 2 /2 2 2(c α a L ) 2 - π α,L + c α B 2 3,L /2 2 2(c α a L ) 2 , ε * α,R = ε α,R + B 2 3,R 2ρ R 1 - ρ * R ρ R + π * α,R + c α B * 3,R 2 /2 2 2(c α a R ) 2 - π α,R + c α B 2 3,R /2 2 2(c α a R ) 2 .
(3.26)

Finally, using previous formulas one can compute the speeds This means that this quantity remains constant through the related contact discontinuities, so that u∂ x (c i π e -c e π i ) = 0 there. For the central discontinuity, u is constant so that u∂ x (c i π e -c e π i ) = ∂ x (u (c i π e -c e π i )) this product is also well defined in the usual weak sense.

Σ 1 = u 1,L - a L ρ L , Σ 2 = u * 1 , Σ 3 = u 1,R + a R ρ R . ( 3

Numerical fluxes

All components of the system except E e and E i are conservative, thus classical computations give the associated numerical fluxes,

F L = (F ρ , F ρu1 , F ρu2 , F Ee L , F Ei L , F B3 ), F R = (F ρ , F ρu1 , F ρu2 , F Ee R , F Ei R , F B3 ), (3.28)
where the conservative part involves the Riemann solution evaluated at x/t=0,

F ρ = (ρu) x/t=0 , F ρu1 = (ρu 2 1 + π e + π i + B 2 3 /2) x/t=0 , F ρu2 = (ρu 1 u 2 ) x/t=0 , F B3 = (u 1 B 3 ) x/t=0 . (3.29)
More explicitly (3.29) yields that the quantities between parentheses are evaluated at " 1). As usual there is no ambiguity when equality occurs in these conditions.

L" if Σ 1 ≥ 0, at "L * " if Σ 1 ≤ 0 ≤ Σ 2 , at "R * " if Σ 2 ≤ 0 ≤ Σ 3 , at "R" if Σ 3 ≤ 0 (see Figure
We complete these formulas by computing the left/right numerical fluxes from (3.14) for the variables E α with α = e,i ,

F Eα L = u 1 E α + π α + c α B 2 3 /2 L + min(0,Σ 1 ) E * α,L -E α,L + min(0,Σ 2 ) E * α,R -E * α,L + min(0,Σ 3 ) E α,R -E * α,R , (3.30)
F Eα R = u 1 E α + π α + c α B 2 3 /2 R -max(0,Σ 1 ) E * α,L -E α,L -max(0,Σ 2 ) E * α,R -E * α,L -max(0,Σ 3 ) E α,R -E * α,R .
(3.31)

Positivity of density and internal energies

We must now provide some sufficient conditions on a L and a R in order to satisfy (3.21)

(a L + a R ) ≥ ρ L (a R (u 1,L -u 1,R ) + π R -π L ) , a R (a L + a R ) ≥ ρ R (a L (u 1,L -u 1,R ) + π L -π R ) ,
i,L + c i B 2 3,L /2| 2c i √ ε i,L , a R ≥ max |π e,R + c e B 2 3,R /2| 2c e √ ε e,R , |π i,R + c i B 2 3,R /2| 2c i √ ε i,R .

Entropy minimum principle

The present section is devoted to prove that the scheme (3.13) -(3.15) satisfies the following discrete entropy inequality

η(U n+1 i ) -η(U n i ) + ∆t ∆x (G(U i ,U i+1 ) -G(U i-1 ,U i )) ≤ 0, (4.1)
where η(U ) = -ρ(s e (U ) + s i (U )), (

is the entropy from (1.8). The numerical entropy flux G(U l ,U r ) satisfies the consistency condition G(U,U ) = -ρu (s e (U ) + s i (U )). The state vector U is defined by (1.10) and belongs to the admissible state space Ω defined as follows

Ω = U ∈ R 6 ; ρ > 0, ε e > 0, ε i > 0 .
First we recall the classical and most general condition from Harten-Lax-van Leer [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] concerning discrete entropy inequalities. Let η(U ) be the entropy defined by (4.2). Let R(ξ, U L , U R ) be the approximate Riemann solver defined in Section 3.1.1. Under CFL condition (3.12), assume the following entropy consistency condition:

1 ∆x ∆x/2 -∆x/2 η (R(x/∆t, U L , U R )) dx ≤ 1 2 (η(U L ) + η(U R )) - ∆t ∆x (η(U R )u R + η(U L )u L ) . (4.3)
Then the scheme (3.13) -(3.15) satisfies the discrete entropy inequality (4.1). We skip the proof of this well-know result (for instance, see [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF]).

The previous result states a criterion which is too weak to be applied on our scheme. Instead we use a stronger condition, based on a local minimum entropy principle. This idea was introduced in [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources[END_REF] for the gas dynamic sytem. A suitable extension of this technique has been derived for the MHD system in [START_REF] Bouchut | A multiwave approximate Riemann solver for ideal MHD based on relaxation I -Theoretical framework[END_REF] and in [START_REF] Berthon | An entropy preserving relaxation scheme for ten-moments equations with source terms[END_REF] for the ten-moments system.

The first step consists in stating a sufficient condition on the intermediate states of the approximate Riemann solver to enforce the required discrete entropy inequality.

(U * L ) ≥ s α,L , s α (U * R ) ≥ s α,R , (4.4)
where

s α,L = s α (U L ), s α,R = s α (U R ),
and s α (U ) is given by (1.9). In addition assume the CFL condition (3.12) holds. Let U n i ∈ Ω for all i ∈ Z. Then U n+1 i defined by (3.13)-(3.14) satisfies the discrete entropy inequality (4.1).

Remark 4.2. The result remains valid when B 3 = 0 and proves that the scheme developed by a Suliciu relaxation approach in [START_REF] Aregba-Driollet | Estibals Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF] also satisfies a discrete entropy inequality.

Proof. Arguing similarly to the proof of the Theorem 6.1 in [START_REF] Berthon | An entropy preserving relaxation scheme for ten-moments equations with source terms[END_REF], the scheme will be proved to satisfy a discrete entropy inequality as soon as we have the following entropy estimates

η(U * L ) ≥ η(U L ), η(U * R ) ≥ η(U R
), and the following relations are satisfied

ρ * L u * 1 -ρ L u 1,L = Σ 1 (ρ * L -ρ L ) , ρ * R u * 1 -ρ R u 1,R = Σ 3 (ρ * R -ρ R ) . Using (4.
2), we sum (4.4) for α = e, i and we obtain the entropy estimate. Moreover, these previous relations are Rankine-Hugoniot relations which are satisfied because the solver resolves exactly the equation on density (3.1).

Our goal is now to obtain the sufficient condition (4.4). In the sequel, we denote Σ = (λ, ε α , B 3 , π) T ∈ R + ×R + ×R×R and we set Σ eq = (λ, ε α , B 3 , p α (λ, ε α )) T . Moreover we introduce three functions

Σ → φ(Σ), Σ → ϕ(Σ), Σ → ψ(Σ) in C 2 (R + × R + × R × R) associated to the Riemann invariants already exhibited in (3.23), as follows φ(Σ) = π + c α B 2 3 /2 + c α a 2 λ, ϕ(Σ) = ε α + λB 2 3 - π + c α B 2 3 /2 2 2 (c e a) 2 , ψ(Σ) = λB 3 , (4.5)
where we have set λ the specific volume as follows λ = 1 ρ .

We now give our central technical statement.

S * α (π) = S * α φ(λ * L , ε * α,L , B * 3,L , π), ϕ(λ * L , ε * α,L , B * 3,L , π), ψ(λ * L , ε * α,L , B * 3,L , π)
then we have

s α (λ * L ,ε * α,L ) = max π∈R S * α (π) ≥ S * α ( π), ∀ π ∈ R.
We fix π = π * α,L to write

s α (λ * L ,ε * α,L ) ≥ S * α (π * L ). (4.7)
Next the functions φ, ϕ, ψ defined by (4.5) are Riemann invariants shared by the eigenvalues u 1 ± a/ρ. Thus they are invariant across the wave speeds

u 1,L -a L /ρ L and u 1,R + a R /ρ R . So we have ϕ λ * L , ε * α,L , B * 3,L , π * α,L = ϕ (λ L , ε α,L , B 3,L , π α,L ) , φ λ * L , ε * α,L , B * 3,L , π * α,L = φ (λ L , ε α,L , B 3,L , π α,L ) , ψ λ * L , ε * α,L , B * 3,L , π * α,L = ψ (λ L , ε α,L , B 3,L , π α,L ) . (4.8)
Last three equations imply that

S * α (π * L ) = S α (φ(λ L , ε α,L , B 3,L , p L ), ϕ(λ L , ε α,L , B 3,L , p L ), ψ(λ L , ε α,L , B 3,L , p L )) .
Using last equality in (4.7) we get

s α (λ * L ,ε * α,L ) ≥ S α (φ(λ L , ε α,L , B 3,L , p L ), ϕ(λ L , ε α,L , B 3,L , p L ), ψ(λ L , ε α,L , B 3,L , p L )) .
In addition, using second equation in (4.6), we have

s α (λ L ,ε α,L ) = S α (φ(λ L , ε α,L , B 3,L , p L ), ϕ(λ L , ε α,L , B 3,L , p L ), ψ(λ L , ε α,L , B 3,L , p L )) .
Thus the expected left minimum principle is reached and the proof is complete.

To complete this section, we now establish Proposition 4.3. To address such an issue, we need the following lemma whose helpfulness is just technical. In the sequel we set σ = (λ, ε) T . Lemma 4.4. Let U defined by (1.10) and φ(Σ), ϕ(Σ), ψ(Σ) by (4.5). Assume that (3.21) holds for all U ∈ Ω. Then there exists three functions, denoted by λ(X,Y,Z), εα (X,Y,Z), B3 (X,Y,Z) such that λ(φ(Σ eq ), ϕ(Σ eq ), ψ(Σ eq )) = λ, εα (φ(Σ eq ), ϕ(Σ eq ), ψ(Σ eq )) = ε and B3 (φ(Σ eq ), ϕ(Σ eq ), ψ(Σ eq )) = B 3 , (4.9)

and the following derivatives are satisfied:

∂ λ ∂X (X,Y,Z) = 1 D(σ) λ - γ -1 c α a 2 p α ( λ,ε α ) + c α B2 3 2 , (4.10) ∂ λ ∂Y (X,Y,Z) = -c α (γ -1) D(σ) , (4.11) ∂ εα ∂X (X,Y,Z) = 1 D(σ) λ B2 3 2 - p α ( λ, εα ) + c α B2 3 (c α a) 2 p α ( λ, εα ) + c α B2 3 2 , (4.12) ∂ εα ∂Y (X,Y,Z) = 1 D(σ) λa 2 -p α ( λ,ε α ) -c α B2 3 , (4.13) ∂ B3 ∂X (X,Y,Z) = -B3 D(σ) 1 - γ -1 λc α a 2 p α ( λ,ε α ) + c α B2 3 2 , (4.14) ∂ B3 ∂Y (X,Y,Z) = c α (γ -1) B3 λD(σ) , (4.15)
where we have set In the second statement, we exhibit a suitable function derived from λ and ε and we consider its local extremum. Lemma 4.5. Let U defined by (1.10) and φ(Σ), ϕ(Σ), ψ(Σ) by (4.5). Assume that (3.21) holds for all U ∈ Ω. We introduce S(Σ) :

R + × R + × R × R → R defined by: S(Σ) = s α λ (φ(Σ), ϕ(Σ), ψ(Σ)) , εα (φ(Σ), ϕ(Σ), ψ(Σ)) , (4.17)
where s α (λ, ε α ) denotes the specific entropy. For sake of clarity we denote λ, εα and B3 instead of λ (ϕ(Σ), φ(Σ), ψ(Σ)), εα (ϕ(Σ), φ(Σ), ψ(Σ)), and B3 (ϕ(Σ), φ(Σ), ψ(Σ)).

Then we have

∂S ∂π (Σ) = p α ( λ, εα ) + B2 3 /2 -π + B 2 3 /2 a 2 εα , (4.18)
with the functions λ, εα and B3 defined in Lemma 4.4.

The last result concerns the study of the extrema of the function S defined by (4.17). Lemma 4.6. Let U defined by (1.10) and S(Σ) defined by (4.17). Assume that (3.21) holds for all U ∈ Ω. Then the function π → S(Σ) admits a unique maximum given by π = p α (λ, ε α ).

Equipped with these results, we can establish Proposition 4.3

Proof of Proposition 4.3. From assumption (3.21), we can apply Lemmas 4.4 and 4.5 to define a function S as follows:

S(φ(Σ),ϕ(Σ),ψ(Σ)) = S(Σ), = s λ (φ(Σ),ϕ(Σ),ψ(Σ)) , εα (φ(Σ),ϕ(Σ), ψ(Σ)) ,
where the function s is nothing but the specific entropy. Now, by definition of λ and εα , we have

λ (φ(Σ),ϕ(Σ), ψ(Σ)) | π=p(λ,εα) = λ and εα (φ(Σ),ϕ(Σ), ψ(Σ)) | π=p(λ,εα) = ε α , which immediately implies S (φ(Σ),ϕ(Σ), ψ(Σ)) | π=p(λ,εα) = s(λ,ε α ).
Next, we study the extrema of the function π → S(Σ). By Lemma 4.6, the unique maximum of the function S is π = p α (λ, ε α ), which completes the proof.

We conclude this section by giving successively the proof of the three intermediate lemmas.

Proof of Lemma 4.4. Let us consider the function

Θ : (λ, ε α , B 3 ) →   φ(λ, ε α , B 3 , p α (λ, ε α )) ϕ(λ, ε α , B 3 , p α (λ, ε α )) ψ(λ, ε α , B 3 , p α (λ, ε α ))   . (4.19)
We remark that the function D(λ,ε α ,B 3 ), defined by (4.16) is nothing but the Jacobian function of Θ. Since for all (λ,ε α ,B 3 ) under consideration (3.21) holds, D(λ,ε α ,B 3 ) does not vanish. Thus we can apply the inverse function theorem to deduce the existence of a reciprocal function

Θ -1 (X, Y, Z) =   λ(X,Y,Z) εα (X,Y,Z) B3 (X,Y,Z)   ,
defined for (X,Y,Z) in the range of Θ and such that D(λ,ε α ,B 3 ) = 0. By definition of the functions λ, εα and B3 , the relation (4.9) is obviously obtained. Now, we evaluate the derivatives of those reciprocal functions. Once again by definition of λ, εα and B3 , we have

ϕ λ(X,Y,Z), B3 (X,Y,Z), p α ( λ, εα ) = X, φ λ(X,Y,Z), εα (X,Y,Z), B3 (X,Y,Z), p α ( λ, εα ) = Y, ψ λ(X,Y,Z), B3 (X,Y,Z) = Z.
By differentiating in X these two relations and using (4.5), we obtain:

c α a 2 - p α ( λ, εα ) λ ∂ λ ∂X + c α (γ -1) λ ∂ εα ∂X + c α B3 ∂ B3 ∂X = 1, B2 3 2 + p α ( λ, εα ) (c α a) 2 λ p α ( λ, εα ) + c α B2 3 2 ∂ λ ∂X + 1 - p α ( λ, εα ) (c α a) 2 εα p α ( λ, εα ) + c α B2 3 2 ∂ εα ∂X + B3 λ - 1 c α a 2 p α ( λ, εα ) + c α B2 3 2 ∂ B3 ∂X = 0, B3 ∂ λ ∂X + λ ∂ B3 ∂X = 0.
With D(λ,ε α ,B 3 ) = 0, this above 3 × 3 system is solvable in the variables

( ∂ λ ∂X , ∂ εα ∂X , ∂ B3 ∂X ).
Then we get the expected definition of the derivatives

∂ λ ∂X (X,Y,Z), ∂ εα ∂X (X,Y,Z) and ∂ B3 ∂X (X,Y,Z)
given by (4.10), (4.12), and (4.14). Proof of Lemma 4.5. Here, we have to compute the derivative of S with respect to π where the function S is defined by (4.17). Using the definition of ψ in (4.5) we notice that ∂ψ ∂π (Σ) = 0, which enables us to write ∂S ∂π

(Σ) = ∂s α ∂λ (σ) ∂ λ ∂X (ϕ(Σ), φ(Σ), ψ(Σ)) ∂ϕ ∂π (Σ) + ∂ λ ∂Y (ϕ(Σ), φ(Σ), ψ(Σ)) ∂φ ∂π (Σ) + ∂s α ∂ε α (σ) ∂ εα ∂X (ϕ(Σ), φ(Σ), ψ(Σ)) ∂ϕ ∂π (Σ) + ∂ εα ∂Y (ϕ(Σ), φ(Σ), ψ(Σ)) ∂φ ∂π (Σ) , (4.20)
where we have set

σ = λ (ϕ(Σ), φ(Σ), ψ(Σ)) , εα (ϕ(Σ), φ(Σ), ψ(Σ)) T .
In the sequel, we will denote λ, εα and B3 instead of λ (ϕ(Σ), φ(Σ), ψ(Σ)), εα (ϕ(Σ), φ(Σ), ψ(Σ)) and B3 (ϕ(Σ), φ(Σ), ψ(Σ)). Moreover we will use the notation σ = λ, εα , B3 . Next using (4.10)-(4.13), we compute

∂ λ ∂X (ϕ(Σ), φ(Σ), ψ(Σ)) ∂ϕ ∂π (Σ) + ∂ λ ∂Y (ϕ(Σ), φ(Σ), ψ(Σ)) ∂φ ∂π (Σ) = 1 D(σ) λ + (γ -1) c α a 2 π + c α B 2 3 2 -p α ( λ,ε) + c α B2 3 2 , (4.21)
and

∂ εα ∂X (ϕ(Σ), φ(Σ), ψ(Σ)) ∂ϕ ∂π (Σ) + ∂ εα ∂Y (ϕ(Σ), φ(Σ), ψ(Σ)) ∂φ ∂π (Σ) = 1 D(σ) λ B2 3 2 - 1 c α π + c α B 2 3 2 + p α ( λ,ε α ) + c α B2 3 (c α a) 2 π + c α B 2 3 2 -p α ( λ,ε) + c α B2 3 2 . ( 4.22)
Then we plug into (4.20) the two previous relations together with

∂s α ∂λ (σ) = (γ -1)
λ , and

∂s α ∂ε α (σ) = 1 ε α ,
to obtain the following equation:

∂S ∂π (Σ) = 1 D(σ) A + B π + c α B 2 3 2 -p α ( λ,ε) + c α B2 3 2 , (4.23)
with

A = (γ -1) + λ c α εα c α B2 3 2 -π -c α B 2 3 2 , B = (γ -1) 2 c α a 2 λ + p α ( λ,ε) + c α B2 3 (c α a) 2 εα .
We now write

A = λ c α εα p α ( λ,ε α ) + c α B2 3 2 -π -c α B 2 3 2 , B = γp α ( λ,ε α ) + c α B2 3 (c α a) 2 εα λ .
Finally we plug these equalities into (4.23) which gives (4.18).

Proof of Lemma 4.6. We apply Lemma 4.4 and Lemma 4.5, which enables us to introduce the function

g(Σ) = p α ( λ, εα ) + B2 3 /2 • (φ(Σ), ϕ(Σ), ψ(Σ)) -π + B 2 3 /2 ,
where Σ = (λ, ε α , B 3 , π) T . In order to prove the lemma, we are going to study the roots of the function g. First we notice that, thanks to relations in (4.9), π 0 = p α (λ, ε α ) is a root of g. Then we will show that for every π0 root of g, we have ∂g ∂π (π 0 ) < 0. Let π0 be a root of g. We denote by g the partial derivative of g with respect to π, and we compute it as follows:

g (Σ) = ∂p α ∂λ (σ) ∂ λ ∂X (ϕ(Σ), φ(Σ), ψ(Σ)) ∂ϕ ∂π (Σ) + ∂ λ ∂Y (ϕ(Σ), φ(Σ), ψ(Σ)) ∂φ ∂π (Σ) + ∂p α ∂ε α (σ) ∂ εα ∂X (ϕ(Σ), φ(Σ), ψ(Σ)) ∂ϕ ∂π (Σ) + ∂ εα ∂Y (ϕ(Σ), φ(Σ), ψ(Σ)) ∂φ ∂π (Σ) + B3 (ϕ(Σ), φ(Σ), ψ(Σ)) ∂ B3 ∂X (ϕ(Σ), φ(Σ), ψ(Σ)) ∂ϕ ∂π (Σ) + ∂ B3 ∂Y (ϕ(Σ), φ(Σ), ψ(Σ)) ∂φ ∂π (Σ) -1, (4.24)
where σ = λ (ϕ(Σ), φ(Σ), ψ(Σ)) , εα (ϕ(Σ), φ(Σ), ψ(Σ))

T .

In the sequel, we use the notation Σ 0 = (λ, ε α , B 3 , π0 ) and we are interested in computing g (Σ 0 ). For sake of clarity we will denote λ, εα and B3 instead of λ ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 ) , εα ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 ) and B3 ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 ) . Moreover we will denote σ = λ, εα , B3 )

T .

First we use the previous computations (4.21), (4.22) and the fact that g(Σ 0 ) = 0 to get

∂ λ ∂X (ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 )) ∂ϕ ∂π (Σ 0 ) + ∂ λ ∂Y (ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 )) ∂φ ∂π (Σ 0 ) = λ D(σ) ,
and

∂ εα ∂X (ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 )) ∂ϕ ∂π (Σ 0 ) + ∂ εα ∂Y (ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 )) ∂φ ∂π (Σ 0 ) = -λp α ( λ,ε α ) c α D(σ) .
Moreover combining g(Σ 0 ) = 0 and relations (4.5), (4.14), (4.15) we get that

∂ B3 ∂X (ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 )) ∂ϕ ∂π (Σ 0 ) + ∂ B3 ∂Y (ϕ(Σ 0 ), φ(Σ 0 ), ψ(Σ 0 )) ∂φ ∂π (Σ 0 ) = -c α B D(σ) .
Then we plug in (4. Thus g (Σ 0 ) < 0 for all roots g. The function g being continuous, π → g(Σ) has a unique root, which is π = p α (λ, ε α ).

To complete the proof we will show that π = p α (λ, ε α ) is a maximum of the function S in Lemma 4.5. Indeed we can rewrite (4.18) as

∂S ∂π (Σ) = g(Σ) a 2 εα (ϕ(Σ), φ(Σ), ψ(Σ))
.

Since π = p α (λ, ε α ) is a unique root of g, g(Σ eq ) = 0 and from the previous relation we deduce that ∂S ∂π (Σ eq ) = 0. Moreover we have

∂ 2 S ∂π 2 (Σ eq ) =
g (Σ eq ) a 2 εα (ϕ(Σ eq ), φ(Σ eq ), ψ(Σ eq )) ,

and we have previously shown that g is negative for all roots of g. So we finally get that ∂S ∂π (Σ eq ) = 0 and ∂ 2 S ∂π 2 (Σ eq ) < 0. Thus π = p α (λ, ε α ) is the maximum of the function S, which concludes the proof.

Numerical tests

In this section we perform numerical approximations in order to evaluate both accuracy and robustness of the scheme. First order in time and space are evaluated. The CFL number is 1/2 in all tests. The computations are performed over interval [0,1]. Boundary conditions are transmissive. For all test case we use the following values for the parameters

k B = 1, γ e = γ i = 5 3 , Z = 1, m e = 10 -3 , m i = 1, u 2 = 0.
The numerical test are organized as follows. The subsection 5.1 is dedicated to the approximation of a smooth analytical solution for the full system (the source term is a non-zero function). The subsection 5.2 is dedicated to the approximation of discontinuous solutions of the homogeneous system (parameters are taken such that the source term is the identically zero function).

Source term approximation: an analytical solution

We take initial data such that

∀x ∈ R, ρ(x, 0) = 1, u 1 (x, 0) = 10, T i (x, 0) + T e (x, 0) = T = 2.
An exact solution of the system (1.1)-(1.6) is available,

∀(x, t) ∈ R × R + , ρ(x, t) = 1, u 1 (x,t) = 10, T e (x, t) = T e (x -u 1 t, 0)e -2µt + a(x,t) 1 -e -2µt + b(x,t) 2µt -1 + e -2µt , (5.1)
where µ = (γ α -1)ν 1,αβ /(n e k B ) and

a(x,t) = T 2 + ν 2,ei 2ν 1 ei (∂ x B 3 (x -u 1 t, 0)) 2 , b(x,t) = ν 2,ei + ν 2,ie 4µ (∂ x B 3 (x -u 1 t, 0)) 2 .
Here we choose T = 2 and for all x ∈ [0,1],

T e (x,0) = T 2 + exp(-200(x -1/2) 2 ), and B 3 (x,0) = exp(-50(x -1/2) 2 ).

Moreover we set ν 1,ei = 1 and ν 2,ei = ν 2,ie . The results for ν 2,ei = 2 and ν 2,ei = 4 are displayed in figure 2. They show a good agreement with analytical solution (5.1).

Resolution of the Riemann Problem

In this section, we consider seven test cases that are inspired from the book of Toro ([29]) in order to highlight the robustness of the numerical method and take into account the magnetic field. In all chosen tests, data consists of two constant T and W R = (ρ R ,u 1,R ,T e,R ,T i,R ,B We compare our Suliciu method with a nonconservative version of the classical HLL method. Like the Sulicu method, it is a nonconservative finite volume type of scheme, with update formula (3.13). In the following we explain how we define the numerical fluxes F l (U l ,U r ) and F r (U l ,U r ). First we use HLL numerical fluxes (denoted by the subscript HLL in the following) to discretize the conservative part of our system. It leads to conservative fluxes (F l = F r = F ) for the conserved quantities ρ, ρu 1 , ρu 2 , B 3 :

F ρ (U l ,U r ) = F ρ HLL (U l ,U r ) (5.2) F ρu1 (U l ,U r ) = F ρu1 HLL (U l ,U r ) (5.3) F ρu2 (U l ,U r ) = F ρu2 HLL (U l ,U r ) (5.4) F B3 (U l ,U r ) = F B3 HLL (U l ,U r ). (5.5)
In addition we use a centered discretisation of the nonconservative terms. In this subsection we will use the notation φ = c i p e -c e p i so that the nonconservative terms now write -u 1 ∂ x φ in (1.4) and u 1 ∂ x φ in (1.5) .This leads to nonconservative fluxes (F l = F r ) for the components E e and E i :

F Ee l (U l ,U r ) = F Ee HLL (U l ,U r ) -u l (φ r -φ l ) (5.6)
F Ee r (U l ,U r ) = F Ee HLL (U l ,U r ) -u r (φ r -φ l ) (5.7)
F Ei l (U l ,U r ) = F Ee HLL (U l ,U r ) + u l (φ r -φ l ) (5.8) F Ei r (U l ,U r ) = F Ee HLL (U l ,U r ) + u r (φ r -φ l ) .
(5.9)

The nonservative HLL method is thus defined by (3.13), (5.2)-(5.9) with φ = c i p e -c e p i .

Numerical results

The results are presented as follows. Figure 3 to Figure 7 show results for Suliciu method. Figure 8 to Figure 10 show results for the nonconservative HLL method. Figure 11 show results aimed at comparing the performance of the nonconservative HLL and the Suliciu methods for isolated, stationnary and slowly moving contact discontinuities.

For Test 1 to Test 5, in presence of magnetic field no exact solutions are available. Thus we compute numerical solutions for 300, 2000, 10000 cells. The quantities shown are density ρ, speed u 1 , electronic and ionic temperatures T e and T i , and magnetic field B 3 . The goal is to investigate qualitative behavior of numerical methods. Figure 3 to Figure 7 show three good properties of the Suliciu method. First, the Suliciu method is able to produce solutions for all tests. Second it converges when the number of cell increases. Third all densities and temperatures computed remain positives. Figure 8 to Figure 10 show that the previous good properties of the Sulicu method are not met when not dealing properly with nonconservative products. First, the nonconservative HLL fails to produce solutions for Test 2 and Test 5. On the other tests, the density remains positive but the ionic temperatures do not on Figure 8 and 10. Moreover, the convergence does not occur. More precisely, on Figure 8, for ionic temperature and magnetic field, a spike grows when the number of cell is increasing. For electronic temperature, unphysical oscillations occur. On Figure 9, a spike grows for density and magnetic field, unphysical oscillations grow on ionic temperature. On Figure 10 or a spike appears and grows for density, magnetic field and ionic temperature.

For Test 6 and Test 7, exact solutions are available. We compare numerical solution for 300 cells and the exact solution. Figure 11 shows a good property that the Suliciu method has and the nonconservative HLL has not. The Sulicu method is exact on isolated stationnary contact discontinuity, and very sharp on isolated slowly moving contact discontinuity. On the other hand, the nonconservative HLL spreads too much those discontinuities.

Conclusion

In this paper we have investigated at the modelling and numerical point of view the bitemperature Euler system with transverse magnetic field.

At the modelling point of view, we introduced a multicomponent BGK kinetic system coupled with Mawell equations. Next by perfoming an hydrodynamic limit, the bitemperature Euler equations with transverse magnetic field has been established.

At the numerical level, we designed a Suliciu relaxation approximation and we showed that the associated scheme is entropic. We compared our method with an a priori not positive nor entropic scheme, a nonconservative version of the classical HLL method. The results highlight the fundamental role of designing positive and entropic schemes to obtain their convergence.

We shall address to forthcoming papers the following different points. Firstly, we are currently working on taking into account transverse electric (TE) fields. Then, we plan to use this work dealing with transverse magnetic field and future work on TE field in order to propose novel high order and multi-dimensional schemes. Moreover, the case of Navier-Stokes asymptotics has also to be considered.

Figure 1 :

 1 Figure 1: Intermediate states in the Riemann solution

Theorem 4 . 1 .

 41 Assume the intermediate states U * L and U * R defined by belong Ω for all U L,R ∈ Ω and satisfy the following entropy estimates for α = e,i s α

Proposition 4 . 3 .

 43 Let U defined by(1.10). Assume that (3.21) holds for all U ∈ Ω. Then there exist a function Σ → S α (φ(Σ), ϕ(Σ), ψ(Σ)) so that max π∈R S α (φ(Σ), ϕ(Σ), ψ(Σ)) = S α (φ(Σ), ϕ(Σ), ψ(Σ)) | π=pα(λ,εα) = s α (λ,ε α ).

(4. 6 )

 6 Before we give the proof of this result, let us illustrate the interest of this technical proposition, by proving Theorem 4.1.Proof of Theorem 4.1. Here we focus on the first inequality of (4.4) since the establishment of the second inequality is similarly obtained. First we apply Proposition 4.3. Thus there exists a function Σ → S α (Σ) such that relation (4.6) holds. As a consequence, if we denote

  σ = ( λ(X,Y,Z), εα (X,Y,Z), B3 (X,Y,Z)) T and D(λ,ε α ,B 3 ) = c α λa 2 -γ α p α (λ, ε α ) -c α B 2 3 . (4.16)

  24) last three previous relations together with get g (Σ 0 ) = -γp α ( λ, εα ) + c α B3 + D(σ) D(σ) .

Figure 2 :

 2 Figure 2: Solution for analytic test case for the temperatures T e , T i computed with 200 and 500 cells. On the left ν 2,ei = 2 and on the right ν 2,ei = 4.

Figure 3 :

 3 Figure 3: Suliciu method applied to Test 1 with x 0 = 0.3. Numerical solutions are compared at computational time t = 0.2.

Figure 4 :

 4 Figure 4: Suliciu method applied to Test 2 with x 0 = 0.5. Numerical solutions are compared at computational time t = 0.15.

Figure 5 :

 5 Figure 5: Suliciu method applied to Test 3 with x 0 = 0.5. Numerical solutions are compared at computational time t = 0.012.

Figure 6 :

 6 Figure 6: Suliciu method applied to Test 4 with x 0 = 0.4. Numerical solutions are compared at computational time t = 0.035.

Figure 7 :

 7 Figure 7: Suliciu method applied to Test 5 with x 0 = 0.8. Numerical solutions are compared at computational time t = 0.012.

Figure 8 :

 8 Figure 8: Nonconservative HLL method applied to Test 1 with x 0 = 0.3. Numerical solutions are compared at computational time t = 0.2.

Figure 9 :

 9 Figure9: Nonconservative HLL method applied to Test 3 with x 0 = 0.5. Numerical solutions are compared at computational time t = 0.012.

Figure 10 :

 10 Figure 10: Nonconservative HLL method applied to Test 4 with x 0 = 0.4. Numerical solutions are compared at computational time t = 0.035.

Figure 11 :

 11 Figure 11: Noncervative HLL method (left) and Suliciu method (right) applied to Test 6 with x 0 = 0.5, and Test 7 with x 0 = 0.3. Numerical and exact solutions are compared at time t = 2.0 for both Test 6 and Test 7.

  .52) with S ei defined by (2.22)-(2.25). To complete the proof we have to recover (1.4)-(1.5) with ν 1,ei = ν 1,ei , ν 1,ei defined by (2.23), and ν 2,ei , ν 2,ie defined by (ρ e u 2,e ) + ∂ x (ρ e u 2,e u 1 ) = ν 3,ei ∂ x B 3 ,

	(2.27), (2.28). In the following we focus on the electron energy equation (1.4),
	the ion energy equation (1.5) is obtained in a similar way. First we notice the
	following equality:
			E e = E e + c e B 2 3 /2 + ρ e	u 2 2 2	-ρ e	u 2 2,e 2	.	(2.53)
	From (1.4) we have a PDE on E e + c e B 2 3 /2, thus in order to use the previous relation we need PDEs on ρ e u 2 2 2 and ρ e u 2 2,e 2 . Thus on the one hand, we use (2.26)
	and (2.38) in order to get
			∂ (2.54)
	with ν 3,ei defined by (2.25). Next we multiply by u 2,e (2.54) and we use (2.13)
	in order to get that
	∂ t ρ e	u 2 2,e 2	+ ∂ x ρ e	u 2 2,e

t

  .17)On the other hand one can check with straightforward computations that the smooth solutions to (1.1) -(1.6) verify ∂ t (ρp e ) + ∂ x (ρp e u 1 ) + γ e ρp e ∂ x u 1 = 0,(3.18)

Table 1 :

 1 3,R ) T , separated by a discontinuity at a position x = x 0 . The states W L and W R are given in Table1. These test cases are Riemann problems in the space domain [0,1]. For each test problem we select a convenient position x 0 of the initial discontinuity and the output time. These are stated in the legend of each figure displaying computational results. Data for seven test problems.Test 1 is a modified Sod test case. The solution is constituted by a right shock wave, a right travelling contact and a left sonic rarefaction wave. Test case 2 consists of two symmetric rarefaction waves travelling in opposite direction and a trivial contact wave. The solution of Test 3 consists of a strong wave of shock, a contact surface and a left rarefaction wave. The solution of Test 4 is composed of three strong discontinuities travelling to the right. The solution of Test 5 corresponds to a left rarefaction rarefaction wave, a right-travelling shock wave and a stationary contact discontinuity. The solution of Test 6 is an isolated stationary contact wave. The solution of Test 7 is an isolated contact moving slowly to the right.

	Test	ρ	u	B 3	T e	T i
	Test 1 left	1	0.75	0.8164966 0.3336667 0.3336667
	Test 1 right 0.125	0	0.2581989 0.2669333 0.2669333
	Test 2 left	1	-2	0.5163978 0.1334667 0.1334667
	Test 2 right 1	2	0.5163978 0.1334667 0.1334667
	Test 3 left	1	0	14.142136 100.10000 100.1
	Test 3 right 1	0	0.2581989 0.0333667 0.0333667
	Test 4 left	5.9999924 19.5975	17.528909 25.630859 25.630860
	Test 4 right 5.9999242 -6.19633 5.5434646 2.5634264 2.5634266
	Test 5 left	1	-19.5975 8.1649658 33.366665 33.366667
	Test 5 right 1	-19.5975 0.2581989 0.0333667 0.0333667
	Test 6 left	1.4	0	0.8164966 0.2383333 0.2383333
	Test 6 right 1	0	0.8164966 0.3336667 0.3336667
	Test 7 left	1.4	0.1	0.8164966 0.2383333 0.2383333
	Test 7 right 1	0.1	0.8164966 0.3336667 0.3336667

Acknowledgement

The autors want to thank Denise Aregba for fruitful discussions that gave rise to this work. This work was partially granted by the Conseil Régional de la région Nouvelle-Aquitaine.


Katharina Scheiter

Scheiter

Peter Gerjets
email: p.gerjets@iwm-kmrc.de

When Less is Sometimes More: Optimal Learning Conditions are Required for Schema Acquisition from Multiple Examples

Keywords: schema acquisition, number of examples, cognitive load, comparison, problem solving

) -that demonstrates that multiple examples may only be helpful for learning when sufficient time for their processing is provided and when learners are guided by an instruction to compare examples. On the other hand, learning from single examples proved to be less dependent on optimal learning conditions. Results are discussed in the light of current instructional design theories.

Schema Acquisition from Multiple Examples

Problem schemas are sophisticated knowledge structures that enable solving problems in knowledge-rich domains. A problem schema "allows problem solvers to group problems into categories in which the problems in each category require similar solutions" (Cooper & Sweller, 1987, p. 348). Once a problem has been identified as belonging to a known problem category the relevant schema is retrieved from memory, information that is specific to the to-be-solved problem (e.g., concrete objects, variable values) is filled into the slots of the schema (schema instantiation), and the solution procedure that is attached to the schema is executed in order to produce a solution to the problem (cf. [START_REF] Vanlehn | Problem solving and cognitive skill acquisition[END_REF].

Schema-based problem solving is very efficient and therefore often seen as a marking feature of experts' problem solving. Because problem schemas are crucial for proficient problem solving, research has often focused on the question of how such schemata can be acquired. Bernardo (1994, p. 379) argues that there is "a consensus that problem-type schemata are acquired through some inductive or generalization process involving comparisons among similar or analogous problems of one type." Therefore, a widely proposed instructional method for fostering the acquisition of problem schemata is to present multiple examples for each problem category conveyed [START_REF] Cooper | Effects of schema acquisition and rule automation of mathematical problemsolving transfer[END_REF][START_REF] Cummins | Role of analogical reasoning in the induction of problem categories[END_REF][START_REF] Gick | Schema induction and analogical transfer[END_REF][START_REF] Reed | Use of examples and procedures in problem solving[END_REF][START_REF] Sweller | The use of worked examples as a substitute for problem solving in learning algebra[END_REF][START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF]. This method enables comparisons of examples that belong to the same problem category and therefore, fosters two processes of abstraction: First, learners can determine features that appear in each of the category's examples (i.e., commonalities) and thus may be properties that a problem must possess in order to be an instance of this particular problem category. Thus, these shared properties of examples may potentially be the structural features that determine a problem's membership to a specific problem category and that cannot be altered without altering the solution procedure that applies to a problem. Second, example comparisons within a problem category may enable learners to identify features that vary between the category's examples (i.e., differences) and that are therefore obviously irrelevant with regard to the applicability of the solution principle that is attached to this particular problem category. Therefore, these varying characteristics of examples are indubitably surface features that concern only the problem's cover story [START_REF] Ross | Distinguishing types of superficial similarities: Different effects on the access and use of earlier problems[END_REF]. Thus, by comparing multiple instances within a category with regard to their commonalities and differences all perceived features of the examples can be hypothetically classified as either being structural or surface features.

Despite the fact that many researchers advocate the provision of multiple examples, there is not much empirical evidence to support this claim. In particular, there is a lack in studies that directly compare single-and multipleexample conditions. An exception to that is a study conducted by [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF], who presented their participants with zero, one, or three example word problems for each of three problem categories from the domain of statistics for studying. Each of the examples was couched into a different cover story. Subsequently, participants had to sort test problems from these three problem categories according to their structural features. The results showed that both example groups outperformed the zero-example group, whereas contrary to the initial expectations there were no performance differences between the single-example and the multiple-example groups. In to account for this lack of difference [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF] argued that it might not be sufficient to merely present multiple examples per problem category but that it is necessary to carefully design example combinations so that they allow for useful inferences with regard to structural and surface features of the examples presented.

Therefore, in Experiment 2 they compared two different instructional example sets that each contained three examples per problem category. In a surface-emphasizing example set the three examples that belonged to the same problem category were all couched into the same cover story and every problem category was illustrated by a different cover story. In contrast, in a structure-emphasizing example set each problem category was illustrated by examples with three different cover stories. The same three cover stories were used for every problem category. The results showed that participants in the structure-emphasizing example condition were more likely to sort the test problems on the basis of the structural similarities among the problems compared to participants in the surfaceemphasizing example condition. This superiority of structure-emphasizing example sets could be demonstrated not only for sorting tasks, but also for problem solving measures where participants later had to solve isomorphic test problems by themselves (Quilici & Mayer, 1996, Experiment 4;[START_REF] Schorr | Designing sets of instruction examples to accomplish different goals of instruction[END_REF]. These experiments support the assumption that some sets of multiple examples may be more helpful than others, in particular, if they are designed to foster specific cognitive processes like inferring the examples' commonalities and differences. However, they do not tell us anything regarding the effectiveness of multiple examples in general.

Based on the idea that the cognitive processes which are enabled by examples are more important predictors for learning outcomes than the mere number of examples itself, we argued in a previous paper [START_REF] Scheiter | The impact of example comparisons on schema acquisition: Do learners really need multiple examples?[END_REF]) that there might be alternative effective exampleprocessing strategies that rely on single examples per problem category, namely comparing examples across problem categories. In the aforementioned second experiment by [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF] Quilici and Mayer's Experiment 3 finally provided some initial support for the suspicion that across-category comparisons may be likewise effective for schema acquisition. In this experiment participants had to study four instructional examples in order to acquire knowledge on two problem categories. These examples were presented in two training sessions. Besides the quality of the example sets (structure-emphasizing versus surface-emphasizing) the manner of presentation (mixed versus blocked) was varied.

In the blocked format the first training session contained two examples belonging to one problem category and the second training session contained two examples belonging to the other category used for this experiment. However, in the mixed presentation mode participants studied one example of each category in the first training session and the remaining two examples in the second session. It was hypothesized that both manners of presentation enable different example comparisons: "In the mixed condition, students have the opportunity to notice which features differ between problem types, whereas in the blocked condition, students have the opportunity to notice which features are the same within a problem type" (Quilici & Mayer, 1996, p. 156). Thus, the blocked conditions afforded within-category comparisons, whereas the mixed conditions enabled acrosscategory comparisons. The results first replicated the finding that structure-emphasizing example sets are superior to surface-emphasizing example sets. Second, participants in the mixed conditions performed comparably well as those in the blocked conditions, indicating that across-category comparisons may be similarly effective for schema acquisition as within-category comparisons.

Moreover, learning from multiple examples per problem category may even be harmful if these examples are not appropriately processed. In particular, studying multiple examples can be very demanding because of the vast amount of information that has to be processed simultaneously in order to identify the commonalities and differences among the examples. That is, although comparing multiple examples may be a very successful way of learning it requires a lot of time and effort to be devoted to the learning task which may in turn result in a substantial amount of cognitive load [START_REF] Sweller | Cognitive architecture and instructional design[END_REF]. On the other hand, if learners process multiple examples only cursorily, they may get easily confused, which will have a negative impact on learning outcomes. Single examples are probably less vulnerable to effects of i.e., not sufficiently intense processing of examples because there is less information that has to be properly processed from the very beginning. This also implies that learning from multiple examples per problem category may require instructional support that aims at fostering the appropriate processing of examples, whereas probably no instructional guidance is needed when single examples are given.

We provided evidence for these claims in an experimental study in which we made use of the materials and experimental setting introduced by [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF].

Necessary Instructional Conditions for Learning from Multiple Examples

In the study by [START_REF] Scheiter | The impact of example comparisons on schema acquisition: Do learners really need multiple examples?[END_REF] The time students took to study the examples was registered and used to distinguish among participants who had studied examples in a cursory manner and those who had studied them intensively by means of a median split with regard to example-study time conducted within each of the four experimental conditions. The resulting variable (i.e., intensity of example study) was used as a third independent variable in the design of the study.

After participants had indicated having studied the examples sufficiently, these were removed from the table and the participants were told to sort test problems according to the features that seemed to be relevant to their solution. The problems were identical to those used by [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF].

Analyzing participants' outcomes by means of a threefactor ANOVA (number of examples x availability of comparison instruction x intensity of example study) revealed that providing multiple examples increased the time needed for learning and was moreover accompanied with improvements regarding the identification of structural similarities only when learners studied the examples intensively and when they further received instructional support by prompting them to compare examples and to identify the commonalities and differences. Under less optimal learning conditions, i.e., when the examples were processed without any additionally instructional guidance, multiple examples even tended to worsen performance. Providing one example per problem category proved to be far less dependent on the presence of optimal instructional conditions and on adequate learning behavior. Multiple examples thus might be recommendable only under rather optimal learning conditions.

There is however a limitation to this previous study which we aim at addressing in the current study. In the previous study we distinguished between participants who either processed examples cursorily or intensively by taking into account the time they decided to devote for learning. There may however be other differences between these learners besides the time spent on examples, which may be responsible for the obtained effects. In the current study we thus assigned students to conditions with fixed learning times in order to investigate the relationship between example-processing time, number of examples, and available instructional support by means of a comparison instruction more thoroughly.

Experiment Method

Participants. Eighty-one students of the University of Tuebingen, Germany, participated either for course credit or payment. Average age was 22.6 years. The participants had already taken or were currently enrolled in a statistics course and were thus familiar with the domain used for experimentation.

Materials and Procedure. The materials were identical to the ones in the study by [START_REF] Scheiter | The impact of example comparisons on schema acquisition: Do learners really need multiple examples?[END_REF]. Participants first had to fill in a multiple-choice questionnaire that contained 12 items dealing with basic concepts and terms of descriptive and inferential statistics (e.g., what is expressed by a correlation between two variables?, Which testing procedure can you apply to frequency data?). After having filled in the questionnaire they were either informed that there would be either one example or three examples illustrating three problem categories from statistics depending on the experimental condition. Half of the students were additionally told to compare the examples and to particularly pay attention to the examples' commonalities and differences. Subsequently, participants received a booklet that contained either one example or three examples for each problem category. In the one-example conditions there was one example problem that was typical for a t-test, one typical for a correlation, and one typical for χ2-test. The cover stories of the examples were the same across problem categories thereby enabling profitable processes of comparison. We used three different kinds cover stories in order to control for possible effects of the surface features of the examples. In the three-examples condition, these three cover stories were used to construct surfaceemphasizing example sets. That is, the three examples of each problem category were embedded in three different cover stories, which were used across problem categories.

Contrary to the previous experiment, participants had a fixed amount of time available for studying the examples that depended on experimental condition. After this time had exceeded, the examples were removed from the table and the participants conducted the sorting task used in the previous experiment. They were instructed to sort 12 problems according to features relevant to their solution. There were always four test problems that belonged to one problem category and each of these problems had a different cover story. The same four cover stories were used across categories. Participants were informed that they could build as many categories as they wanted and that the problems might not divide evenly upon the categories. Participants who had been told to compare examples with regard to their commonalities and differences received this comparison instruction again for the sorting task. Participants who had been told to compare examples with regard to their commonalities and differences received this comparison instruction again for the sorting task. There were no time limits for working on the sorting task.

Design and Dependent Measures. As a first independent variable we varied the number of examples per problem category (one versus three) that were presented to participants. The second independent variable consisted in the presence or absence of the comparison instruction for learning as well as for the sorting task. Learning time was fixed at three different levels as a third independent variable. Learners in the one-example conditions were either allowed 150 or 300 seconds for studying the examples, learners in the three-examples conditions either had available 300 or 600 seconds. These time limits correspond to the mean times used by learners in the previous experiment [START_REF] Scheiter | The impact of example comparisons on schema acquisition: Do learners really need multiple examples?[END_REF], who had studied one example cursorily (150 sec.) or intensively (300 sec.) or three examples cursorily (300 sec.) or intensively (600 sec.).

This research design allowed for answering three questions: (1) How do learning time and a comparison instruction affect learning from one example? ( 2 As dependent variables we registered the sorting time and the quality of the sorting performance by determining the structure and the surface score introduced by [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF]. The rationale for determining these scores is as follows: From the 12 test problems (12 * 11)/2 = 66 pairs of problems can be build, with 18 pairs containing problems that are members of the same problem category and 12 pairs containing problems that share the same cover story. The remaining 36 pairs contained problems that share neither structure nor surface features. In order to determine the structure score for a person one has to count the number of problem pairs that have been correctly identified as sharing the same structural features by putting them into the same category and dividing this number by 18 (i.e., the highest possible score). The structure score expresses a participant's ability to categorize problems according to their structural similarities and can therefore be seen as a measure for the successful acquisition of problem schemata. On the contrary, the surface score indicates a participant's tendency to sort problems according to surface similarities and is determined by counting the number of problem pairs that have been assigned to the same problem category although only sharing the same cover story and by dividing this number by 12. For the ease of interpretation the scores were transformed into percentages.

Results

The results section is divided into three parts each addressing one of the three research questions. First, we thus analyzed the one-example conditions for effects of learning time and the comparison instruction, and then we conducted the same analyses for the three-example conditions. We end by contrasting the outcomes of the one-and the threeexample conditions with identical learning times.

How do learning time and a comparison instruction affect learning from one example per category? A n ANOVA (learning time x comparison instruction) for the one-example conditions (columns one and two in Table 1) revealed that neither learning time nor the comparison instructed affected the participants' ability to sort problems according to their structural features as reflected in the structure score (learning time: F < 1; comparison instruction F(1, 38) = 1.39; MSE = 773.53; p > .20; interaction: F < 1). However, providing more learning time slightly decreased learners' tendency of being misled by the problems' surface features in the sorting task (F(1, 38) = 2.88; MSE = 506.01; p < .10). The comparison instruction had no effect on the structure score (F < 1) nor was there an interaction (F(1, 38) = 1.57; MSE = 506.01; p > .20). Finally, learning time affected the efficiency by which participants sorted problems: Participants, who were allowed to spend more time processing the examples, also took more time to complete the sorting task (F (1, 38) = 8.91; M S E = 124184.77; p = .005). There was no main effect of the comparison instruction or an interaction for the sorting time (both Fs < 1).

How do learning time and a comparison instruction affect learning from three examples per category? The same analyses were conducted for the three-examples conditions (columns three and four in Table 1). Learning time positively influenced participants' ability to sort problems according to their structural features (F(1, 35) = MSE = 790.51; p < .05). While it seemed that the comparison instruction also increased the structure score, the respective contrast failed to reach statistical significance (F(1, 35) = 2.06; MSE = 790.51; p > .15). There was no interaction between the two factors (F < 1). Participants' tendency to sort problems according to relevant surface features was decreased by both, the comparison instruction (F(1, 35) = 6.53; MSE = 658.68; p < .02) and, though less pronounced, by learning time (F(1, 35) = 3.42; MSE = 658.68; p < .10). There was no interaction between the two factors (F < 1). Finally, there were no significant effects for sorting time (comparison instruction: F(1, 35) = 1.12; MSE = 89329.37; p > .25; learning time: F < 1; interaction: F(1, 35) = 1.07; MSE = 89329.37; p > .30).

How do the number of examples and a comparison instruction affect learning when the learning time is fixed? This final analysis assessed the effects of the number of examples whose processing took place within a fixed amount of time (i.e., 300 sec) and either was or was not guided by the comparison instruction (columns two and three in Table 1). An ANOVA (number of examples x comparison instruction) revealed no significant effects for the structure score (number of examples: F < 1; comparison instruction: F(1, 34) = 1.85; MSE = 713.74; p > .15; interaction: F(1, 34) = 1.26; MSE = 713.74; p > .25). Additionally, there were no significant main effects for the surface score (number of examples: F(1, 34) = 1.95; MSE = 537.06; p > .15; comparison instruction: F(1, 34) = 2.30; MSE = 537.06; p > .10). However, a significant interaction between the two factors (F(1, 34) = 4.82; MSE = 537.06; p < .05) indicated that without a comparison instruction three examples compared to one example per problem category slightly increased the participants' tendency to sort problems according to their surface features (t(15) = -1.87; p = .08). There were no differences between the two example conditions when additional instructional support was given (t(15) = 0.87; p = .40). Finally, learners needed less sorting time when multiple examples were provided F(1, 34) = 4.09; MSE = 137900.66; p = .05). Sorting time was unaffected by the comparison instruction and there was no interaction (both Fs < 1).

Summary and Discussion

Many researchers have claimed that learning from multiple examples is superior to having only one example for the illustration of each problem category. However, direct tests of this assumption have yet failed to provide empirical support for this claim [START_REF] Gerjets | Resourceadaptive selection of strategies in learning from workedout examples[END_REF][START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF][START_REF] Scheiter | The impact of example comparisons on schema acquisition: Do learners really need multiple examples?[END_REF]. Moreover, it can be assumed that learning from multiple examples more heavily relies on optimal learning conditions compared to learning from single examples.

In the current study learning from one example per problem category was improved the more learning time was available in that learners tended to be less easily misguided by the problems' surface features. However, these improvements in sorting performance were accompanied by the fact that learners took more time to complete the sorting task. Accordingly, providing more learning time for studying one example per problem category did not prove to be a very efficient means of fostering schema acquisition. On the other hand, performance was left unaffected by the availability of a comparison instruction. These results partially resemble findings from the [START_REF] Scheiter | The impact of example comparisons on schema acquisition: Do learners really need multiple examples?[END_REF] study, where sorting performance proved to be rather stable across different one-example conditions. This robustness of learning from single examples may be especially helpful for situations in which full control of possible moderating factors (e.g., the time available for learning, the time learners are willing to spend with the instructional materials, the absence or presence additional instructional guidance) cannot be guaranteed.

In the three-example conditions favorable learning conditions like extended learning times or additional instructional support improved learning outcomes to a larger extent than in the one-example conditions. In particular, they also increased the effectiveness of the instruction by inducing a problem schema based on the problems' structural features rather than on their irrelevant surface features. The latter tendency of assigning meaning to surface features was reduced to zero percent, once learners were given sufficient time to process multiple examples and an instruction guided them to compare the examples. On the other hand, when not given sufficient time and without Comparing the conditions with single or multiple examples to each other furthermore revealed that there were no other benefits achieved by presenting multiple examples besides reducing the time needed to sort the problems. That is, the recognition of a problem's assumed category membership was faster when learning from multiple examples, which may indicate a more stable or higher automated schema. However, this schema was by no means more accurate than when learning from single examples only. Moreover, when not accompanied by an instruction to compare examples multiple examples even tended to increase the participants' tendency to be misled by surface features. This again confirms findings from the [START_REF] Scheiter | The impact of example comparisons on schema acquisition: Do learners really need multiple examples?[END_REF] study that only under optimal instructional conditions learning from multiple examples may be superior to learning from single examples. In fact, as simple contrasts for the structure score revealed the one-example condition in which only 150 seconds were provided as learning time and in which there was no instructional guidance was outperformed only by the two three-example conditions in which the learning time was four times as long. Note that this is a time span only half of the learners had been willing to invest when being given the choice to determine their own learning time in the Scheiter et al. study (2004).

We found this lack of a general advantage of multiple examples in both our studies, although we used structureemphasizing example sets that were carefully designed to allow for useful inferences [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF]. Accordingly, we recommend presenting less instructional materials to learners, which should however be carefully adjusted to the cognitive processes assumed to foster schema acquisition. Presenting multiple examples may result in a so-called redundancy effect [START_REF] Sweller | Cognitive architecture and instructional design[END_REF], because information on structural features is repeatedly presented across all examples of a problem category. Nevertheless, a learner has to process and compare all the information first, before he or she will recognize that -after the first example has been processed -subsequent examples do not contain any new relevant information. Processing redundant information thus demands cognitive resources (e.g., time) no longer available for other, more helpful cognitive processes and thus may even hinder learning. This reasoning sheds some doubts on instructional design recommendations given in constructivist approaches according to which experience with multiple instances illustrating a common principle from different perspectives is required to develop higher-level knowledge structures (e.g., Cognitive Flexibility Theory, Spiro & Jehng, 1991). According to our findings instructional designers have to carefully decide whether the new aspects delivered through multiple examples are worth the danger that can result from presenting partially redundant information.

  ) How do learning time and a comparison instruction affect learning from three examples per category? (3) How do the number of examples and a comparison instruction affect learning when the time for studying the examples is fixed?

  students received either one or three examples illustrating each of three problem categories. The problem categories and the examples were identical to the ones used by Quilici and Mayer (1996). In both example conditions the cover stories of the examples were the same across problem categories thereby enabling profitable processes of comparison. In the three-examples condition the examples of each problem category were embedded in three different cover stories, which were used across problem categories (i.e., surface-emphasizing example sets). Participants could study the examples as long as they liked to. In addition to varying the number of examples per problem category, we manipulated whether there was an additional instruction to compare the examples or not. Whereas some participants only received the instruction to study the examples carefully in order to understand them, others were additionally told to compare the examples and to particularly pay attention to the examples' commonalities and differences. This comparison instruction left open whether to compare examples across categories or within categories (in the three-examples condition).

Table 1 :

 1 Performance as a function of the number of examples, learning time, and the presence of a comparison instruction

				Number of examples
			1		3
			Learning time	Learning time
			150 sec	300 sec	300 sec	600 sec
	Without	Structure score (%)	38.4	46.9	36.1	63.3
	comparison	Surface score (%)	29.9	9.3	36.5	14.2
	instruction	Sorting time (sec)	565.0	977.1	742.4	847.0
	With	Structure score (%)	56.7	49.0	57.8	67.7
	comparison	Surface score (%)	17.5	14.4	8.3	0.0
	instruction	Sorting time (sec)	755.3	996.3	740.2	645.4

Acknowledgments

We thank Antonia Baumeister, Eva Schmetz, and Brigitte Vollmann for conducting the experiment.