R Belaouar

T Colin

G Gallice

Numerical coupling of Landau damping and Raman amplification

In this paper, we present a numerical model for laser-plasma interaction involving Raman instability and Landau damping. This model exhibits three main difficulties. The first one is the coupling of PDE's posed both in Fourier space and in physical space. The second one is a three wave resonance condition that has to be verified. The third one is the boundary conditions. We overcome these difficulties using respectively a splitting scheme, a numerical dispersion relation and absorbing boundary conditions. We present some comparison between several phenomena that are involved and the influence of the Raman amplification and the Landau damping.

Introduction and physical context

The interaction of an intense laser pulse with a plasma is a complex physical phenomenon. Numerical simulation plays a key role in its understanding. One of the main goal is to simulate nuclear fusion by inertial confinement in a laboratory. We therefore need some accurate and reliable numerical models of laser-plasma interactions. Vlasov or particle-incell (PIC) simulations have been used for a complete description of the problem. However, these kinetic simulations have difficulties in studying weak instabilities and long time behaviors because they need to resolve very small spatial and temporal scales. For the same reasons, it is not possible to use Euler-Maxwell equations.

Recently, M. Colin and T. Colin [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF], starting from [START_REF] Russel | Nonlinear saturation of simulated Raman scattering in laser hot spots[END_REF], derived a complete set of quasi-linear Zakharov equations describing the interactions between the laser fields, the stimulated Raman processes, the electronic plasma waves and the low-frequency variations of density of the ions. The system involves four Schrödinger equations coupled by quasilinear terms and a wave equation and describes a three-waves interaction. Physically, the lasers interacts with the plasma, part of it backscattered through a Raman-type process to create an electron plasma wave. These three waves interact in order to create a lowfrequency variation of density which has itself an influence on the three preceding waves. However, this model that is obtained starting from the fluid equations does not take into account the kinetic effects such the Landau damping effect which is a wave-particle process which occurs in under-dense plasma. The Landau damping process is especially important in the context of fusion by inertial confinement by lasers because electrons are accelerated to high energy and this induces a preheat of the fusion fuel and reduces the target gain. This wave-particle process corresponds to a resonant effect between the electrons of the plasma and the plasma electronic waves. This effect implies an exchange of energy between electrons and the plasma waves. As a result, the plasma waves are damped.

Of course many description of the Landau damping phenomenon exists in the literature starting at the kinetic level (see Glassey-Schaeffer [START_REF] Glassey | On time decay rates in Landau damping[END_REF], P. Degond [START_REF] Degond | Spectral theory of the linearized Vlasov-Poisson equation[END_REF] for example). Here we do not try to obtain such precise models. We try to couple a simpler one with the Raman process in order to be able to simulate the main feature of the Raman-Landau interaction.

In order to obtain a system describing this wave-particle process we complete the system used in [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF] by using the model derived in [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF]. The aim of this paper is to perform mathematically and numerically the coupling of these models that describes the interaction of the variation of the density of ions with the slowly varying envelope of the plasma electronic waves, the spatial mean value of the distribution function of the electrons, the laser field and the Raman component. We want to achieve two goals. The first one is to investigate what is the influence of the Landau damping process on the saturation of the Raman amplification. The second question we want to address is the influence of the Raman instability on the model [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF] in terms of the number of accelerated electrons.

For that study, we use the scheme introduced in [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF], a time-splitting discretization for the Landau damping term and a implicit finite difference scheme for the distribution function of the electrons. The main difficulties are the following : i) First we have to couple the equations of the Raman model of [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF] with those of the Landau model of [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF]. This is done numerically by using a splitting strategy in section 3.1. The Landau damping model consist in two partial differential equations, one is posed in the physical space, the other one in Fourier space. The Fourier transform of some field occurs explicitly in the partial differential equations. The coupling of such models in the context of boundary value is not obvious especially because of the electronic plasma waves have to be considered in a periodic framework in the model [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF].

ii) The second difficulty is the three-wave interaction condition. Indeed, it is shown in [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF] that the Raman system that is obtained relies on an interaction condition. In our context, this condition means that the couple (k 1 , ω 1 ) involved in the system is such that e i(k 1 x-ω 1 t) is an exact solution to a linear Schrödinger equation. It is a phase matching condition. After discretization, one obtains a numerical phase matching condition that is different from that of the continuous case. In order to handle this difficulty, we define and use ω 1d , the frequency given by the numerical dispersion relation. This is done in section 3.2.

iii) The third difficulty is linked to the spatial box. For physical considerations, we can not use periodic boundary conditions since we want that once a pulse ( the laser part or the Raman part) hits the boundary, it does not interact anymore with the remaining part of the system. We therefore introduce some kind of absorbing boundary conditions. It is the object of section 3.3.

The outline of the paper is the following one. Section 2 is devoted to a complete presentation of the model and we introduced a dimensionless form. In section 3, in order to solve the problem, we introduce an efficient numerical scheme and show some of its stability properties. Finally, in the last section, we will provide some numerical results in order to see how the coupling between Raman amplification and Landau damping process works.

2 The model and its properties.

2.1 The equations and their non-dimensional form.

In this section, we introduce the one dimensional system describing the Raman amplification and the Landau damping process. We consider here an homogeneous plasma where collisions between the particles (electrons and ions) and the gravitational field are neglected. We want to describe the interaction of a laser field with this plasma and the physical phenomenon quoted previously. We use the following model (see [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF]) :

i ∂ t A 0 + k 0 c 2 ω 0 ∂ y A 0 + c 2 2ω 0 1 - k 2 0 c 2 ω 2 0 ∂ 2 y A 0 = ω 2 pe 2n 0 ω 0 δnA 0 - e 2m e ω 0 (∂ y • E) A R e -i(k 1 y-ω 1 t) , (2.1)
i ∂ t A R + k R c 2 ω R ∂ y A R + c 2 2ω R 1 - k 2 R c 2 ω 2 R ∂ 2 y A R = ω 2 pe 2n 0 ω R δnA R - e 2m e ω R (∂ y • E * ) A 0 e i(k 1 y-ω 1 t) , (2.2)
i(∂ t E + ν * E) + 3v 2 th 2ω pe ∂ 2 y E = ω pe 2n 0 δnE + eω pe 2c 2 m e ∂ y A * R A 0 e i(k 1 y-ω 1 t) , (2.3)
∂ 2 t -c 2 s ∂ 2 y δn = 1 4πm i ∂ 2 y |E| 2 + ω 2 pe c 2 |A 0 | 2 + |A R | 2 , (2.4) ν(ξ, t) = - πω 3 pe 2ξn 0 |ξ| ξ ∂ v F e ( ω pe ξ ), (2.5)
∂ t F e = ∂ v (D(v, t)∂ v F e ) , D(v, t) = e 2 2m 2 e |v| E ξ = ω pe v , t 2 . ( 2

.6)

Here A 0 is the envelope of the vector potential of the incident electromagnetic laser field, A R is the envelope of the vector potential of Raman backscattered light, E is the slowly varying amplitude of the high-frequency electronic plasma waves, δn the low-frequency variation of the density of the ions, F e the spatially averaged electron distribution function, ν the spatial fourier transform of ν corresponding to the Landau damping rate and u * is the complex conjugate of u. In this work we consider that the laser propagates in the positive y direction and we stay in the one dimensional framework. This system involves three Schrödinger equations coupled by quasi-linear terms and the low frequency fluctuation of density given by the wave equation (2.4). The electron distribution function satisfies a heat equation where the diffusion coefficient D(v, t) depends on the density spectral energy of electron plasma waves. The constants are defined by:

• c is the speed of light in the vacuum, e is the elementary electric charge,

• m e and m i are respectively the electron's and ion's mass,

• n 0 is the mean background density of the plasma,

• T e is the electronic temperature,

• ω pe , v the and c s are respectively the electronic plasma pulsation, the thermal velocity of electrons and the acoustic velocity of ions given by

ω pe = 4πe 2 n 0 m e , v the = T e m e , c s = T e m i ,
• ω 0 , ω R , ω pe + ω 1 are respectively the laser pump frequency, the Raman component frequency and the electronic plasma wave frequency,

• k 0 , k R , k 1 are respectively the laser pump wave number, the Raman component wave number and the electronic plasma wave number.

Note that (ω 0 , ω R , ω 1 ) and (k 0 , k R , k 1 ) have to satisfy the three-waves resonance condition

ω 0 = ω R + ω pe + ω 1 , (2.7
)

k 0 = k R + k 1 , (2.8)
in order to have an efficient process. Here (k 0 , ω 0 ), (k R , ω R ) correspond to electromagnetic waves while, (k 1 , ω pe + ω 1 ) corresponds to electronic plasma waves and the dispersion relations are therefore

ω 2 0 = ω 2 pe + c 2 k 2 0 , (2.9
)

ω 2 R = ω 2 pe + c 2 k 2 R , (2.10)
(ω pe + ω 1 ) 2 = ω 2 pe + 3v 2 the k 2 1 .
(2.11)

Note that, the last relation can be written approximatively

ω 1 ≈ 3v 2 the k 2 1 2ω pe .
The full electric field can then be recovered as follows

E f (t, x) = i ω 0 c 0 A 0 e i(k 0 y-ω 0 t) + i ω R c A R e i(k R y-ω R t) + Ee -iωpet + c.c.
where c.c. denotes the complex conjugate.

With this model, we can recover the model used in [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF] by taking ν = 0 in (2.3) to obtain system (2.1)-(2.4) which was derived from a bi-fluid Euler-Maxwell system. We can also recover the system used in [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF] by fixing the potentials A R and A 0 to obtain system (2.3)- (2.6) where in (2.3), we have a fixed source term given by ∂ y A * R A 0 e i(k 1 y-ω 1 t) . Concerning the wave-particle process, the model is valid for bounded velocity that are also bounded away from zero (see [START_REF] Delcroix | Physique des plasmas 1, 2[END_REF]). Let Ω v be the velocity domain on which equation (2.6) has to be satisfied and

Ω ξ = {ξ ∈ R s.t. ∃v ∈ Ω v , v = 1 ξ }.
The domain Ω ξ will therefore be taken under the form, Ω ξ = [-A, -a] ∪ [a, A] with 0 < a < A (see section 2). Note the term ν is only defined on Ω ξ by (2.5) and is extended by 0 outside the domain Ω ξ .

Dimensionless system

We now introduce a dimensionless form of (2.1)-(2.6).

We use T = 1 ω 0 as time scale and L = 1 k 0 as space scale and introduce

A 0 = e m e c 2 A 0 , A R = e m e c 2 A R , E = e m e v e ω pe E, ν = 1 ω pe ν, k 1 = k 1 k 0 , ω 1 = ω 1 ω 0 , F e = v the n 0 F e , δn = 1 n 0 δn.
Omitting the tildes, we get the following system :

i (∂ t A 0 + v 0 ∂ y A 0 ) + 1 2 v 0 (1 -v 0 )∂ 2 y A 0 = ω 2 pe 2ω 2 0 δnA 0 - k 0 k De ω 2 pe ω 2 0 (∂ y E) A R e -i(k 1 y-ω 1 t) , (2.12
)

i (∂ t A R + v R ∂ y A R ) + ω 0 2ω R (1 - k 2 R c 2 ω 2 R )∂ 2 y A R = ω 2 pe 2ω R ω 0 δnA R - k 0 k De ω 2 pe ω 0 ω R (∂ y E * ) A 0 e i(k 1 y-ω 1 t) , (2.13)
i ∂ t E + ω pe ω 0 ν * E + 3k 2 0 2k 2 De ω pe ω 0 ∂ 2 y E = ω pe 2ω 0 δnE + c 2 k 0 k De 4ω 0 ω pe ∂ y A * R A 0 e i(k 1 y-ω 1 t) , (2.14)
∂ 2 t -v 2 s ∂ 2 y δn = m e 4m i k 2 0 k 2 De ω 2 pe ω 2 0 ∂ 2 y |E| 2 + m e 4m i v s ∂ 2 y |A 0 | 2 + |A R | 2 .
(2.15)

ν(t, ξ) = - πk 2 De 2k 2 0 ξ |ξ| ∂ v F e (v = k De k 0 ξ ), (2.16
)

∂ t F e = ∂ v (D(v, t)∂ v F e ) , D(v, t) = k De ω pe k 0 ω 0 |v| E ξ = k De k 0 v , t 2 .
(2.17)

where

v 0 = c 2 k 2 0 ω 2 0 is the group velocity of A 0 , v R = c 2 k 0 k R ω 0 ω R is the group velocity of A R , v s = c 2 s k 2 0 ω 2 0
is the velocity of the ion acoustic waves and k De = 1 λ De where λ De denotes the Debye length. In order to study the quasi-linear diffusion equation (2.17), it is more convenient to use the variable ξ = k De k 0 v instead of v. Then denoting

H(t, ξ) = F e (t, k De k 0 ξ ),
the distribution function, (2.17) becomes

∂ t H - ω pe k 2 0 ω 0 k 2 De ξ 2 ∂ ξ (|ξ| 3 | Ê(t, ξ)| 2 ∂ ξ H) = 0, (2.18)
and the Landau damping rate reads

ν(t, ξ) = sgn(ξ) π 2 k 3 De k 3 0 ∂ ξ H e (t, ξ). (2.19)
2.3 Some basic properties.

One first have an energy conservation given by Proposition 2.1. For any regular solution of (2.12) -(2.17), one has

d dt 2 ω 0 ω pe |A 0 | 2 + ω R ω pe |A R | 2 + 4v 2 the c 2 |E| 2 dy + 4k De v 2 the k 0 c 2 ν(t, ξ)| E| 2 (t, ξ)dξ = 0.
Proof : Multiply (2.12) by 2A * 0 , (2.13) by A * R , (2.14) by E * , integrating over R, summing the results, taking the imaginary part, using the Plancherel formula for the Landau damping term and denoting by θ = k 1 x -ω 1 t, give

1 2 ∂ t 2 ω 0 ω pe |A 0 | 2 + ω R ω pe |A R | 2 + 4v 2 the c 2 |E| 2 dy + 4k De v 2 the k 0 c 2 ν(t, ξ)| E| 2 (t, ξ)dξ = ℑ -2∂ y EA R A * 0 e -iθ -∂ y E * A * R A 0 e iθ + ∂ y (A * R A 0 e iθ )E * dy, = ℑ 2∂ y E * A * R A 0 e iθ -∂ y E * A * R A 0 e iθ -A * R A 0 ∂ y E * e iθ dy, = 0.
Moreover, one has some maximum principle on (2.16)-(2.17) that show that the convolution term in (2.14) is indeed a damping term. More precisely, we recall the following result imported from [2] :

Proposition 2.2. If ν(0, ξ) ≥ 0 for all ξ, then one have

ν(t, ξ) = sgn(ξ)∂ ξ H(t, ξ) ≥ 0,
for all ξ ∈ R and t > 0.

Remark 2.1. If the the initial distribution function of electron is a Maxwellian then one has ν(0, ξ) ≥ 0. Proposition (2.1) expresses the decay of the electromagnetic energy due to the Landau damping process between the electron plasma waves and electrons.

3 Numerical approximation.

In this section, we present an numerical scheme for system (2.12)-(2.17) endowed with the following initial conditions

A 0 (0, y) = A 00 (y), A R (0, y) = A R0 (y), E(0, y) = E 0 (y), (3.1)
δn(0, y) = δn 0 (y), ∂ t δn(0, y) = δn 1 (y), H(0, ξ) = H 0 (ξ).

We work on the spatial domain [0, L] and we use a regular mesh in space. The spatial mesh size is h = δy with h = L/N for N = 2M being an even number, the time step being δt > 0 and let the grid points and the time step be y j = jh, t k = kδt, j = 0, 1, . . . , N, k = 0, 1, 2 . . . with y 0 = 0 and y N = L. We also define the sequence (ξ j = 2πj L ) j=-N 2 ,...,0,..., N 2 -1 as the regular mesh grid for the frequencies with δξ = 2π L . Furthermore, let A k 0j ,A k Rj , E k j , δn k j , and ν k j be the approximations of A 0 (t k , x j ), A R (t k , x j ), E(t k , x j ), δn(t k , x j ), ν(t k , ξ j ). Then, in order to be consistent with the evaluation of ν(t k , ξ j ) = sgn(ξ j )∂ ξ H(t k , ξ j ), we approximate H on the grid (ξ j+ 1 2 ) j defined by ξ j+ 1 2 = 2π(j+1/2) L . The numerical scheme used in [START_REF] Russel | Nonlinear saturation of simulated Raman scattering in laser hot spots[END_REF] for the wave part of our model is a pseudo-spectral discretization. The authors observed some aliasing errors due to the nonlinear and quasilinear terms and they were obliged to truncate the Fourier transform of the different fields. In [START_REF] Colin | A numerical model for the Raman amplification for laser-plasma interaction[END_REF], a fractional-step, Crank-Nicholson-type scheme with relaxation directly inspired by that of C. Besse for NLS (see [START_REF] Besse | Schéma de relaxation pour l'équation de Schrödinger non linéaire et les systèmes de Davey et Stewartson[END_REF]) is proposed for the quasi-linear system. For the acoustic part, they used an energy-preserving finite difference scheme introduced by Glassey (see [START_REF] Glassey | Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension[END_REF]).

We present our scheme in three parts. In the first part, we give the scheme in itself in the case of periodic boundary condition (section 3.1). In the second part, we deal with the three-waves resonance conditions and we explain how we overcome this difficulty (section 3.3). In the third part, we explain how one can construct some kind of transparent boundary conditions for our problem (section 3.4).

The numerical scheme.

The Landau damping rate in the Schrödinger equation evolving the electronic plasma waves is given in Fourier space and it is given by a diffusion equation while the Zakharov part of the model is posed in the physical space. Therefore we have to use a spectral method to evaluate this convolution operator. In the other hand we can not use a spectral method for the linear part of (2.12)-(2.13) since we deal with transport operators. To overcome this difficulty, we introduce a splitting technique on the Landau damping process in order to separate the Raman amplification process and the Landau damping. Therefore, as we will see in numerical applications, since the group velocity of the electronic plasma waves is closed to zero, this allow us to use a spectral method (periodic boundary conditions for E) to solve the Landau damping part. Finally, for the Raman amplification, we use the numerical scheme introduced in [START_REF] Colin | A numerical model for the Raman amplification for laser-plasma interaction[END_REF]. We now describe more precisely the different step of our method. If at time t k , we know A k 0j ,A In a first step, we use a scheme for the quasi-linear diffusion equation. In a second one, we introduce the scheme used in [START_REF] Colin | A numerical model for the Raman amplification for laser-plasma interaction[END_REF] for (2.12)-(2.14) without the convolution operator describing the Landau process and finally, using a fast Fourier transform, we compute the modification given by the Landau damping rate on the electronic plasma waves.

Step 1 : The diffusion.

In order to evaluate the approximation of

H(t k , ξ j+ 1 2
), we use an implicit difference scheme for the diffusion equation :

∂ t H -ξ 2 ∂ ξ |ξ| 3 | E| 2 ∂ ξ H = 0, ξ ∈ Ω, where Ω = [-ξ 2 , -ξ 1 ] ∪ [ξ 1 , ξ 2 ], (ξ 2 > ξ 1 > 0), with ξ 1 = 2π(j 1 +1/2) L > 2π L , ξ 2 = 2π(j 2 +1/2) L < 2π(M -1) L . The boundary conditions are ∂ ξ H(., ±ξ 1 ) = 0, H(., -ξ 2 ) = H(0, -ξ 2 ), H(., ξ 2 ) = H(0, ξ 2 ).
The finite difference scheme reads :

1 δt (H k+1 j+ 1 2 -H k j+ 1 2 ) + (AH) k+1 j+ 1 2 = 0, (3.3)
where (AH) k+1

j+ 1 2 is a discretization of -ξ 2 ∂ ξ (β(t, ξ)∂ ξ H) in a conservative form at the point ξ j+ 1 2
and time t k+1 . We choose A such that :

(AH) k j+ 1 2 = - ξ 2 j+ 1 2 ξ j+1 -ξ j   β k-1 j+1 H k j+ 3 2 -H k j+ 1 2 ξ j+ 3 2 -ξ j+ 1 2 -β k-1 j H k j+ 1 2 -H k j-1 2 ξ j+ 1 2 -ξ j-1 2   , (3.4)
where

β k j is the approximation of β(t, ξ) = |ξ| 3 | E| 2 .
Then, with this scheme, we can evaluate ν(ξ j , t k+1 ) on the frequency grid with the following centered finite difference scheme :

ν k+1 j = sgn(ξ j ) H k+1 j+ 1 2 -H k+1 j-1 2 ξ j+ 1 2 -ξ j-1 2 . (3.5)
As shown in [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF], one take

β k j = ξ 2 j+ 1 2 ξ 2 j-1 2 |ξ j+ 1 2 + ξ j-1 2 | | Êk j | 2
in order to ensure the following conservation relation

j 1 ξ 4 j+ 1 2 H k+1 j+ 1 2 = j 1 ξ 4 j+ 1 2 H k j+ 1 2 + δt j ν k+1 j | Êk j | 2 (3.6)
which corresponds to the energy exchange between electrons and the electronics plasma waves. One also have a numerical maximum principle for ν. If ν 0 satisfies

ν 0 (ξ j ) ≥ 0, j = - N 2 , . . . , N 2 -1, then for all k > 0 ν k j ≥ 0, j = - N 2 , . . . , N 2 -1. (3.7)
In order to illustrate how the quasi-linear diffusion works on the solution, we have computed the diffusion equation with a diffusion coefficient given by a fixed electric field

E(t, x) = e i(k 1 x-ω 1 t) e - (x-L 2 ) 2 2△L 2 ,
with L = 2000, △L = 50 and k 1 = 0.45. The initial electron distribution function is assumed to be a Maxwellian,

F e0 (v) = 1 √ 2π exp (- v 2 2 )
which gives the following initial condition for the Landau damping rate We can remark on figure 1 that the electron distribution function is flattened near the

ν(0, ξ) = π 8 1 |ξ| 3 e - 1 2ξ
× × × × × × × × × × × × × × × × × × × × × × ×× ××× ××××××× ×××××××× ×××× ×× ×× × × × × × × × × × × × × × × × × × × × × × 1.
××××××××××× ××××× ×××× ××× ××× ××× ××× ××× ××× ×× ×× ×× × × × × × × × × × × × × × × × × × × ×× ××× ×××××××××× ×× × × × × × × × × × × × × × × × × × ×
phase velocity v φ = 1 k 1 = 2.
22 and since the Landau damping rate depends on the slope of the electron distribution, we can see that ν tends toward zero near ξ = k 1 . We will see in section 4, what happens when we take into account the time evolution of the diffusion coefficient.

Step 2 : The Raman amplification.

For the Raman amplification, we introduce a fractional-step, Crank-Nicolson-type scheme with relaxation introduced by of C. Besse for NLS equation (see [START_REF] Besse | Schéma de relaxation pour l'équation de Schrödinger non linéaire et les systèmes de Davey et Stewartson[END_REF]). For the acoustic part, we use the scheme introduced by Glassey (see [START_REF] Glassey | Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension[END_REF]). We consider centered discretization for each spatial differential operator. Therefore, ∂ y is approximated by the centered finite difference operator D 0 :

(D 0 E) i = E i+1 -E i-1 2δy ,
and

∂ 2 y by D + D -: (D + D -E) i = E i+1 -2E i + E i-1 δy 2 .
For this first step of the splitting, the scheme reads:

i A k+1 0 -A k 0 δt + (iv 0 D 0 + α 0 D + D -) A k+1 0 + A k 0 2 = β 0 δn k+1 + δn k 2 A k+1 0 + A k 0 2 - γ 0 2 φ k+ 1 2 A k+1 R + A k R 2 e -iθ k+ 1 2 - γ 0 2 ψ k+ 1 2 D 0 E k+1 + D 0 E k 2 e -iθ k+ 1 2 , (3.8) i A k+1 R -A k R δt + (iv R D 0 + α R D + D -) A k+1 R + A k R 2 = β R δn k+1 + δn k 2 A k+1 R + A k R 2 -γ R (φ k+ 1 2 ) * A k+1 0 + A k 0 2 e iθ k+ 1 2 , (3.9
)

i E k+1 -E k δt + α E D + D - E k+1 + E k 2 = β E δn k+1 + δn k 2 E k+1 + E k 2 + γ E D 0 (ψ k+ 1 2 ) * A k+1 0 + A k 0 2 e iθ k+ 1 2 , (3.10
)

δn k+1 -2δn k + δn k-1 δt 2 -v 2 s D + D - δn k+1 + δn k-1 2 =δ 1 D + D -(|E k | 2 )+ δ 2 D + D -|A k 0 | 2 + |A k R | 2 , (3.11)
for the time step of length δt.

The constants (α

0 , β 0 , γ 0 ), (α R , β R , γ R ), (α E , β E , γ E ), (δ 1 , δ 2 )
, are the following one

α 0 = 1 2 v 0 (1 -v 0 ), β 0 = ω 2 pe 2ω 2 0 , γ 0 = k 0 ω 2 pe k De ω 2 0 , α R = ω 0 2ω R (1 - c 2 k 2 R ω 2 R ), β R = ω 2 pe 2ω 0 ω R , γ R = k 0 ω 2 pe k De ω 0 ω R , α E = 3k 2 0 ω pe 2k 2 De ω 0 , β E = ω pe 2ω 0 , γ E = c 2 k 0 k De 4ω 0 ω pe , δ 1 = m e m i k 2 0 k 2 De β 2 E , δ 2 = m e 4m i v s .
The auxiliary functions φ and ψ are given by

φ k+ 1 2 + φ k-1 2 2 = D 0 E k , ψ k+ 1 2 + ψ k-1 2 2 = A k R . (3.12)
φ k+ 1 2 and ψ k+ 1 2 are prediction respectively of ∂ y E and A R at time (k + 1 2 )δt. Moreover the value φ -1 2 and ψ -1 2 are obtained by explicit integration of the system on one half-time step backward. Therefore the discretization of the phase θ k+ 1 2 is given by

θ k+ 1 2 = k 1 y -ω 1 (k + 1 2 )δt.
We now denote by A 0 , A R , E, δn the solutions of (3.8)- (3.11).

Step 3 : The Landau damping.

In the following step, we have to solve

∂ t A 0 = 0, A 0 (0, y) = A 0 (y), (3.13)
∂ t A R = 0, A R (0, y) = A R (y), (3.14)
∂ t E + ν * E = 0, E(0, y) = E(y), (3.15)
which corresponds to the modification of the electronic plasma waves due to the Landau process. Using the Fourier transform for (3.15), one gets exactly

A 0 (t k+1 , y j ) = A 0 (y j ), (3.16)
A R (t k+1 , y j ) = A R (y j ), (3.17)

E(t k+1 , ξ j ) = E(ξ j )exp - t k+1 t k ν(s, ξ j )ds (3.18)
and using a trapezoidal rule, we obtain

E(t k+1 , ξ j ) ≈ E(ξ j )exp - δt 2 ( ν k+1 j + ν k j ) . (3.19)
Using the inverse discrete Fourier transform, we get

E k+1 j = M -1 l=-M E k+1 l
e iξ l x j , j = 1, .., N.

(3.20)

L 2 stability result

The first stability result concerns the semi-discretization in time of the model. This semidiscretization in time is obtained by discretization of (2.12)-(2.17) only in time and using the same time-splitting as described before. In order not to be confuse, in this part, we will denote by u, r, w, µ, f the semi-discretized version of the physical unknowns A 0 , A R , E, ν, δn. Let u 0 , r 0 , w 0 , µ 0 and f 0 be the initial conditions of u, r, w, µ and f . The semi discretization scheme reads

• Step 1 : The diffusion
In the first step, we solve the semi-discretization in time of the quasi-linear diffusion evolving the electron distribution function on the interval [t k , t k+1 ]. By using the usual implicit Euler scheme in time, the first step reads

H k+1 -H k δt -ξ 2 ∂ ξ (β(t k , ξ)∂ ξ H k+1 ) = 0, ξ ∈ Ω ξ (3.21) µ k+1 (ξ) = sgn(ξ)∂ ξ H k+1 . (3.22)
Note that µ k+1 (ξ) is extended by zero outside the domain Ω ξ .

• Step 2 : The ion density fluctuation

The semi-discretization in time of the wave equation reads

f k+1 -2f k + f k-1 δt 2 -v 2 s ∂ 2 y f k+1 + f k-1 2 = δ 1 ∂ 2 y |w k | 2 + δ 2 ∂ 2 y |u k | 2 + |r k | 2 . (3.23)
• Step 3 : The three wave interaction Here, we find ( u, r, w) in terms of (u k , r k , w k ) by solving

i u -u k δt + (iv C ∂ y + α 0 ∂ 2 y ) u + u k 2 = β 0 f k+1 + f k 2 u + u k 2 - γ 0 2 φ k+ 1 2 r + r k 2 e -iθ k+ 1 2 - γ 0 2 ψ k+ 1 2 ∂ y w + ∂ y w k 2 e -iθ k+ 1 2 , (3.24) i r -r k δt + (iv R ∂ y + α R ∂ 2 y ) r + r k 2 = β R f k+1 + f k 2 r + r k 2 -γ R (φ k+ 1 2 ) * u + u k 2 e iθ k+ 1 2 , (3.25) i w -w k δt + α E ∂ 2 y w + w k 2 = β E f k+1 + f k 2 w + w k 2 + γ E ∂ y (ψ k+ 1 2 ) * u + u k 2 e iθ k+ 1 2 , (3.26)
where the auxiliary functions φ and ψ are defined by

φ k+ 1 2 + φ k-1 2 2 = ∂ y w k , ψ k+ 1 2 + ψ k-1 2 2 = r k . (3.27)
• Step 4 : The Landau damping Finally, in the last step, in order to compute the contribution of the Landau damping process, we solve

∂ t u = 0, u(t k , y) = u(y), (3.28)
∂ t r = 0, r(t k , y) = r(y), (3.29)
∂ t w + µ * w = 0, w(t k , y) = w(y), (3.30)
w(t k+1 , ξ) = w(ξ)exp - t k+1 t k µ(s, ξ)ds (3.33)
and we recover w(t k+1 , y) by taking the inverse Fourier transform of (3.33) and using a trapezoidal rule.

Proposition 3.1. ( L 2 stability) If the initial data µ 0 satisfy µ 0 (ξ) ≥ 0 for all ξ, then for all k > 0, any solution (u k , r k , w k ) given by the semi-discrete scheme

(3.21)-(3.33) satisfies R 2α 1 |u k | 2 + α 2 |r k | 2 + α 3 |w k | 2 dy ≤ R 2α 1 |u 0 | 2 + α 2 |r 0 | 2 + α 3 |w 0 | 2 dy,
where

α 1 = ω 0 ω pe , α 2 = ω R ω pe and α 3 = 4 v 2 the c 2 .
Proof : Using (3.31), (3.32) and the Parseval formula for w k+1 , we have

R 2α 1 |u k+1 | 2 + α 2 |r k+1 | 2 + α 3 |w k+1 | 2 dy = R 2α 1 | u| 2 + α 2 | r| 2 dy + α 3 R | w| 2 exp -δt( µ k (ξ) + µ k+1 (ξ)) dξ.
(3.34)

In order to conclude, we need the following version of the maximum principle. Proof : see [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF]. Using the lemma 3.1 and again the Parseval formula, one gets

R 2α 1 |u k+1 | 2 + α 2 |r k+1 | 2 + α 3 |w k+1 | 2 dy ≤ R 2α 1 | u| 2 + α 2 | r| 2 + α 3 | w| 2 dy. (3.35)
In order to conclude, we need to prove that

R 2α 1 | u| 2 + α 2 | r| 2 + α 3 | w| 2 dy = R 2α 1 |u k | 2 + α 2 |r k | 2 + α 3 |w k | 2 dy.
To this aim, we compute

2α 1 R (3.24) u + u k 2 * + α 2 R (3.25) r + r k 2 * + α 3 R (3.26) w + w k 2 *
, and take the imaginary part. This yields

1 2δt R 2α 1 | u| 2 + α 2 | r| 2 + α 3 | w| 2 - 1 2δt R 2α 1 |u k | 2 + α 2 |r k | 2 + α 3 |w k | 2 = -Im R φ k+ 1 2 r + r k 2 e -iθ k+ 1 2 u + u k 2 * -Im R ψ k+ 1 2 ∂ y w + ∂ y w k 2 e -iθ k+ 1 2 u + u k 2 * -Im R (φ k+ 1 2 ) * u + u k 2 e iθ k+ 1 2 r + r k 2 * + Im R ∂ y (ψ k+ 1 2 ) * u + u k 2 e iθ k+ 1 2 w + w k 2 It is clear that I = -III. Moreover IV = -Im R (ψ k+ 1 2 ) * u + u k 2 e iθ k+ 1 2 ∂ y w + w k 2 * = II.
The result follows since

R 2α 1 | u| 2 + α 2 | r| 2 + α 3 | w| 2 = R 2α 1 |u k | 2 + α 2 |r k | 2 + α 3 |w k | 2 .
(3.36)

Remark 3.1. Proposition 3.1 holds for the full discretization of the system in a periodic framework. This can be shown by using the fact that if we assume μ(t, ξ) = 0 for all (t, ξ), then it is shown in [START_REF] Colin | A numerical model for the Raman amplification for laser-plasma interaction[END_REF] that any solution of (3.24)-(3.26) satisfies

2α 1 |u k | 2 2 + α 2 |r k | 2 2 + α 3 |w k | 2 2 = 2α 1 |u 0 | 2 2 + α 2 |r 0 | 2 2 + α 3 |w 0 | 2 2 ,
where

|f | 2 2 = N j=1 |f j | 2
is the l 2 discrete norm. In our case, we have to include the Landau damping term μ(t, ξ).

The key point is that we have used a time splitting discretization which allows us to write

2α 1 |u k+1 | 2 2 + α 2 |r k+1 | 2 2 + α 3 |w k+1 | 2 2 = 2α 1 | u| 2 2 + α 2 | r| 2 2 + α 3 N j=1 |w k+1 j | 2 , (3.37)
where w k+1 j is defined by (3.20). By using twice the Parseval formula, the maximum principle (3.7) and (3.19), we obtain that

N j=1 |w k+1 j | 2 ≤ N j=1 | w j | 2
and the result follows :

2α 1 |u k+1 | 2 2 + α 2 |r k+1 | 2 2 + α 3 |w k+1 | 2 2 ≤ 2α 1 |u k | 2 2 + α 2 |r k | 2 2 + α 3 |w k | 2 2 .
(3.38)

The three wave resonance condition.

As noted in the introduction, we expect an exponential growth on A R which corresponds to the decomposition of the laser A 0 into a backscattered electromagnetic wave A R and an electronic plasma waves E. The classical matching conditions for this three waves resonance interactions is

k 0 = k R + k 1 , ω 0 = ω R + ω pe + ω 1 ,
where

ω 2 0 = ω 2 pe + k 2 0 c 2 , ω 2 R = ω 2 pe + k 2 R c 2 , (ω pe + ω 1 ) 2 = ω 2 pe + 3v 2 the k 2 1 .
As recalled in the introduction and since ω 1 ≪ ω pe (see [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF]), this last condition implies

ω 1 ≈ 3 2 ω pe (k 1 λ De ) 2 where λ De = v the ω pe is the Debye's length.
This means that (k 1 , ω 1 ) satisfies the dispersion relation of the linear part of (2.14) and therefore the term A * R A 0 e i(k 1 y-ω 1 t) is resonant and the process will be efficient. More precisely, the equation on E can be written in (neglecting δn and considering A R and A 0 as given)

i∂ t E + λ 2 D ω pe 2 ∂ 2 y E = Ae i(k 1 y-ω 1 t) . (3.39)
Let L be a characteristic scale in space, T in time. The dimensionless form of (3.39) reads:

i∂ t E + λ 2 D ω pe T 2L 2 ∂ 2 y E = AT e i((k 1 L)y-(ω 1 T )t) .
Introduce

k 1 L = k ε , ω 1 T = ω ε , and replace AT by A. Since ω 1 = 3 2 ω pe λ 2 D k 2 1 and ω 1 T = ω ε (with ω, k = O(1)), one gets ω pe λ 2 D T L 2 = ω 1 T k 2 1 L 2 = ε 2 ω k 2 ε = ω k 2 ε = O(ε).
The dimensionless form of (3.39) is therefore

i∂ t E + εω k 2 ∂ 2 y E = Ae i (ky-ωt) ε . (3.40)
A standard WKB expansion shows that

E = Be i (ky-ωt) ε + O(ε), with i(∂ t B + ω k ∂ y B) = A.
This implies a linear growth in time for B that travels at the group velocity ω k . On the other hand, if the equation is

i∂ t E + εω k 2 ∂ 2 y E = Ae i (ky-ω ′ t) ε ,
with ω ′ = ω, then E will stay of size O(ε). Of course, nonlinear versions of these results can be proved.

For numerical consideration, we can not take

ω 1 = 3 2 ω pe (k 1 λ De ) 2 .
In fact, if we consider the following scheme

i u k+1 j -u k j δt + αD + D - u k+1 + u k 2 j = βe i(k 1 x j -w 1 t k+ 1 2 ) (3.41)
then the source term on the r.h.s of (3.41) will be not resonant with the discretization of the linear Schrödinger equations and the growth of the plasma electronic waves will be not efficient numerically. To overcome this difficulty, we take the frequency ω 1 given by the numerical dispersion relation of the Crank-Nicholson scheme

i u k+1 j -u k j δt + αD + D - u k+1 + u k 2 j = 0. (3.42)
By seeking the solution (u k j ) of (3.42) under the form u k j = γe i(k 1 x j -w 1 t k+ 1 2 ) with γ = 0, we find that the discrete pulsation w 1d := ω 1 has to satisfy the following dispersion relation

ω 1d = 2 δt arctan αδt 1 -cos (k 1 δy) δy 2 , (3.43)
which is not the same relation than for the continuous linear Schrödinger equation ω 1 = αk 2 1 . Nevertheless, when δt → 0 and δy → 0, one finds the continuous dispersion relation, since

ω 1d = αk 2 1 - 1 12 (α 3 k 6 δt 2 + αk 4 δy 2 ) + 1 48 α 3 k 8 δy 2 δt 2 + O(δy 3 ) + O(δt 3 ). (3.44)
However, for finite values of δt and δy, (3.43) gives much better results than the continuous version, see below.

Numerical illustration

Here, in order to illustrate numerically this phase matching condition, we solve the following equation We take u 0 = 0, α = 2, β = 1, L = 4π, k 1 = 1. Note that in this case, the exact solution of (3.45)-(3.47) is

i∂ t u + α∂ 2 y u = βe i(k 1 y-w 1 t) , y ∈ [0, L], (3.45)
u(0, y) = u 0 (y), y ∈ [0, L], (3.46
u(t, y) = te i(k 1 y-ω 1 t) if ω 1 = 2k 2 1 , 1 ω 1 -αk 2 1 e i(k 1 y-ω 1 t) if ω 1 = 2k 2 1 .
In figure 2, the maximum amplitude of the solution u is plotted with δt = δy = 0.01. We can see that when we use the numerical dispersion relation, the growth of the amplitude of the solution is an accordance with the growth of the exact solution. In this case, when we use the theoretical dispersion relation, the growth of amplitude of u stops near the time t = 200, that means that the source term e i(k 1 y-w 1 t) becomes non-resonant for large time for the Crank-Nicholson scheme. In figure 3, in order to see the influence of the time step, we take δt = δy 2 = 0.0001. In this case the numerical dispersion relation is closer to the theoretical dispersion relation than in the case where δt = δy. In fact, the plot corresponding with the theoretical dispersion relation shows us that the growth of the amplitude stops later than in the first case (δt = δy). Anyway, on large time-scale, the results are different. We will see in the numerical results for the full system that this choice of ω 1d will be important in order to create electronic plasma waves that will exchange energy with electrons.

The boundary conditions.

For physical considerations, we use absorbing boundary conditions for A 0 and A R and δn. A lot of work involving transparent boundary conditions are available (see for example Di Menza [START_REF] Menza | Transparent and artificial boundary conditions for the linear Schrödinger equation[END_REF], X. Antoine-C. Besse [START_REF] Antoine | Mathematical and numerical analysis of nonlinear artificial boundary conditions for the one-dimensional Schrödinger equation[END_REF]). Here we take into account the particular physical setting and we use a very simple version of absorbing boundary conditions. In fact, in order to model a realistic plasma slab, non-periodic boundary conditions in (3.8)-(3.9) are imposed. This condition will ensure that if the Raman backscattered field, the laser field and the density fluctuation leave the simulation box no reflection can influence the propagation of the laser field and the growth of the Raman field. It appears physically that it is very important to impose absorbing boundary conditions for A 0 and A R and δn. In order to explain our choice of boundaries conditions, we introduce two simple independent (one for the Raman and laser fields and one for the density fluctuation) models on which one can explain our strategy.

Boundary condition for the Schrödinger equation

For the first model, we focus on the equations evolving the laser potential and the Raman backscattered potential :

i (∂ t A 0 + v 0 ∂ y A 0 ) + α 0 ∂ 2 y A 0 = ω 2 pe 2ω 2 0 δnA 0 - k 0 k De ω 2 pe ω 2 0 (∂ y E) A R e -i(k 1 y-ω 1 t) , (3.48) i (∂ t A R + v R ∂ y A R ) + α R ∂ 2 y A R = ω 2 pe 2ω R ω 0 δnA R - k 0 k De ω 2 pe ω 0 ω R (∂ y E * ) A 0 e i(k 1 y-ω 1 t) . (3.49)
where

α 0 = 1 2 v 0 (1 -v 0 ) and α R = ω 0 2ω R (1 - c 2 k 2 R ω 2 R
). The key point is that in physical applications, |v 0 | and |v R | are close and the dispersion coefficient α 0 , α R are closed to zero (α 0,R ≈ 10 -3 ). It follows that the linear part of equations (3.48)-(3.49) is a dispersive perturbation of a linear transport equation. (see [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF]- [START_REF] Colin | A numerical model for the Raman amplification for laser-plasma interaction[END_REF]). Therefore we will focus with the following Schrödinger equation For numerical illustration of this proposition, we use a Crank-Nicholson type scheme for (3.50)- (3.53). With the notations previously used, the scheme reads

i(∂ t u + ∂ y u) + ε∂ 2 y u = 0, 0 ≤ y ≤ 1, ( 3
u * = ε 2 |∂ y u(t, 1)| 2 -ε 2 |∂ y u(t, 0)| 2 . Remark 3.
i u k+1 j -u k j δt + (iD 0 + εD + D -) u k+1 + u k 2 j = 0, for 2 ≤ j ≤ N -1. (3.57)
We discretize the boundary condition (3.53) by using the following discretization of the linear transport equation

u k+1 N -u k N δt + D + u k+1 + u k 2 N = 0. ( 3
  N -1 j=2 u k+1 j+1 -u k+1 j δy 2 - N -1 j=2 u k j+1 -u k j δy 2   = - u 2 -u 1 δy 2 + u N -u N -1 δy 2 where u j = u k+1 j + u k j 2 .
Proof : The proof of this proposition follows the same line than that of the continuous case. We perform it by multiplying (3.57) by u k+1 -u k * , we take the real part of the result and get

ℜ N -1 j=2 u j+1 -u j-1 2δy i u k+1 j -u k j * + ℜ N -1 j=2 u j+1 -2 u j + u j-1 δy 2 u k+1 j -u k j * = 0
(3.59) which is the sum of two terms (I + II = 0). In order to compute the first term I, we substitute

i(u k+1 j -u k j ) * = -iδt u * j+1 -u * j-1 2δy + εδt u * j+1 -2 u * j + u * j-1 δy 2
in I as in the continuous case. This gives

I = δt 2δy 3 ℜ N -1 j=2 ( u j+1 -u j-1 )( u * j+1 -2 u * j + u * j-1 ), (3.60) = δt 2δy 3 N -1 j=2 | u j+1 | 2 -| u j-1 | 2 - δt δy 3 ℜ N -1 j=2 u * j ( u j+1 -u j-1 ) , (3.61) = δt 2δy 3 | u N | 2 + | u N -1 | 2 -(| u 1 | 2 + | u 2 | 2 ) -2ℜ( u N u * N -1 ) + 2ℜ( u 1 u * 2 ) , (3.62) = δt 2δy u N -u N -1 δy 2 - u 2 -u 1 δy 2 . (3.63)
Note that, the contribution of I is the same that in the continuous case. The second term reads

II = 1 δy 2 ℜ N -1 j=2 ( u j+1 -2 u j + u j-1 ) u k+1 j -u k j * , (3.64) II = 1 δy 2 ℜ N -1 j=2 ( u j+1 -u j ) u k+1 j -u k j * - 1 δy 2 ℜ N -1 j=2 ( u j+1 -u j ) u k+1 j+1 -u k j+1 * , (3.65) II = 1 δy 2 ℜ N -1 j=2 ( u j+1 -u j ) (u k+1 j -u k j ) * -(u k+1 j+1 -u k j+1 ) * + 1 δy 2 ℜ( u N -u N -1 )(u k+1 N -u k N ) * , (3.66) II = - 1 2 N -1 j=2   u k+1 j+1 -u k+1 j δy 2 - N -1 j=2 u k j+1 -u k j δy 2   + 1 δy 2 ℜ( u N -u N -1 )(u k+1 N -u k N ) * . (3.67)
Now, using the boundary condition (3.58) for the last term in (3.67), we get The initial condition is taken as u 0 (y) = e -50(y-0.5) 2 , we take ε 1 = 1.510 -2 , ε 2 = 10 -3 , ε 2 = 10 -4 , N = 512 and δt = δx.

II = - 1 2 N -1 j=2   u k+1 j+1 -u k+1 j δy 2 - N -1 j=2 u k j+1 -u k j δy 2   - δt δy u N -u N -1 δy 2 . ( 3
In figures 4,5 and 6, we see that the solution u propagates in the y-positive direction and leaves the simulation box with some reflections of size ε. Moreover the dispersive effects are more important when ε is greater. Finally in the last plot (figure 7, discrete L 2 -norm in function of time), we see that, when the solution leaves the domain, the energy decays with time which is in accordance with estimate (3.54). We will see that for the full system, since these reflections are small, the behavior of the solutions will be not affected in the interior of the domain by the boundary. We perform the same computations using Neumann boundary conditions ∂ y u = 0 at the point y = 1 for ε = 0.001. Figure 8 shows that the solution u leaves the domain at the point y = 1 with more reflections than in the case where we use the boundary condition (3.58

Boundary condition for the wave equation

We now focus on the wave equation (2.15) describing the evolution of the fluctuation density of ion

∂ 2 t -v 2 s ∂ 2 y δn = m e 4m i k 2 0 k 2 De ω 2 pe ω 2 0 ∂ 2 y |E| 2 + m e 4m i v s ∂ 2 y |A 0 | 2 + |A R | 2 , (3.69)
and more particularly on the source term (the ponderomotive force). This source term contains three terms that propagate at different velocities. The first term ∂ 2 y |E| 2 describes the fluctuation due to the propagation of the longitudinal electronic plasma waves. The second and the third terms

∂ 2 y |A 0 | 2 + |A R | 2
describe the fluctuation of density due to the propagation of the electromagnetic laser field and the Raman backscattered wave. The key point is that in physical applications, the velocity v s of the acoustic waves is small compared to, the group velocity v 0 of the laser and the group velocity v R of the Raman field. Moreover the group velocity of electronic plasma waves is also small compared to the electromagnetic waves. Therefore the significant part is

∂ 2 t δn -v 2 s ∂ 2 y δn = ∂ 2 y f 0 (y -v 0 t) + ∂ 2 y f R (y -v R t), 0 ≤ y ≤ 1, (3.70) δn(0, y) = 0, 0 ≤ y ≤ 1, (3.71)
∂ t δn(0, y) = 0, 0 ≤ y ≤ 1, (3.72)
where v 0 > 0, v R < 0 and v s is a small positive parameter such that v s ≪ v 0 , |v R |. Here, f 0 and f R are given functions and they refer to the fields A 0 and A R respectively. The

δn(t, y) = αf 0 (y -v s t) + βf 0 (y + v s t) + γf 0 (y -v 0 t) + δf R (y -v R t), (3.73)
where α, β, γ and δ are real constant. This explicit solution for δn shows us that the perturbations of density due to the source terms ∂ 2 y f 0,R (y -v 0,R t) propagate more rapidly than αf 0 (y -v s t) + βf 0 (y + v s t). We work on a time scale for the full system for which this perturbation at velocity v s does not have the time to leave the computational domain. It is therefore sufficient to build absorbing boundary conditions for the perturbations which leave the domain at velocity v 0 respectively v R at point y = L respectively at point y = 0. To ensure this, the boundary conditions for δn consists in our case in the following first order boundary conditions

∂ t δn + v R ∂ y δn = 0, y = 0, (3.74)
∂ t δn + v 0 ∂ y δn = 0, y = 1. (3.75)
In order to validate this kind of boundary condition, we use the scheme introduced by Glassey [START_REF] Glassey | Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension[END_REF]. It reads

δn k+1 -2δn k + δn k-1 δt 2 -v 2 s D + D - δn k+1 + δn k-1 2 = D + D -(f k 0 + f k R ), (3.76)
where the discret operator D + D -is defined by

(D + D -δn) i = δn i+1 -2δn i + δn i-1 δy 2 , i = 2, N -1. (3.77)
The quantities δn k 1 , δn k N are given by the following discretization of the boundary conditions (3.74)-(3.75) at each time step k

δn k+1 1 = (1 + v R δt δy )δn k 1 - v R δt δy δn k 2 , (3.78)
δn k+1 N = (1 - v 0 δt δy )δn k N + v 0 δt δy δn k N -1 . (3.79)
For the numerical illustrations of this problem, we take v s = 0.15, v 0 = 0.9, v R = -0.8,

f 0,R (t, y) = 0.02e - (y-1 2 -v 0,R t) 2 2
, N = 1024 and δt = δy min( 1 v 0 , 1 |vr| ). The initial conditions n 0 and n 1 are set to zero. On figure 9, we can see the evolution of the density fluctuation with time. At the beginning of the simulation, a perturbation due to the force f 0 , f R causes a sink of density. This perturbation propagates at four velocities which are v R , -v s , v s , v 0 . It is obvious that the perturbations propagating at velocity v R , v 0 go more quickly than the others propagating at velocity -v s , v s . The part of the perturbation traveling at velocity v R (v 0 ) leaves the simulation box at point y = 0 (y = 1), (see figure 10). The perturbations going at velocity ±v s still propagate. Remark 3.3. At this point, we gave some efficient absorbing boundary conditions for a linear Schrödinger equation and for a wave equation. For the linear Schrödinger equation endowed by the boundary condition (3.50), we found an estimate which shows us that the energy of the solution decrease with time. Unfortunately, we are not able to give such a theoretical result on a nonlinear version of this kind of system. Nevertheless, in the next part, we show that we can apply this kind of boundary conditions on a Zakharov model with a satisfactory numerical accuracy.

The boundary conditions for the Zakharov system.

We consider the following Zakharov model

i (∂ t u + v∂ y u) + ε∂ 2 y u = nu, 0 < y < 1, (3.80)
∂ 2 t δn -v 2 s ∂ 2 y δn = ∂ 2 y (|u| 2 ), 0 < y < 1, ( 3

Boundary condition for the full system

We now adapt our set of boundary conditions to

∂ t A 0 + v 0 ∂ y A 0 = 0, y = L, (3.87)
∂ t A R + v R ∂ y A R = 0, y = 0, (3.88)
∂ t δn + v 0 ∂ y δn = 0, y = L, (3.89)
∂ t δn + v R ∂ y δn = 0, y = 0. (3.90)
Concerning the plasma waves, since the group velocity of the electronic plasma waves is small with respect to v 0 and v R (see [START_REF] Colin | On a quasilinear Zakharov System describing laser-plasma interactions[END_REF], [START_REF] Colin | A numerical model for the Raman amplification for laser-plasma interaction[END_REF]), we use periodic boundary condition for E, that is E(0) = E(L). These boundary conditions are discretized as we described previously using a spectral method.

The detailed numerical results are given in the next section.

4 Numerical results

Basic simulations

In this part, we use the dimensionless quasi-linear type Zakharov model. We use the following set of values of parameters that are representative of the physics that is involved.

• the velocity of light is c = 3.10 8 ms -1 ,

• the thermal velocity of electrons is taken to be equal to v the = 0.1c,

• the mass ratio is taken to be m e m i = 1 2000 ,

• the sound velocity is c s = m e m i v the ,

• the plasma frequency ω pe = 3.10 15 s -1 ,

• the wave number of the laser field k 0 = 3.10 6 ,

• the Debye's length λ De = v the ω pe .

With this parameters, we compute the frequency ω 0 thanks to the dispersion relation

ω 0 = ω 2 pe + k 2 0 c 2 ,
and search for k R , k 1 , ω R and ω 1d with the following matching conditions for the three waves resonance condition

k 0 = k R + k 1 , ω 0 = ω R + ω pe + ω 1d ,
where

ω R = ω 2 pe + k 2 R c 2 , ω 1d = 2 δt arctan αδt 1 -cos (k 1 δy) δy 2
with a dichotomy process and where α = 3k 2 0 2k 2

De

. We compute the system on the spatial domain [0, Ly] with Lk 0 = 250. For the initial conditions, we consider an gaussian initial data for A 0 of the form A 0 (0, x) = 0.3e -0.01(x-40) 2 .

Since we deal with simulated Raman instability, we have to begin with a small perturbation on A R and we take A R (0, x) = 0.005A 0 (0, x). Furthermore, E, δn and ∂ t δn are taken equal to 0 at time t = 0, (we considerer that the plasma is at equilibrium when t = 0). The initial electron distribution function is assumed to be a Maxwellian :

F e (0, v) = 1 √ 2π e -v 2 2 ,
which gives the following initial condition for the Landau damping rate

ν(0, ξ) = π 8 k 3 De k 3 0 1 |ξ| 3 e - k 2 De 2k 2 0 ξ 2 .
The number of discretization points in space is N y = 2048 and we choose δt = δy min( 1 v 0 , 1 |vr| ). We perform the simulation on the time interval [0, T max ] with T max = 100. At the beginning of simulation, the laser field and the stimulated Raman field interact in order to create an electronic plasma waves. These three fields combine in order to create a perturbation on the low-frequency density δn. The Laser field propagates on the positive y-direction and create on its way Raman backscattered wave which propagate on the y negative direction. Concerning the evolution of the spatial profile of the electronic plasma wave created by the three wave resonance, we can see that its amplitude grows with time and therefore we confirm that its velocity is small with respect to the velocity of A R and A 0 as we can see in its dispersion relation. The figure 4.1 shows us that the energy exchanges between the different fields. In a first stage, there is a transfer of energy from the laser field to the Raman field and to the electronic plasma waves until an amplification threshold is reached. This stage occurs before time ω 0 t = 15. In a second stage, the evolution starts to be non-linear and the Landau damping acts. A new plasma waves is created and a transfer of energy between this plasma wave and the electrons take place. This lead to a creation of hot electron tail as we can see on the

Influence of the boundary conditions

It is clear that for the Schrödinger equation or even for the Zakharov system, some boundary conditions like Neumann ones give rise to reflexions that change deeply the result of the computation. It is not clear that this still the case for the complete system. The aim of this part is to quantify this effect on (2.12)-(2.17

∂ y A C = 0, y = L (4.2)
which give at each time step k

A k R 1 = A k R 2 , (4.3)
A k C N = A k C N-1 . (4.4)
The boundary condition for δn is given by

δn k+1 1 = (1 + v R δt δy )δn k 1 - v R δt δy δn k 2 , (4.5
)

δn k+1 N = (1 - v 0 δt δy )δn k N + v 0 δt δy δn k N -1 . (4.6)
We use the same parameters than the first simulation. As we can see on the left plot of figure 18, when the Raman field leaves the domain at time t = 70, some reflexions appear and this is clearly caused by the fact that we use Neumann boundary conditions for A R at point y = 0. These reflexions are propagated in the y positive direction and modify the electronic plasma waves (and also the electronic plasma waves). So these reflexions modify the electron distribution function as we can see on the figure [START_REF] Sanbomatsu | Quantitative comparison of reduced-description particle-incell and quasilinear-Zakharov models for parametrically excited Langmuir turbulence[END_REF]. In this case, more electrons are accelerated.

Neumann boundary conditions for δn

We investigate the symmetric case of previous section. We keep the boundary conditions (3.88)-(3.89) for A 0 and A R and we put Neumann boundary condition for the density's fluctuation δn. In figure 20, we can see reflexions on the density fluctuation. This occurs exactly when the Raman field leaves the spatial domain at time t = 60 -70. Note that the reflexions are more important in this case than the previous case. In figure 21, as in the previous case, more electrons are accelerated.

Influence of the dispersion relation

In this part, we emphasize the fact that it is important to use the numerical dispersion relation (3.43). For this, we compare the growth of the maximum amplitude of the electronic plasma waves in two case. The first one concerns the numerical dispersion relation and the second one deals with the theoretical dispersion relation. As in the case of the linear Schrödinger equation, figure 22 show us that when we use the theoretical relation dispersion, we underestimate the growth of the amplitude of the electronic plasma waves and it has an influence on the number of accelerated electrons as we can see in figure 23. In this plot, we can see that when we use the numerical dispersion relation then more electron are accelerated which is an accordance with the fact that the maximum of amplitude of E is more higher than in a non-resonant case.

Influence of the Landau damping process

In order to show how the Landau damping process works on the saturation of the Raman amplification, we compare the results with the case where the Landau damping rate is close to zero. We focus on the evolution in time of the discret l 2 norms |E| 2 2 . In the case where the Landau damping process can be neglected, we remark (in figure 24 ) that the growth of the electronic plasma amplitude and the Raman amplification is the same than the case where we take into account the Landau process. This means that the Landau damping process plays no role in this amplification process. After this first step, we observe that during the saturation of the electronic plasma wave (non linear process), the energy of plasma waves is greater. This corresponds with the fact, in this case, electrons play no role and plasma waves can not give some energy. It follows ( it depends on the parameter k1 k De , see [START_REF] Belaouar | Quasilinear electron acceleration in a driven plasma wave[END_REF] for more details) that the level of the Raman saturation decreases. v 2 F e (t, v)dv -1 2 for the full system (circle line) and for fixed A 0 , A R (solid line).

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

Figure 1 :

 1 Figure 1: The left plot corresponds with the electron distribution function F e at different time in function of the velocity v and the right plot corresponds with the Landau damping rate at different time in function of the frequency ξ.

  which give us u(t k+1 , y) = u(y), (3.31) r(t k+1 , y) = r(y), (3.32)

Lemma 3 . 1 .

 31 If µ 0 (ξ) ≥ 0 for all ξ, then for all k > 0, any solution µ k of (3.21)-(3.22) satisfy µ k (ξ) ≥ 0 for all ξ.

δt arctan αδt 1 -

 1 ) u(t, 0) = u(t, L). (3.47) by using the scheme (3.41) and we compare the results obtained with the theoretical dispersion relation ω 1 = αk 2 1 and the numerical dispersion relation ω 1 := ω 1d = 2 cos (k 1 δy) δy 2 .

Figure 2 :

 2 Figure 2: The maximum of amplitude of |u| in function of time. The circle-line correspond to the exact solution, the triangle-line correspond to the discrete dispersion relation and the solid-line with the continuous dispersion relation for δt = δy = 0.01. The triangle and circle lines are superposed.

Figure 3 :

 3 Figure 3: The maximum of amplitude of |u| in function of time. The circle-line correspond to the exact solution, the triangle-line correspond to the discrete dispersion relation and the solid-line with the continuous dispersion relation for δt = δy 2 = 0.0001. The triangle and circle lines are superposed.

2 .

 2 Using Poincare's lemma, proposition 3.2 expresses the fact that with the boundary condition (3.53), the energy (L 2 -norm) decreases with time. Therefore, the estimate (3.54) show us that we can define the boundary trace of ∂ y u at points y = 0 and y = 1.

2 Proposition 3 . 3 .

 233 .58) We have a discret version of proposition 3.Any solutions u k of (3.57)-(3.58) satisfies δy δt

  .68) and the result follows since I + II = 0. Now, we give some numerical illustrations for this boundary condition and we compare the results with the results obtained with the usual Neumann boundary condition ∂ y u = 0 at point y = 1. The Neumann boundary condition is expressed in terms of the unknown u k+1 N by u k+1 N = u k+1 N -1

Figure 4 :

 4 Figure 4: Modulus of the solution u(t) of the linear Schrödinger equation (3.50) endowed with the boundary condition (3.53) at time : t = 5, 510 -2 (A), t = 0.5 (B), t = 0.77 (C), t = 1.6 (D) for ε = 0.015. The scale of picture D is not the same than that of (A)-(B)-(C).

Figure 5 :Figure 6 :

 56 Figure 5: Modulus of the solution u(t) of the linear Schrödinger equation (3.50) endowed with the boundary condition (3.53) at time : t = 5, 510 -2 (A), t = 0.5 (B), t = 0.77 (C), t = 1.6 (D) for ε = 0.001. The scales of figures (C),(D) are not the same than that of (A) and (B).

Figure 7 :Figure 8 :

 78 Figure 7: L 2 -norm in function of time for three different values of ε of the solution u of (3.50) endowed by the boundary condition (3.53).

Figure 9 :

 9 Figure 9: Snapshop of density fluctuation δn solution of (3.70)-(3.72) endowed with the boundary conditions (3.74)-(3.75) at time t = 11, 6 for A), t = 35 for B), t = 47 for C), t = 59 for D).

Figure 10 :

 10 Figure 10: Snapshop of density fluctuation δn solution of (3.70)-(3.72) endowed with the boudary conditions (3.74)-(3.75) at time t = 95 for E), t = 126 for F ), t = 162 for G), t = 208 for H).

∂

  t u + v∂ y u = 0, y = 1, (3.84) δn(0, y) = 0, ∂ t δn(0, y) = 0, (3.85) ∂ t δn + v∂ y δn = 0, y = 1. (3.86) Even if we are not able to justify rigorously our set of boundary conditions for the full Zakharov system (3.80)-(3.86), the numerically accuracy is satisfactory. Indeed, in Fig 11, we have plotted the modulus of electric field u and the variation of density δn with respect to time. As seen on the snapshots, no visible reflexions can be seen on any of the curves. For the simulation described in Fig 11, we have taken δt = δy = 0.01. Of courses, these results has to be confirmed on the complete system

Figure 11 :

 11 Figure 11: |u|(y) (left )and δn(y) (right) solution of (3.80),(3.86) at different time for ε = 5.10-3.

figure 4 . 1 .Figure 12 :Figure 13 :Figure 14 :Figure 15 :

 4112131415 Figure 12: Snapshots of the modulus of the laser field at time t = 10n for n = 1, 3, 5, 7, 9, 10 in the resonant case.

Figure 16 : 10 Figure 17 :

 161017 Figure 16: l 2 discret norm in function of time for A C corresponding with the solid line, A R corresponding with the circle points and E corresponding with the square line.

  ). We will perform two different tests. In the first one, we put Neumann boundary conditions for A C and A R and we keep the boundary conditions (3.74)-(3.75) for the density fluctuation of density δn. In the second one, we use Neumann boundary condition for the density fluctuation and we keep the boundary conditions (3.88)-(3.89) for A C and A R .

4. 2 . 1

 21 Neumann boundary conditions for A C and A R The Neumann boundary conditions for A C and A R read ∂ y A R = 0, y = 0, (4.1)

Figure 18 :

 18 Figure 18: The modulus of the Raman component in space at time (from left to right) at time t = 10n for n = 6, 7, 9, 10 by using the Neumann boundary conditions on A 0 and A R .

10 Figure 19 :

 1019 Figure 19: The log of the spatially averaged electron distribution function against kinetic energy v 2 2 at initial time (solid line) and at the end of simulation : the circle point corresponds to the Neumann boundary conditions on A 0 and A R and the dashed line corresponds to the boundary conditions (3.80)-(3.86).

Figure 20 :

 20 Figure 20: The density fluctuation in space at time (from left to right) at time t = 10n for n = 6, 7, 9, 10 by using the Neumann boundary condition on δn.

Figure 21 :

 21 Figure 21: The log of the spatially averaged electron distribution function against kinetic energy v 2 2 at initial time (solid line) and at the end of simulation : the circle point corresponds to the Neumann boundary condition on δn and the dashed line corresponds to the boundary conditions (3.80)-(3.86).

Figure 22 :

 22 Figure 22: The maximum of amplitude of |E| in function of time. The dotted-line correspond with the numerical dispersion relation, and the circle line correspond with the theoretical dispersion relation.

10 Figure 23 :

 1023 Figure 23: The spatially averaged electron distribution function against kinetic energy, for the numerical dispersion relation (dashed line) and for the theoretical dispersion relation (circle line).

Figure 25 :

 25 Figure 25: Time evolution the electron energy 1 2

  ).

							ε = 0.015
	1.0										1.0
	0.9									A)	0.9										B)
	0.8										0.8
	0.7										0.7
	0.6										0.6
	0.5										0.5
	0.4										0.4
	0.3										0.3
	0.2										0.2
	0.1										0.1
	0.0										0.0
	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	1.0										0.012
	0.9									C)											D)
											0.010
	0.8
	0.7
											0.008
	0.6
	0.5										0.006
	0.4
											0.004
	0.3
	0.2
											0.002
	0.1
	0.0										0.000
	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0

Comparison with a first study

In this part, we want to compare our results with those obtained in [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF]. To this aim, in order to recover the results of [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF], we fixed the potential A 0 , A R as being localized Maxwellian functions A 0,R (x) ≈ e -ax 2 where a is a constant and we only solve equations (2.14)-(2.17) with the following initial conditions E(0, y) = 0, δn(0, y) = 0, ∂ t δn(0, y) = 0, H(0, ξ) = π 8

The energy is brought into this system by the source term of the Schrödinger equation (2.14) ∂ y (A 0 A * R e i(k 1 y-ω 1 t) ) and since we choose a such that √ a ≪ k 1 , the more important part of this source term is k 1 A 0 A * R e i(k 1 y-ω 1 t) . We take the same physical parameters that the first basic simulation and we want to compare the energy gain of the electrons in each case. We fixed the amplitude of A 0 and A R at the values found at the end of the basic simulation.

Since at time t = 0, the electron distribution function is a Maxwellian, the electron energy is v 2 F e (0, v)dv = 1. At the beginning of the simulation, Landau damping plays no role in both cases and this corresponds to the first stage of the simulation (figure 25). For the full system (2.12)-(2.17), the second stage starts at time t = 20 when a significant part of energy is given to electrons. This corresponds to the decay of the amplitude of the electronic plasma waves due to the interaction between the wave-wave and wave-particle processes (see 4.1).

The saturation of the electron energy (figure 25) occurs almost simultaneously with the wave energy saturation (see 4.1).

Concerning the case where we fixed the potential A 0 , A R , the behavior in time of the electron energy is about the same than for the full system. Nevertheless, since the amplitude of the source term k 1 A 0 A * R e i(k 1 y-ω 1 t) of the Schrödinger equation (2.14) does not not evolves with time, the amplitude of the electronic plasma wave created in this case is more important. So the electron energy increases earlier and saturates with a higher level than for the full system. This means that the system used in [START_REF] Belaouar | Theorical and numerical study of a quasilinear Zakharov system describing Landau damping[END_REF] overestimates the exchanged energy between the electronic plasma waves and the electrons.


François Dross
email: dross@lirmm.fr

Fractional triangle decompositions in graphs with large minimum degree

A triangle decomposition of a graph is a partition of its edges into triangles. A fractional triangle decomposition of a graph is an assignment of a non-negative weight to each of its triangles such that the sum of the weights of the triangles containing any given edge is one. We prove that every graph graph on n vertices with minimum degree at least 0.9n has a fractional triangle decomposition. This improves a result of Garaschuk that the same conclusion holds for graphs with minimum degree at least 0.956n. Together with a recent result of Barber, Kühn, Lo and Osthus, this implies that for all > 0, every large enough triangle divisible graph on n vertices with minimum degree at least (0.9 + )n admits a triangle decomposition.

Introduction

Decomposition and packing problems are central and classical problems in combinatorics, and, in particular, in design theory. Kirkman's theorem [START_REF] Kirkman | On a problem in combinatorics[END_REF] from the middle of 19th century gives a necessary and sufficient condition on the existence of a Steiner triple system with a certain number of elements. In the language of graph theory, Kirkman's result asserts that every complete graph with an odd number of vertices and a number of edges divisible by three can be decomposed into triangles. Note that if a graph can be decomposed into triangles, then its vertex degrees are even and the total number of edges is divisible by three. Barber, Kühn, Lo and Osthus [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF] showed that the same conclusion is true for large graphs satisfying these divisibility conditions if their minimum degree is not too far from the number of their vertices. In this short paper, we study the fractional variant of the problem and we use it, together with a result of Barber, Kühn, Lo and Osthus [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF], to improve the best known bound.

Let us fix the terminology we are going to use. A graph is a pair of sets (V, E) such that elements of E are unordered pairs of elements of V . The elements of V are called vertices and the elements of E are called edges. We denote by uv (or vu) the edge with vertices u and v. We denote by |G| the number of vertices of G. Two distinct vertices contained in the same edge are said to be adjacent or to be neighbours. Two edges that share a vertex are said to be adjacent. The degree of a vertex v is equal to the number of neighbours of v. Let gcd (G) denote the greatest common divisor of the degrees of the vertices of G.

Two graphs

G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) are isomorphic if there exists a bijection b from V 1 to V 2 such that uv is an edge of G 1 if and only if b (u) b (v) is an edge of G 2
for every two vertices u and v of G 1 . The complete graph K k is the graph with k vertices all mutually adjacent. The graph K 3 is also called a triangle.

A graph G 1 = (V 1 , E 1 ) is a subgraph of G 2 = (V 2 , E 2 ) if V 1 ⊆ V 2 and E 1 ⊆ E 2 .
The subgraphs of G 2 isomorphic to G 1 will be referred to as copies of G 1 .

Let H be a graph. An H-decomposition of a graph G is a set of subgraphs of G isomorphic to H that are edge disjoint such that each edge of G is contained in one of them. A graph is H-decomposable if it admits an H-decomposition. A K 3 -decomposition is also called a triangle decomposition and a graph is

triangle decomposable if it is K 3 - decomposable. A graph G is H-divisible if gcd (G)
is a multiple of gcd (H) and the number of edges of G is a multiple of the number of edges of H. It is easy to see that every Hdecomposable graph is H-divisible. However, the converse is not true. For example a cycle on six vertices is K 3 -divisible, but not K 3 decomposable. As noted previously, Kirkman [START_REF] Kirkman | On a problem in combinatorics[END_REF] proved that every K 3 -divisible complete graph is K 3 -decomposable. The fact that, for all H, every H-divisible complete graph is H-decomposable remained an open problem for over one hundred years before it was solved by Wilson [START_REF] Wilson | Decomposition of complete graphs into subgraphs isomorphic to a given graph[END_REF].

The first generalisation to graphs that are near complete is due to Gustavsson [START_REF] Gustavsson | Decompositions of large graphs and digraphs with high minimum degree[END_REF]. He proved that for every graph H, there exist n 0 (H) and (H) such that every Hdivisible graph with n ≥ n 0 (H) vertices and minimum degree at least (1 -(H))n is Hdecomposable. This has been generalised to hypergraphs in a recent result of Keevash [START_REF] Keevash | The existence of designs[END_REF]. The best that is known to date for a general graph H is due to Barber, Kühn, Lo and Osthus [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF], who gave a way to turn a fractional decomposition into an exact one. They proved the following: Let F be a graph with minimum degree δ(F ) and let H be a graph with chromatic number χ(H) and e(H) edges. Let C := min{9χ(H) 2 (χ(H) -1) 2 /2, 10 4 χ(H) 3/2 } and let t := max{C, 6e(H)}. Then for all > 0, there exists an n 0 such that every Hdivisible graph G on n ≥ n 0 vertices with δ(G) ≥ (1 -1/t + )n has an H-decomposition. For some particular classes of graphs, the exact asymptotic minimum degree threshold is known [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF] [START_REF] Yuster | The decomposition threshold for bipartite graphs with minimum degree one[END_REF].

A fractional H-decomposition of a graph G is an assignment of non-negative weights to the copies of H in G such that for an edge e, the sum of the weights of the copies of H that contain e is equal to one. A graph is fractionally H-decomposable if it admits a fractional H-decomposition. A graph can be fractionally H-decomposable without being H-divisible. A fractional K 3 -decomposition is also called a fractional triangle decomposition and a graph is fractionally triangle decomposable if it is fractionally K 3 -decomposable. For all r ≥ 2, Yuster [START_REF] Yuster | Asymptotically optimal K k -packings of dense graphs via fractional K kdecompositions[END_REF] proved that every graph on n vertices with minimum degree at least 1 -1 9r 10 n is fractionally K r -decomposable, and Dukes [3][4] proved that the same result holds for sufficiently large graphs on n vertices with minimum degree at least 1 -2 9r 2 (r-1) 2 n. Our paper already led to further research: Barber, Kühn, Lo, Montgomery and Osthus [START_REF] Barber | Fractional clique decompositions of dense graphs and hypergraphs[END_REF] proved, building on our combinatorial approach, that every graph on n ≥ 10 4 r 3 vertices with minimum degree at least (1 -1/10 4 r 3/2 )n has a fractional K r -decomposition.

In this paper we will focus on triangle decompositions of graphs with large minimum degree. The following conjecture is due to Nash-Williams [START_REF] Nash-Williams | An unsolved problem concerning decomposition of graphs into triangles[END_REF]: Conjecture 1 (Nash-Williams [START_REF] Nash-Williams | An unsolved problem concerning decomposition of graphs into triangles[END_REF]). Let G be a K 3 -divisible graph with n vertices and minimum degree at least 3 4 n.

If n is large enough, then G is K 3 -decomposable.
The best result towards a proof of Conjecture 1 is due to the combination of results of Garaschuk [START_REF] Garaschuk | Linear methods for rational triangle decompositions[END_REF] and Barber, Kühn, Lo and Osthus [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF].

Theorem 2 (Garaschuk [5], Barber, Kühn, Lo and Osthus [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF]). There exists an n 0 such that every K 3 -divisible graph G on n ≥ n 0 vertices with minimum degree at least 0.956n is K 3 -decomposable.

The proof of Theorem 2 relies on a result on fractional K 3 -decomposability, which we now state. The following appears as a conjecture in [START_REF] Garaschuk | Linear methods for rational triangle decompositions[END_REF]. Note that for K 3 -divisible graphs, this is a consequence of Conjecture 1.

Conjecture 3 (Garaschuk [5]). Let G be a graph with n vertices and minimum degree at least 3 4 n. If n is large enough, then G is fractionally K 3 -decomposable. The best known result towards proving Conjecture 1 was established by Garaschuk [START_REF] Garaschuk | Linear methods for rational triangle decompositions[END_REF].

Theorem 4 (Garaschuk [5]). Let G be a graph with n vertices and minimum degree at least 0.956n. The graph G admits a fractional triangle decomposition.

In this paper we use a different method to prove the following.

Theorem 5. Every graph with n vertices and minimum degree at least 9 10 n admits a fractional triangle decomposition.

In [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF], a particular case of Theorem 11.1 and Lemma 12.3 imply the following. Theorem 6 (Barber, Kühn, Lo and Osthus [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF]). Suppose there exist n 0 and δ such that every graph on n ≥ n 0 vertices with minimum degree at least δn is fractionally K 3decomposable. For all > 0, there exists n 1 such that every K 3 -divisible graph on n ≥ n 1 vertices with minimum degree at least max δ, 3 4 + n vertices is K 3 -decomposable. Together with Theorem 6, our result improves Theorem 2. Theorem 7. Let > 0. There exists an n 0 such that every K 3 -divisible graph on n ≥ n 0 vertices with minimum degree at least ( 9 10 + )n is K 3 -decomposable.

Proof of Theorem 5

Let δ = 1 10 . Fix a graph G with n vertices and minimum degree at least (1 -δ) n. Suppose the graph G has at least one triangle with three vertices of degree at least (1 -δ) n+2. Let G be the graph G where the edges of one such triangle are removed. Observe that G has minimum degree at least (1 -δ) n and that if G has a fractional triangle decomposition, then G has one too. By doing this operation several times, we can assume that G has no triangle with three vertices of degree at least (1 -δ) n + 2. Let m be the number of edges of G.

Initially, we give the same weight w ∆ to every triangle such that the sum of the weights of the triangles is equal to m 3 . We will modify the weights of the triangles to obtain a fractional triangle decomposition. We will do so in a way that the total sum of the weights is preserved.

We define the weight of an edge e to be the sum of the weights of the triangles that contain e. Given H a copy of K 4 in G, and two non-adjacent edges e 1 and e 2 in H, let -- us call (H, {e 1 , e 2 }) a rooted K 4 of G. We will use the following procedure to modify the weights of the edges of a rooted K 4 of G: Let (H, {e 1 , e 2 }) be a rooted K 4 of G. By removing a weight w from the two triangles of H that contain e 1 and adding the same weight w to each of the other two triangles (i.e. those that contain e 2 ), we transfer a weight of 2w from e 1 to e 2 . The weights of all the other edges of the graph remain unchanged (see Figure 1).

+ - + - + + + - + -
To prevent the weight of any triangle from becoming negative, we have to restrict how much weight we can transfer using the procedure above. If for some w we use the procedure to transmit a weight of 2w from an edge to another one, then any triangle's weight is lowered by at most w for triangles that are in the K 4 , and does not change for other triangles. Moreover, since every triangle contains a vertex with degree at most (1 -δ)n + 1, any triangle is in at most (1 -δ)n -1 copies of K 4 , and thus in at most 3(1 -δ)n -3 rooted K 4 (since for each K 4 there are three possible choices for the pair of edges). Since each triangle has an initial weight of w ∆ , if each rooted K 4 containing that triangle is used to transfer weight of at most 2w ∆ 3(1-δ)n-3 between its labelled edges, then its final weight will be non-negative.

For each edge e, let T e be the number of triangles of G containing e. We express redistributing the weights as a flow problem in an auxiliary graph, which is denoted by G. The vertices of G are the edges of G, plus two special vertices, called the supersource and the supersink. Two edges of G are adjacent as vertices in G if they form a pair in a rooted K 4 . The edge between them is set to have the capacity c = We will show that G has a flow of value M from the supersource to the supersink.

If G does not have a flow of value M , then it has a vertex cut (A 0 , B 0 ) such that the supersource is contained in A 0 , the supersink in B 0 and the sum of the capacities of the edges from A 0 to B 0 is less than M . Let A be the edges of G corresponding to the vertices of A 0 and B the edges corresponding to the vertices of B 0 . Note that |A| = |A 0 | -1 and

|B| = |B 0 | -1. Finally, let k = |A|, and observe that |B| = m -k.
Let T A and T B be the average T e for e in A and in B respectively. Let e = uv be an edge of G. Let W e be the set of the vertices w such that uvw is a triangle. By the definition of T e , |W e | = T e . Each vertex of W e is non-adjacent to at most δn vertices of G, and thus is non-adjacent to at most δn vertices of W e . So each vertex of W e is adjacent to at least T e -δn vertices of W e . Therefore e is in at least Te(Te-δn) The inequalities ( 1) and ( 2) can be rewritten using that Since the summand is a convex function of T e , we obtain the following.

(T A (T A -δn) -2k)c -2(T A w ∆ -1) < 0 (3) (T B (T B -δn) -2(m -k))c -2(1 -T B w ∆ ) < 0 (4)
The inequality (3) implies that

T A (T A -δn) + 2 c (1 -T A w ∆ ) < 2k. (5)
The inequality (4) implies that

2k < 2m -T B (T B -δn) + 2 c (1 -T B w ∆ ). ( 6
)
We now combine the inequalities ( 5) and ( 6) and we substitute c = 2w ∆ 3(1-δ)n-3 to get the following.

T A (T A -δn) -(3n(1 -δ) -3)T A < 2m -T B (T B -δn) -(3n(1 -δ) -3)T B ( 7
)
Let e be an edge of G. Each end-vertex of e is non-adjacent to at most δn vertices of G. Hence, the edge e is contained in at least n -2δn triangles. Since e cannot be contained in more than n triangles, we get that n -2δn ≤ T e ≤ n. Consequently, we have n -2δn ≤ T A , T B ≤ n.

A standard analytic argument shows that the left hand side of ( 7) is minimized when T A = n and the right hand side is maximized when T B = n -2δn. Consequently, it must hold that

n(n -δn) -3(1 -δ)n 2 + 3n < 2m -(n -2δn)(n -3δn) -3n(1 -δ)(n -2δn) + 3n -6δn. Therefore (2 -12δ + 12δ 2 )n 2 < 2m -6δn. (8)
Recall that we assumed that in G there is no triangle with three vertices of degree at least (1-δ)n+2. Let V b be the set of vertices of degree at least (1-δ)

Assume n < 20. Since G has minimum degree at least (1 -δ)n = 0.9n, the graph G is a complete graph. Then giving the same weight to every triangle leads to a fractional triangle decomposition of G.

Therefore we can assume that n ≥ 20, and thus 4 + n -12δn ≤ 0. We get from ( 9) that (1 -10δ)(1 -δ) = 1 -11δ + 10δ 2 < 0. Since δ = 0.1, this leads to a contradiction. Therefore, there must exist a flow of value M in G, and hence, as described previously, the weights of the triangles can be adjusted in such a way that these weights now form a fractional decomposition of G. This finishes the proof of Theorem 5.

Conclusion

In this paper we proved that every graph on n vertices with minimum degree at least 9 10 n is fractionally triangle decomposable. Together with a result of Barber, Kühn, Lo and Osthus [START_REF] Barber | Edge-decompositions of graphs with high minimum degree[END_REF], this implies that, for all > 0, there exists a constant n 0 such that every triangle divisible graph on n ≥ n 0 vertices with minimum degree at least (0.9 + ) n is triangle decomposable.

Figure 1 :

 1 Figure1: By removing some weight w from the two triangles containing the thick edge and adding w to the two triangles containing the dashed edge, we remove 2w from weight of the dashed edge and add 2w to the weight of the thick edge.

  δ)n-3 . Let E c be the set of these edges. If T e w ∆ > 1, then the vertex of G corresponding to e is joined to the supersource and the capacity of the corresponding edge of G is T e w ∆ -1. Likewise, if T e w ∆ < 1, then the vertex of G corresponding to e is joined to the supersink and the capacity of the corresponding edge is 1 -T e w ∆ . The vertices of G adjacent to the supersource are referred to as sources and those adjacent to the supersink as sinks. Let M = e source (T e w ∆ -1) = e sink (1 -T e w ∆ ) .

2 distinct copies of K 4 . 2 -( 1 -( 1 -( 1 -

 242111 Let e be a vertex of A. It is adjacent to at least Te(Te-δn) k vertices of B. Therefore the cut contains at least e∈A T e (T e -δn) 2 -k edges of E c . Similarly, it contains at least e∈B T e (T e -δn) 2 -(m -k)edges of E c . Moreover, for each source e that is in B and each sink e that is in A, the cut contains the edge between e and the supersource or the supersink. Recall that the capacities of the edges ofE 1 is c = 2w ∆ 3(1-δ)n .Therefore the sum of the capacities of the edges of G is at least e∈A T e (T e -δn) 2 -k c + e source∈B (T e w ∆ -1) + e sink∈A T e w ∆ ) . At the same time, it is also at least e∈B T e (T e -δn) 2 -(m -k) c + e source∈B (T e w ∆ -1) + e sink∈A T e w ∆ ) . Since the sum of the capacities of the edges in the considered cut is less than M , we get that e∈A T e (T e -δn) 2 -k c + e source∈B (T e w ∆ -1) + e sink∈A (1 -T e w ∆ ) < M (1) and e∈B T e (T e -δn) 2 -(m -k) c + e source∈B (T e w ∆ -1) + e sink∈A T e w ∆ ) < M. (2)

( 1 - 2 e∈B( 1 -

 121 T e w ∆ ) respectively as follows.e∈A (T e (T e -δn) -2k) c -2 e∈A (T e w ∆ -1) < 0 e∈B (T e (T e -δn) -2(m -k)) c -T e w ∆ ) < 0

2 4

 2 n+2 in G, and let n b be the number of vertices in V b . Let us prove that n b ≤ 2δn -4. Assume by contradiction that n b ≥ 2δn -3. Since every vertex in V b has at most δn -3 non-neighbours in V (G), every vertex in V b has at most δn-3 non-neighbours in V b , and thus has at least n b -δn+2 neighbours in V b . The graph induced by V b is triangle-free, has n b ≥ 2δn -3 vertices, and has minimum degree at least n b -δn + 2. A triangle-free graph on k vertices has at most k edges, thus it has minimum degree at most k 2 . This implies that n b -δn + 2 ≤ n b /2, i.e. n b /2 ≤ δn -2, and thus δn -1.5 ≤ δn -2, a contradiction. Therefore there are at most 2δn -4 vertices in V b . We have 2m ≤ (2δn -4)n + ((1 -2δ)n + 4)((1 -δ)n + 1) = (1 -δ + 2δ 2 )n 2 + 4 + n -6δn and thus, with (8), (2 -12δ + 12δ 2 )n 2 < (1 -δ + 2δ 2 )n 2 + 4 + n -12δn.

Acknowledgements

This work was done during the author's visit to the group of Dan Král' at the University of Warwick; the visit was partially supported from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259385. I am deeply grateful to Tereza Klimošová for her helpful comments and to Dan Král' for his careful reading and many suggestions. Special thanks to Ben Barber and Richard Montgomery for pointing out that we can assume that every triangle has a vertex with low degree.


Végétalisation des projets de construction

Quelle valorisation dans les bilans environnementaux par Analyse de Cycle de Vie

Qu'est-ce que l'Analyse du Cycle de Vie?

L'ACV est un outil d'évaluation des performances environnementales d'un système, dont les normes de la série ISO 14040 harmonisent sa réalisation. Un cadre méthodologique constitué de quatre étapes régit la réalisation d'une ACV (illustration 1): la définition des objectifs et du système étudié, l'inventaire des flux entrant et sortant du système (réalisation d'un bilan matière et énergie), la quantification des impacts environnementaux par des indicateurs adaptés et enfin l'interprétation des résultats (JOLLIET et al., 2010).

Illustration 1: Schéma d'interaction des étapes de l'ACV (Jolliet et al., 2010)

Influence de la végétation sur les indicateurs sélectionnés

Nous avons concentré notre attention sur l'influence de trois catégories de végétation : les prairies, les arbustes et les arbres. Notre analyse porte sur une recherche bibliographique pour quatre indicateurs identifiés comme pertinents pour évaluer l'impact des espaces végétalisés : le potentiel de changement climatique, l'épuisement des ressources biotiques et abiotiques, l'usage des sols et la biodiversité.

Changement climatique. L'indicateur de changement climatique en ACV a pour objet de mesurer l'impact de l'objet étudié sur le système climatique. Nous avons choisi d'exposer l'influence de la végétation sur le changement climatique au travers de plusieurs effets : le stockage de carbone dans la végétation, le changement d'albédo de surface, les propriétés hydrothermales de la végétation (e.g. l'évapotranspiration). Depuis plusieurs décennies, la communauté scientifique est consciente et préoccupée par le changement climatique, aussi nous avons pu trouver de nombreux éléments de réflexion sur le sujet. Des valeurs générales existent sur les stocks de carbone urbains dans la végétation ainsi que sur son albédo de surface, qu'il est possible d'inclure directement dans les ACV. La conclusion quant aux propriétés hydrothermales est plus nuancée : sous des climats chauds et secs, la régulation thermique qu'offre la végétation permet de réaliser des économies d'énergie conséquentes. Il y a donc là un fort potentiel de valorisation.

Biodiversité. L'indicateur biodiversité fait encore l'objet de débats dans la communauté de l'ACV. Dans une méthode de caractérisation, appelée méthode CML, deux moyens sont proposés : la quantification d'espèces végétales sur une unité de surface et la production de biomasse d'une parcelle. Cependant, nos lectures nous ont amenés à penser que ces indicateurs sont hautement contestables pour plusieurs raisons. En effet, la biodiversité ne consiste pas en une maximisation de la densité des espèces sur une surface et elle doit s'évaluer au regard de sa qualité et des services écosystémiques rendus. De plus, elle est variable entre les régions.

Pour l'instant, l'ACV ne propose pas de méthode de caractérisation pertinente pour évaluer la biodiversité mais d'autres méthodes restent mieux adaptées. L'inventaire d'espèces dites « cibles », représentatives du bon état écologique d'une strate végétale est la meilleure méthode d'évaluation. Entre autres, l'indicateur de Singapour, élaboré en 2009, semble faire consensus auprès des collectivités territoriales pour évaluer la qualité des espaces verts urbains.

Ressources biotiques et abiotiques. Même si il n'y a pas de consensus aujourd'hui sur l'apport et la consommation en éléments biotiques et abiotiques des espaces végétalisés, notre étude bibliographique permet de montrer qu'il est possible de trouver des chiffres pertinents à ce sujet. Ainsi, nous avons pu estimer à 110 m3/ha/an la production d'une eau de qualité par les prairies ou les forêts. Dans des conditions particulières, il a aussi été estimé entre 30 et 140 kg/ha/an la quantité de nitrates éliminés des eaux souterraines sous prairie. Le potentiel de production de biomasse des différentes strates végétales est répertorié dans le tableau 1. Nous avons également répertorié ou estimé des éléments concernant la consommation en engrais et en eau de ces espaces ainsi que l'énergie dépensée pour leur entretien. Cette partie de notre analyse montre qu'il existe des études, mais elles nécessitent d'être approfondies pour pouvoir être appliquées au cas des espaces végétalisés.

Usage des sols. L'indicateur d'usage des sols permet d'évaluer l'impact environnemental que peut avoir une activité du fait de l'occupation et de la transformation du sol nécessaire à son fonctionnement. Cet indicateur est une catégorie d'impact intermédiaire puisqu'il a notamment des répercussions sur la production biotique, la biodiversité ou encore le changement climatique. Différentes méthodes d'évaluation de cet indicateur existent mais aucune ne fait l'objet d'un consensus dans la communauté de l'ACV. Parmi ces méthodes, les approches fonctionnelles des sols, l'établissement de classes de sol ou encore l'utilisation d'indicateurs clés sont proposés. Toutes soulèvent l'importance d'une mesure de qualité d'usage des sols. Afin d'évaluer l'impact environnemental d'usage des sols, certaines études proposent de mesurer leur qualité à travers sa teneur en matière organique (MOS). En effet, elle joue un rôle important dans les services éco-systémiques rendus par les sols.

Cette teneur est intimement liée au type de végétation qui y est implanté et nous savons ainsi, selon les études menées par le Gis Sol, que les sols des forêts et des prairies permanentes renferment le plus de matière organique.

Indicateurs identifiés

Préconisations

Notre analyse à la fois scientifique et d'enquête nous a permis d'identifier des pistes de réflexion à mener pour améliorer l'outil ACV, dans le cas précis des projets végétalisés mais aussi de façon plus globale pour sa meilleure intégration dans le secteurs des projets de construction.

Une analyse scientifique qui interroge sur l'utilisation de l'ACV pour les espaces végétalisés

• Préciser les valeurs mises en avant par de nouvelles recherches par des mesures directes sur des projets de construction

•

Mutualiser les inventaires des analyses socio-économiques et des ACV

•

Multiplier les retours d'expériences en ACV sur les projets de construction Un manque de gouvernance pour l'ACV dans le domaine de la construction

•

Mener une réflexion sur le financement des travaux relatifs à l'ACV des infrastructures

•

Ne pas opposer l'ACV aux autres méthodes

•

Créer une gouvernance institutionnelle de l'outil ACV pilotée par le CGDD et l'IDRRIM

Conclusion

Notre analyse montre que le blocage lié à l'utilisation de l'ACV pour valoriser la végétalisation est essentiellement scientifique. Le sujet de la prise en compte de la végétalisation des infrastructures par la méthode ACV n'est donc pas encore mûr pour faire l'objet d'une action publique, si ce n'est proposer un investissement bien ciblé dans la re-cherche de la part des pouvoirs publics. Cet investissement pourrait permettre dans l'avenir, d'éviter les écueils d'une concurrence difficile à arbitrer entre des constructeurs toujours plus à même de proposer des projets verts sans en démontrer la réelle plus-value environnementale. GASTON (2011). « Mapping an urban ecosystem service : quantifying aboveground carbon storage at a city-wide scale ». In : Journal of Applied Ecology 48.5, p. 1125-1134. (2011). L'état des sols de France. Groupement d'intérêt scientifique sur les sols. (2009). Biomasse forestière, populicole et bocagère disponible pour l'énergie à l'horizon 2020. Rapport. Agence de l'environnement et de la Maîtrise de l'Énergie.

Bibliographie DAVIES, EDMONSTON, HEINEMEYER, LEAKE ET

GIS SOL

RANTIEN, COLIN, THIVOLLE-CAZAT, COULON et COUTURIER

  Rapport du Groupe d'Analyse d'Action Publique pour le master PAPDD, année universitaire 2015-2016. Pour le compte de la chaire Eco-conception. Pierre-Baptiste DELPUECH, Alban FARCHI, Florian GIRAUD, Valérie WIART Encadré par Mesdames Adélaïde Féraille, chercheur à l'ENPC au laboratoire Navier et Anne de Bortoli, ingénieure de recherche au laboratoire ville mobilité transport à l'ENPC. Dans le secteur de la construction, il est courant de mettre en avant la valeur environnementale des projets, mais rarement de manière chiffrée. Pourtant, mesurer l'apport d'un projet à l'environnement est une question intéressante, voire indispensable lorsque l'on cherche à comparer objectivement et quantitativement plusieurs options d'un projet. Dans un contexte de verdissement des projets de construction, il apparaît nécessaire de rechercher les éléments scientifiques permettant de vérifier les allégations données et d'interroger les acteurs concernés sur l'intérêt de la végétalisation des projets de construction. a végétalisation des projets d'infrastructures apparaît aujourd'hui comme la solution pour continuer à mener des projets de construction tout en tenant compte des aspirations des citoyens et des engagements de la France pour un développement plus durable. Mais tous les projets parsemés d'espaces verts ne sont pas forcément respectueux de l'environnement. Les labels de qualité, la responsabilité sociétale des entreprises, les bilans carbone et les déclarations environnementales sur les L produits de construction sont autant de démarches qui ne permettent pas de chiffrer les impacts de la végétalisation. Il semble nécessaire que les études comparatives des projets puissent être objectives et pertinentes en se basant sur des chiffres et des indicateurs établis sur une base scientifique solide. L'Analyse du Cycle de Vie (ACV), en tant qu'outil d'évaluation environnementale, semble pertinente et propose plusieurs indicateurs pour mesurer les impacts des espaces verts.

  INA P-G (2003). Les prairies. Rapport. INA P-G. JOLLIET, SAADE ET CRETTAZ, 2010. Analyse du cycle de vie, Comprendre et réaliser un écobilan. Presses polytechniques et universitaires romandes. MADGUNI et SINGH (2013). « Potential of forest biomass for energy conversion ». In : International Journal of Research in Applied, Natural and Social Sciences 1.1, p. 53-58.

espaces végétalisés par quelques acteurs de la construction

	Les acteurs interrogés et l'ACV
	L'ACV reste très méconnue des acteurs que nous
	avons interrogés. Les études qui sont réalisées sont
	celles exigées par les réglementations (études d'im-
	pact ou études socio-économiques). L'ACV est
	plus connue dans les domaines d'éco-conception.
	Ainsi, les ACV contribuent à valider les innova-
	Prairies tions de la recherche dans des secteurs comme les	Arbustes	Forêt
	MOS (Gis Sol, 2011) nouveaux matériaux, les types d'ouvrage ou les 70 t C/ha nouvelles chaussées. Certaines collectivités territo-	30 t C/ha	70 t C/ha
	riales de grande envergure comme la Ville de Paris Stockage de carbone 1,44 t C/ha ont les moyens d'avoir leurs propres experts à dis-(Davies et al., 2011) position. Mais aucune compétence en ACV n'est	118 t C/ha	283 t C/ha
	4t / ha / an (INA-PG, actuellement présente dans leurs services et elles ne Biomasse produite 2003) possèdent pas d'outil d'aide à la décision sur la	4t / ha / an (Ratien et al, 2009)	~100 t / ha / an (Mad-guni et Singh, 2013)
	question des atouts écologiques apportés par les es-
	paces verts. Leurs compétences reposent sur des
	connaissances empiriques, sur des bureaux d'études
	et des partenariats entretenus avec le monde uni-
	versitaire.
	Les maîtres d'ouvrage. La végétalisation des pro-	projets de construction comme des espaces supplé-
	jets de construction s'impose pour faire accepter	mentaires de refuge pour le faune et la flore sau-
	plus facilement aux riverains les nouvelles infra-	vage. Néanmoins, elle reconnait que la fonction de
	structures. Pour SNCF Réseau et VNF par	stockage de carbone ou de réutilisation de la bio-
	exemple, l'aménagement d'espaces verts, la préser-	masse n'est pas considérée.
	vation de la biodiversité ou la valorisation de la bio-masse ne sont pas des priorités. La sécurité et le coût d'investissement sont les deux arguments dé-cisifs pour réaliser des aménagements. Ils se contentent d'appliquer les réglementations en vi-gueur concernant notamment, la protection des es-pèces protégées en phase de travaux, la réalisation des études d'impact et des mesures compensa-toires. Les partenariats sur les déchets d'élagage restent anecdotiques et n'ont pas pour ambition d'être généralisés. Les services techniques de la Ville de Paris se montrent plus actifs pour l'implan-tation de nouvelles surfaces végétalisées. En effet, celles-ci participent à lutter contre l'effet îlot de chaleur urbaine et à répondre aux exigences du Plan Local d'Urbanisme, ce que confirme notre étude bibliographique. Au-delà de sa fonction réga-lienne et de respect de la règlementation, la DREAL Rhône-Alpes perçoit la végétalisation des	Le monde académique. Les chercheurs sont les plus à même de répondre aux questions concernant la valorisation de la végétalisation. Les recherches en bio-climatologie montrent que la végétalisation a un effet sur l'îlot de chaleur urbaine ou sur une meilleure performance énergétique des bâtiments équipés d'une toiture ou d'un mur végétalisé. De plus, les changements de consommation énergé-tique qui en découlent sont facilement mesurables par la réalisation d'une ACV. Dans le domaine du génie civil, selon le Cerema, il manque une réelle volonté de valoriser les espaces végétalisés dans les projets. En effet, actuellement,. la végétalisation est uniquement perçue comme un élément de sécuri-té, de maintien des trames vertes et bleues ou de qualité paysagère. De plus, la végétalisation est per-çue comme problématique pour l'exploitant car une réflexion sur son entretien devient obligatoire.

Tableau 1: Ordre de grandeur des apports de la végétalisation recensés dans la littérature scientifique

Valorisation des