Analysis of an Asymptotic Preserving Scheme for Relaxation Systems

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DOI:10.1051/m2an/2012042 www.esaim-m2an.org

ANALYSIS OF AN ASYMPTOTIC PRESERVING SCHEME FOR RELAXATION SYSTEMS

^∗

Francis Filbet

¹

and Am´ elie Rambaud

¹

Abstract. We consider an asymptotic preserving numerical scheme initially proposed by F. Filbet and S. Jin [J. Comput. Phys.229 (2010)] and G. Dimarco and L. Pareschi [SIAM J. Numer. Anal.

49(2011) 2057–2077] in the context of nonlinear and stiﬀ kinetic equations. Here, we propose a convergence analysis of such a scheme for the approximation of a system of transport equations with a nonlinear source term, for which the asymptotic limit is given by a conservation law. We investigate the convergence of the approximate solution (u^ε_h, v^ε_h) to a nonlinear relaxation system, whereε >0 is a physical parameter andhrepresents the discretization parameter. Uniform convergence with respect to εandhis proved and error estimates are also obtained. Finally, several numerical tests are performed to illustrate the accuracy and eﬃciency of such a scheme.

Mathematics Subject Classification. 35L02, 82C70, 65M06.

Received January 5, 2012. Revised August 7, 2012 Published online January 15, 2013.

1. Introduction

The numerical resolution of kinetic and hyperbolic equations is a challenging task because these models often contain small parameters describing multiscale phenomena. It is particularly difficult to capture the behavior of the solution characterized by time multiscale features since a straightforward discretization leads typically to stiff differential system of equations. Such problems are encountered in many physical applications, for example rarefied gas dynamics [9,10], semiconductor modeling [8], quasi-neutral plasma simulations [7], the list of possible applications being not exhaustive. The motivation of this work is closely related to rarefied gas dynamics simulations [10]. In this case the source term is represented by the nonlinear and non local Boltzmann operator modelling collision mechanism of particles, the intensity of collisions 1/εreaching several orders of magnitude. The difficulty with these stiff problems is that they become singular in the limit ε →0, where ε is a small parameter responsible of the stiffness. A straightforward discretization of such problems (using, for example, explicit Runge–Kutta schemes) generates a very restrictive condition on the time step Δt =O(ε), which becomes unfeasible forε1. In this paper we present a new approach based on the so called Asymptotic

Keywords and phrases.Hyperbolic equations with relaxation, ﬂuid dynamic limit, asymptotic-preserving schemes.

∗Both authors are partially supported by the European Research Council ERC Starting Grant 2009, project 239983-NuSiKiMo.

1 Universit´e de Lyon, UMR5208, Institut Camille Jordan, Universit´e Claude Bernard Lyon 1 43 boulevard 11 novembre 1918, 69622 Villeurbanne Cedex, France.[email protected]; [email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2013

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F. FILBET AND A. RAMBAUD

Preserving reformulation introduced initially in [16]. This work presents an important improvement in the rigorous convergence analysis of an asymptotic preserving scheme proposed in [9,10] in the framework of kinetic equations with stiﬀ operators. Of course, a complete analysis seems to be tricky for fully nonlinear kinetic equations like the Boltzmann equation. Therefore, we focus on a simple hyperbolic relaxation system, which is the standard benchmark model for all relaxation problems : for all (t, x) in R⁺×T:

⎧⎪

⎨

⎪⎩

∂_tu^ε+∂_xv^ε= 0, u^ε(0, x) =u^ε₀(x)

∂_tv^ε+a ∂_xu^ε=−1

εR(u^ε, v^ε), v^ε(0, x) =v^ε₀(x), (1.1) where a > 0 is a constant coeﬃcient to be discussed later, εis the relaxation parameter, which can be either large or small (leading to a stiﬀ source term) andR: R×R → Ris a nonlinear function such thatR(0,0) = 0.

The system of equations (1.1) is often referred as a two velocity kinetic model and it has been intensively studied for many years [3]. Here we will assume that the functionR ∈ C¹(R×R,R) and its partial derivative with respect to the second variable is bounded from below, then from the implicit function theorem it possesses a unique local equilibrium,

R(u, v) = 0 ⇔ v=A(u), (1.2)

where A is a locally Lipschitz continuous function with A(0) = 0. Therefore, under some assumptions on R and on the initial data, the solution (u^ε, v^ε) to (1.1) converges to (u, v) with v =A(u) and usolution to the conservation law [6,18] ⎧

⎨

⎩

∂_tu+∂_xA(u) = 0, inR⁺×T,

u(t= 0) =u₀, (1.3)

where the initial datumu₀ is given by

u₀ = lim

ε→0u^ε₀. (1.4)

Developing robust numerical schemes for (1.1) that work in various asymptotic regimes (for a wide range of values of ε > 0) becomes challenging. It has been done in the framework of Asymptotic-Preserving (AP) schemes [10,16]. From a numerical point of view, the treatment of the stiffness cannot be done with explicit schemes, so that semi-implicit or fully implicit schemes have to be applied, but this procedure becomes time consuming and should be avoided when the operatorRis nonlinear and non local since it requires an additional step for the numerical resolution of a nonlinear problem. One solution offered by E. Gabetta, L. Pareschi and G. Toscani [12] to design an uniformly stable scheme, was to penalize the nonlinear collision operatorRby a linear functionβ v and then absorb the linearly stiff part into the time variable to remove the stiffness.

More recently, F. Filbet and S. Jin [10] proposed to penalize the operatorR by the BGK type operator β(A(u)−v) in order to build stable schemes with respect to ε > 0. If such schemes are now numerically validated and extensively used to discretize various equations [9–12,16], their mathematical study has only been done in some particular cases [13,14]. Indeed, few works are devoted to the mathematical analysis of asymptotic preserving schemes for the approximation of hyperbolic systems with source terms. Actually most of the theoretical works concern the numerical analysis of Relaxation scheme where (1.1) is only solved in the asymptotic limit ε → 0 and for simple source terms. We refer for instance to the series of paper [1,2] and then [5,17,19,23] on relaxation schemes. The requirement of asymptotic preserving scheme is to be uniformly accurate and stable for all range values ofε >0.

Therefore, the goal of the present paper mainly consists in providing error estimates between the approximate solution (u^ε_h, v_h^ε) and the exact solution to (1.1) which are uniform with respect to εand the numerical parameterh. Here we propose a rigorous analysis of the Asymptotic Preserving scheme proposed by F. Filbet and S. Jin [10] and G. Dimarco and L. Pareschi [9] for a nonlinear relaxation operator.

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Concerning the mathematical study of the system (1.1), we refer for instance to [20] for convergence proof and to [21,22] for error estimates.

Theorem 1.1. Assume that the initial datum(u^ε₀, v₀^ε)is bounded independently of εin BV(T). Consider R ∈ C¹(R×R,R), withR(0,0) = 0and∇Ris uniformly bounded with respect tov and locally bounded with respect tousuch that, for any (u, v) in[−U₀, U₀]×R:

⎧⎨

⎩

|∂_uR(u, v)| ≤ g(U₀),

0 < β₀(U₀) ≤ ∂_vR(u, v) ≤ h(U₀), (1.5) where β₀, g and h are some constants depending only on U₀. Then (1.2) is satisﬁed and there exists a char- acteristic speeda >0 large enough (see the condition (2.6)below) such that the system (1.1) admits a unique solution (u^ε, v^ε)in C

[0,∞[, L¹(T)²

and there exists a constantC >0 which only depends on aand (u^ε₀, v^ε₀), such that for anyε >0:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

v^ε(t)±√a u^ε(t)

∞+T V(u^ε(t)) + T V(v^ε(t)) ≤ C ∀t >0, u^ε(t+τ)−u^ε(t)₁ ≤ C τ, ∀t∈R⁺, τ ∈R⁺,

v^ε(t+τ)−v^ε(t)1 ≤ C

ε τ, ∀t∈R⁺, τ ∈R⁺, v^ε(t+τ)−v^ε(t)₁ ≤ C_ντ, ∀t ≥ ν, τ ∈R⁺,

(1.6)

whereν >0, andC_ν only depends on a,(u^ε₀, v^ε₀)andν. Moreover, there exists β₀>0 such that, v^ε(t,·)−A(u^ε(t,·))₁ ≤ C

⎡

⎣e⁻ β₀t

ε v₀^ε−A(u^ε₀)₁ + ε

⎤

⎦. (1.7)

Finally, ifu^ε₀converges, asεgoes to zero, tou₀deﬁned by (1.4), then the sequence(u^ε, v^ε)converges to(u, A(u)) whenε goes to0, such that, for any ν >0: for all T >0,

u^ε(t)−u(t)_L¹ + v^ε(t)−A(u(t))_L¹≤C_t√

ε, ν≤t≤T, (1.8)

where C_T > 0 depends on the initial data and T and u is the unique entropic solution to the Cauchy prob- lem (1.3).

The paper is organized as follows. We present in Section2the asymptotic preserving scheme for the relaxation model, and state the both convergence results of the Asymptotic Preserving scheme when the relaxation pa- rameterεgoes to zero (Thm.2.2) and next when the discretization parameterhgoes to zero (Thm.2.3). Then, we establish some a priori estimates inL^∞ and BV on the numerical solution to the Asymptotic Preserving scheme in Section3in order to prove both the zero relaxation limit (Sect.4) and the convergence of the scheme (Sect.5). Notice that thesea prioriestimates are uniform both with respect to the relaxation parameterεand the time stepΔtand space stepΔx. Finally, we present some numerical results in Section6.

2. Numerical schemes and main results

We remind that whenR(u, v) =v−A(u), whereA∈ C¹(R,R) is a given function, the necessary and suﬃcient stability condition is given by the so called sub-characteristic condition [3,16]:

|A(u)| < √a. (2.1)

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It means that the propagation speed of the equilibrium problem has to be bounded by the speeds of the relaxation system, which is therefore dissipative. For more details about this case, we refer to [6,18,20].

Hence the sub-characteristic condition reads, in our case:

∂_uR(u, v)

∂_vR(u, v) < √

a. (2.2)

For the sequel, we deﬁne for any N > 0 andα >0,

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

U(N, α) :=

1 + √¹α

N,

F(N, α) := sup

|ξ|≤U(N,α)|A(ξ)|,

V(N, α) :=U(N, α) + √¹αF(N, α).

(2.3)

We also denote byI(N, α) the compact set I(N, α) :=

−√a V(N, α),√a V(N, α)₂

. (2.4)

Moreover, we assume that the initial conditionsu^ε₀, v₀^εare bounded independently ofεinL^∞(T), such that:

N₀:= max

supε>0u^ε₀_∞,sup

ε>0v₀^ε_∞

< ∞. (2.5)

Consider any a₀ >0 and assume that the functionR ∈ C¹(R×R,R) satisﬁes (1.2) and (1.5). We choose the characteristic speed√a >0 and the parameterβ >0 such that

√a > max

√a₀, g(V(N₀, a₀)) β₀(V(N₀, a₀))

and β = h(V(N₀, a₀)), (2.6) whereV is given by (2.3).

We present here the splitting Asymptotic Preserving scheme and its relaxed version. To this aim, we introduce a space time discretization based on a uniform grid of points (x_j_{+ 1/2})_j_∈_Z ⊂ T, with space stepΔx, and discrete timetⁿ=n Δt,n∈N, for which the time stepΔtsatisﬁes the CFL condition:

0 < λ:=

√a Δt

Δx < 1. (2.7)

We denote byh= (Δt, Δx) the discretization parameter.

2.1. An Asymptotic Preserving scheme for the relaxation system

In this section, we design a numerical scheme for system (1.1), by introducing a splitting between the linear transport part, and the nonlinear relaxation part, for which we will take advantage of the knowledge of the equilibrium (1.2). In the asymptotic regime ε → 0, the second equation of (1.1) becomes stiff and explicit schemes are subject to a stability constraintΔt=O(ε). Of course, implicit schemes allow larger time steps, but new difficulty arises in computing the numerical solution of a fully nonlinear problem at each time step. Here we want to combine both advantages of implicit and explicit schemes, that is, large time step for stiff problems and low computational complexity of the numerical solution at each time step. This is done, as said in the introduction, in the spirit of Asymptotic Preserving schemes for nonlinear relaxation problems introduced by F. Filbet and S. Jin [10] and G. Dimarco and L. Pareschi [9].

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Thus we construct a numerical solution (u^ε_h, v_h^ε) to (1.1) for (t, x)∈R⁺×Tas follows

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

u^ε_h(t, x) =

n∈N

j∈Z

uⁿ_j 1_C_j(x)1_[tn,tⁿ⁺¹[(t),

v^ε_h(t, x) =

n∈N

j∈Z

vⁿ_j 1_C_j(x)1_[tn,tⁿ⁺¹[(t),

(2.8)

whereC_j=]x_j−1/2, x_j+1/2[ are the space cells and the sequences (uⁿ_j)_n,j and (v_jⁿ)_n,j depend onεand are given below.

First, initial data are computed as the averaged values of the initial data through each space cell: for all j in Z,

u⁰_j= 1 Δx

Cj

u^ε₀(x) dx and v⁰_j= 1 Δx

Cj

v^ε₀(x) dx. (2.9)

Then, in order to discretize the system (1.1), we apply a splitting strategy into a linear transport part and a stiff ordinary differential part as follows. The first part consists to apply a time explicit scheme combined with a finite volume method to the following linear differential system

⎧⎨

⎩

∂_tu+∂_xv= 0,

∂_tv+a ∂_xu= 0, (2.10)

and then the second part deals with the stiﬀ ordinary diﬀerential equations

⎧⎪

⎨

⎪⎩

∂_tu= 0,

∂_tv=−1

εR(u, v). (2.11)

We ﬁrst approximate the linear transport part (2.10), that is, for a given (uⁿ, vⁿ), we compute u^n+1/2, v^n+1/2

at timetⁿ⁺¹ with a standard Finite Volume scheme, that is, for allj ∈ Z,

⎧⎪

⎨

⎪⎩

u^n+1/2_j = uⁿ_j − Δt D_hv_jⁿ, v_j^n+1/2 = v_jⁿ − Δt a D_huⁿ_j,

(2.12)

where D_hvⁿ_j and a D_huⁿ_j are discrete derivatives with respect to x of v and u, given for instance by a Lax–Friedrichs scheme, namely:

⎧⎪

⎪⎨

⎪⎪

⎩

D_hv_jⁿ= 1 2Δx

v_j+1ⁿ −v_j−1ⁿ

−√a

uⁿ_j+1−2uⁿ_j +uⁿ_j−1 , a D_huⁿ_j = 1

2Δx a

uⁿ_j+1−uⁿ_j−1

−√a

v_j+1ⁿ −2v_jⁿ+v_j−1ⁿ .

(2.13)

Remark 2.1. Of course, there is a wide range of possible choices for the numerical ﬂuxes. As we will see below, the main property of the numerical scheme for the linear transport term that we require is the TVD (Total Variation Diminishing) property, namely, for alln ∈N,

T V(u^n+1/2) ≤ T V(uⁿ) and T V(v^n+1/2) ≤ T V(vⁿ), whereT V(u) :=

j∈Z

|u_j+1−u_j|.

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Hence, the second part of the splitting only consists in approximating the nonlinear ordinary diﬀerential equation (2.11), for allj ∈ Z, starting from

u^n+1/2_j , v_j^n+1/2

. To this aim, we use the decomposition R(u, v) = [R(u, v)−β (v−A(u))] + β (v−A(u)),

where β >0 is given by (2.6). Then, we apply a time exponential scheme on the dissipative part and get the following numerical scheme:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

uⁿ⁺¹_j =u^n+1/2_j , v_jⁿ⁺¹=v_j^n+1/2−

v^n+1/2_j −A

u^n+1/2_j 1−

1 +β Δt ε

e^{−β Δt/ε}

−Δt

ε e^{−β Δt/ε}R

u^n+1/2_j , v_j^n+1/2 .

(2.14)

This numerical scheme (2.12)–(2.14) allows to deﬁne the sequence (uⁿ_j, vⁿ_j)_(j,n)∈Z×N.

2.2. Convergence results

We ﬁrst establish a convergence result on the asymptotic behavior of the numerical solution to (2.12)–(2.14) whenεtends to zero andh= (Δt, Δx) such that the CFL condition (2.7) is satisﬁed.

Theorem 2.2 (discrete asymptotic limit:hfixed andε→0). Under the assumption of Theorem 1.1 whereR satisfies(1.2)and(1.5),N₀ is given by(2.5), the characteristic speed√a >0and the parameterβ >0are given by (2.6). We also assume that the CFL condition (2.7) is verified. Then, the solution (u^ε_h, v^ε_h) given by (2.8) to the scheme (2.12)–(2.14) with the initial data(2.9), converges in L¹(T), as ε →0, to a numerical solution (u_h, v_h), that is,

u^ε_h(t)−u_h(t)1+v_h^ε(t)−v_h(t)1 ≤ C_te^−β⁰^Δt/ε

1 +δ⁰

1

+u^ε_h(0)−u_h(0)1+v^ε_h(0)−v_h(0)1, where C_t>0 depends on the initial data and t >0,(u_h, v_h)is a consistent approximation to the conservation laws (1.3) withv_h=A(u_h),

u_h(t, x) :=

n∈N

j∈Z

u¯ⁿ_j1_C_j(x)1_[tn,tⁿ⁺¹[(t), and the sequence

u¯ⁿ⁺¹_j = ¯uⁿ_j −Δt D_hA(¯uⁿ_j), j∈Z, n≥1, (2.15) where the initial datau¯⁰_j is the local average of u₀ is given by (1.4)andδ⁰=u^ε₀−A(v₀^ε).

The proof of this theorem is analogous to the one corresponding to the continuous problem. The fact that we are able to establish uniformBV bounds on the numerical solution allow to get error estimates with respect toε.

On the other hand, we propose a convergence result of the Asymptotic Preserving scheme whenh= (Δt, Δx) goes to zero andεis ﬁxed:

Theorem 2.3 (convergence analysis :εﬁxed andh→0). Under the assumption of Theorem 2.2, the solution (u^ε_h, v_h^ε)to the scheme (2.12)–(2.14)and the initial data(2.9)veriﬁes for all T >0

u^ε_h(t)−u^ε(t)1 +v_h^ε(t)−v^ε(t)1 ≤ C_t ε

Δtδ⁰ ε 1 + 1

+Δx^1/2

, t∈[0, T].

whereC_T >0 depends on the initial data andT,δ⁰=u^ε₀−A(v₀^ε).

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Gathering these results together with Theorem1.1, we get the following error estimates proving the uniform accuracy of the numerical scheme (2.12)–(2.14).

Theorem 2.4 (uniform error estimates with respect toεandh). Under the assumption of Theorem 2.2, we denote by

E_h,ε=u^ε_h(t)−u^ε(t)₁ +v_h^ε(t)−v^ε(t)₁,

where(u^ε_h, v_h^ε)is given by the scheme(2.12)–(2.14)and the initial data(2.9). Then we have for allT >0 Eh,ε ≤ C_Tmin

Δt ε

δ⁰ ε 1 + 1

+ Δx^1/2

ε ,e^−β⁰^Δt/ε

1 +δ⁰

1

+√ ε+√

Δx

, ν≤t≤T, whereC_T >0 depends on the initial data andT,δ⁰=u^ε₀−A(v₀^ε).

Proof. To prove this result we show that both error estimates are valid. On the one hand the classical convergence analysis in Theorem2.3gives for allT >0 that

E_h,ε ≤ C_T Δt

ε

δ⁰ ε 1 + 1

+ Δx^1/2 ε

, ν≤t≤T.

On the other hand, we splitEh,εas

Eh,ε=E_h,ε¹ +Eh+Eε,

with the numerical errorE_h,ε¹ =u^ε_h(t)−u_h(t)1 + v_h^ε(t)−A(u_h(t))1, where (u_h, A(u_h)) corresponds to the numerical solution in the asymptotic limit ε → 0 and is given by (2.15), the term Eh = u(t)−u_h(t)1 + v(t)−A(u_h(t))1, where (u, A(u) is the solution to the conservation laws (1.3) and the continuous error Eε=u^ε(t)−u(t)1 +v^ε(t)−A(u(t))1. From Theorem2.2, we have

Eh=u(t)−u_h(t)1 + v(t)−A(u_h(t))1≤C_Te^−β⁰^Δt/ε

1 +δ⁰

1

, ν≤t≤T.

Then, the errorEh represents theL¹error between the solution to the conservation laws (1.3) and its approximation by a ﬁrst order Lax-Friedrichs scheme, which givesE_h≤C_T√

Δxand ﬁnally from Theorem1.1, we have E_ε≤C_T√ε. Gathering these latter estimates, we get the second error estimate onE_h,ε.

3.

_{A PRIORI}

estimates

We ﬁrst make sure that the sub-characteristic condition (2.2) is always satisﬁed to ensure the stability of the scheme (2.12)–(2.14) and prove estimates on the solution to the relaxation problem which are uniform with respect to ε. In following section, we drop the subscriptsεfor sake of clarity.

3.1. A priori estimate on the relaxation operator

Let us focus on the second part of the scheme devoted to the approximation of the relaxation source term and give a technical lemma, which establishes a quasi-monotonicity property on the operatorG_ε,s with s >0.

In order to do this, we will rather consider the equivalent formulation on the diagonal variables wand z. Let us rewrite the splitting scheme on these variables. Foruand vgiven, we set

w = −v − √a u and z = +v −√a u. (3.1)

Therefore, the linear transport scheme (2.12) written for (w, z) exactly coincides with an upwind ﬁnite volume method: for allj ∈ Z, ⎧

⎪⎪

⎨

⎪⎪

⎩

w^n+1/2_j = w_jⁿ −√ a Δt

Δx

wⁿ_j −wⁿ_j−1 , z_j^n+1/2 = z_jⁿ +√a Δt

Δx

z_j+1ⁿ −zⁿ_j ,

(3.2)

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whereas the nonlinear stiﬀ part (2.14) yields, for allj ∈ Z

⎧⎪

⎪⎨

⎪⎪

⎩

wⁿ⁺¹_j = w^n+1/2_j +G_ε,Δt

w_j^n+1/2, z_j^n+1/2 , z_jⁿ⁺¹ = z_j^n+1/2 −G_ε,Δt

w_j^n+1/2, z_j^n+1/2 ,

(3.3)

with

G_ε,Δt(w, z) =

z−w 2 −A

−w+z

2√a 1−

1 +β Δt ε

e^{−β Δt/ε}

(3.4)

+Δt

−w+z 2√a,z−w

2

.

The main result of this section is to establish the quasi-monotonicity of the operatorG_ε,s, which will lead to L^∞ anBV estimates.

Lemma 3.1 (Comparison principle). Assume the function R satisﬁes (1.2), (1.5) and choose a > 0, β > 0 such that (2.6)is veriﬁed. Then,

(i) the sub-characteristic condition is satisﬁed for all(w, z)∈I(N₀, a₀), that is, ∂_uR

∂_vR(u, v)

< √a, (3.5)

where2√a u := −(w+z)and2v := z−w;

(ii) for allε, s >0, the source term operator G_ε,s is quasi-monotone on the compact setI(N₀, a₀), that is,

⎧⎨

⎩

−1 ≤ ∂_wG_{ε, s}(w, z) ≤ 0, ∀(w, z) ∈ I(N₀, a₀), 0 ≤ ∂_zG_{ε, s}(w, z) ≤ 1, ∀(w, z) ∈ I(N₀, a₀);

(3.6)

(iii) consider for i = 1,2, (w_iⁿ⁺¹, z_iⁿ⁺¹) two solutions to (3.3) corresponding to two initial data w_i^n+1/2, z^n+1/2_i

∈I(N₀, a₀). Then there exist wandz∈Rsuch that (w, z)∈I(N₀, a₀)and

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

wⁿ⁺¹₁ −wⁿ⁺¹₂ =

w^n+1/2₁ −w₂^n+1/2 1 +∂_wG_ε,s

w, z^n+1/2₁ +

z^n+1/2₁ −z₂^n+1/2

∂_zG_ε,s

w^n+1/2₂ , z , z₁ⁿ⁺¹−zⁿ⁺¹₂ =

z₁^n+1/2−z₂^n+1/2 1−∂_zG_ε,s

w^n+1/2₂ , z

−

w₁^n+1/2−w^n+1/2₂

∂_wG_ε,s

w, z^n+1/2₁ .

(3.7)

Proof. For any N₀ > 0 anda₀ >0, we ﬁrst observe that for (w, z) ∈ I(N₀, a₀), |u| ≤ V(N₀, a₀). Therefore, using the assumption (1.5) and the deﬁnition (2.3), we get that

∂_uR

∂_vR(u, v)

≤ g(V(N₀, a₀)) β₀(V(N₀, a₀)) < √

a, which proves the ﬁrst assertion (i).

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Now we prove (ii) the quasi-monotonicity property of G_ε,s. Computing the partial derivatives of G_ε,s, it yields for alls >0,

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

∂_wG_ε,s = −1 2

1−A(u)

√a 1−

1 + β s ε

e^{−β s/ε}

− s

2εe^{−β s/ε}

∂√_uRa +∂_vR

,

∂_zG_ε,s = +1 2

1 +A(u)

√a 1−

1 + β s ε

e^{−β s/ε}

+ s

2εe^{−β s/ε}

−√∂_uaR+∂_vR

.

Hence, from the implicit function theorem we obtain

A(u) =−∂_uR(u, A(u))

∂_vR(u, A(u))

and the sub-characteristic condition (3.5), we obtain that for all (u, v) ∈ I(N₀, a₀), ∂_wG_ε,s(w, z) ≤ 0 and

∂_zG_a,s(w, z)≥0. Moreover, still using condition (3.5), we also get for all (w, z)∈I(N₀, a₀)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

∂_wG_ε,s(w, z) ≥ −

1−βs ε e^−βs/ε

− ∂_vR(u, v)s εe^−βs/ε,

∂_zG_ε,s(w, z)≤

1−βs ε e^−βs/ε

+ ∂_vR(u, v)s εe^−βs/ε.

Now since |u| ≤ V(N₀, a₀) and from the choice of the parameter β in (2.6), it yields that|∂_vR(u, v)| ≤β.

Therefore, we conclude that−1≤∂_wG_ε,s(w, z) and∂_zG_ε,s(w, z)≤1, for all (w, z)∈I(N₀, a₀).

Finally (iii) follows from a ﬁrst order Taylor expansion ofG_ε,s.

This Lemma allows to obtain the following comparison principle.

Corollary 3.2. Consider fori= 1,2, two initial data

w^n+1/2_i , z_i^n+1/2)∈I(N₀, a₀

satisfying the monotonicity condition w^n+1/2₁ ≤w^n+1/2₂ and z₁^n+1/2 ≤ z^n+1/2₂ . Then, the numerical solution (wⁿ⁺¹_i , z_iⁿ⁺¹), given by (3.3) corresponding to the initial data(w^n+1/2_i , z_i^n+1/2)for i= 1,2, satisﬁes

wⁿ⁺¹₁ ≤wⁿ⁺¹₂ and z₁ⁿ⁺¹≤z₂ⁿ⁺¹.

3.2. L

^∞

estimates

We establish a uniform bound on the numerical solution to the scheme (2.12)–(2.14) with respect to the time-space steph= (Δt, Δx) such that (2.7) is satisﬁed.

Proposition 3.3. Consider any a₀ > 0 and N₀ given by (2.5). We assume that the function R satis- fies (1.2), (1.5) and choose a > 0, β > 0 such that (2.6) is verified. Moreover, the time step Δt satisfies the CFL condition (2.7). Then, for all n∈N,uⁿ∞ ≤ V(N₀, a₀)andvⁿ∞≤√a V(N₀, a₀).

Proof. We proceed in two steps and ﬁrst construct a particular solution (wⁿ, zⁿ)∈R²to the scheme (3.2)–(3.3) which is uniformly bounded, then we apply the comparison principle on the compact setI(N₀, a₀) to prove an L^∞ bound on (wⁿ, zⁿ).

To this aim we choose R₀ = (1 +√a)N₀ and construct a numerical solution (wⁿ, zⁿ) to (3.2)–(3.3) corresponding to the initial data (w⁰, z⁰) = (R₀, R₀), which does not depend on the space variable so that the

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transport step (3.2) is invariant. Then we apply the relaxation scheme (3.3), which yields wⁿ = −vⁿ − √a uⁿ andzⁿ = +vⁿ − √a uⁿ, where (uⁿ, vⁿ) are only given by

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

uⁿ=u⁰=−R₀

√a =

1 + 1

√a

N₀,

vⁿ =

1 + β Δt ε

e^−βΔt/εvⁿ⁻¹ +

1−

1 + β Δt ε

e^−βΔt/ε

A u⁰

−Δt

ε e^−βΔt/εR

u⁰, vⁿ⁻¹ .

Then, we proceed by induction to show that for alln ∈ {0, . . . , N}, we have (wⁿ, zⁿ)∈I(N₀, a₀).

We assume that (wⁿ⁻¹, zⁿ⁻¹)∈I(N₀, a₀), for somen≥1. Let us prove that (wⁿ, zⁿ)∈ I(N₀, a₀). On the one hand since uⁿ=u⁰, it yieldsuⁿ_∞≤ u⁰_∞≤U(N₀, a₀), whereU(N₀, a₀) is given by (2.3).

On the other hand, using a ﬁrst order Taylor expansion of the source termR(u⁰, .), we get that there exists v˜ⁿ⁻¹∈Rsuch that

vⁿ=

1 + β−∂_vR(u⁰,˜vⁿ⁻¹)

ε Δt

e^−βΔt/εvⁿ⁻¹ +

1−

1 +β−∂_vR(u⁰,v˜ⁿ⁻¹)

ε Δt

e^−βΔt/ε

A u⁰

. Therefore, denoting byλ_k∈R, the real number such that

λ_k:=

1 + β−∂_vR(u⁰,˜v^k)

ε Δt

e^−βΔt/ε, ∀k∈N,

with|˜v^k| ≤F(N₀, a₀), hence we getvⁿ=λ_n−1vⁿ⁻¹ + (1−λ_n−1)A(u⁰) and sincev⁰= 0, we have vⁿ =

1−ⁿ⁻¹

k=0

λ_k

A(u⁰).

Moreover, using that|u⁰| ≤U(N₀, a₀) and ˜v^k ≤√a V(N₀, a₀), for allk∈N, we get from (1.5) and (2.6), 0 <

1 +β−β₀ ε Δt

e^−βΔt/ε ≤ λ_k ≤

1 + β Δt ε

e^−βΔt/ε < 1, ∀k∈N. Therefore,vⁿ∞≤F(N₀, a₀) and (wⁿ, zⁿ)∈I(N₀, a₀).

Furthermore, starting from the following initial datum (w⁰, z⁰) = (−R₀,−R₀), we can also construct another particular solution (wⁿ, zⁿ)∈I(N₀, a₀) for alln∈ {0, . . . , N}.

Now, we proceed to the second step which consists in applying the comparison principle of Corollary3.2 to prove anL^∞ estimate for any initial datau⁰,v⁰ ∈L^∞(T) given by (2.9). From the deﬁnition of N₀, we have u⁰_∞andv⁰_∞≤N₀. Then, we have for the initial dataw⁰_∞,z⁰_∞≤(1+√a)N₀=R₀ ≤√a V(N₀, a₀), that is,

w⁰≤w⁰≤w⁰, and z⁰≤z⁰≤z⁰.

Thus, we proceed by induction and assume thatwⁿ ≤ wⁿ ≤ wⁿ and zⁿ ≤ zⁿ ≤ zⁿ. We ﬁrst consider the linear transport step (3.2) to (wⁿ, zⁿ) and get that wⁿ ≤ w^n+1/2 ≤ wⁿ andzⁿ ≤ z^n+1/2 ≤ zⁿ. Thus, we apply Corollary 3.2 to the two solutions to (3.3) associated to the initial conditions

w^n+1/2₁ , z₁^n+1/2 =

w^n+1/2, z^n+1/2 and

w₂^n+1/2, z₂^n+1/2

= (wⁿ, zⁿ) (and then (wⁿ, zⁿ)), we have wⁿ⁺¹≤wⁿ⁺¹≤wⁿ⁺¹ and zⁿ⁺¹≤zⁿ⁺¹≤zⁿ⁺¹,

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which ﬁnally gives for all n ∈ N, that (wⁿ, zⁿ) ∈ I(N₀, a₀). By construction of (uⁿ, vⁿ) we have proved that

uⁿ_∞≤V(N₀, a₀) andvⁿ_∞≤√a V(N₀, a₀).

3.3. BV estimates

In this section, we obtain a BV estimate on the numerical solution to the scheme (2.12)–(2.14) with the time-space steph= (Δt, Δx) such that (2.7) is satisﬁed.

Proposition 3.4. Assume that u₀, v₀ are uniformly bounded with respect to εin BV(T). For any a₀>0 and N₀ given by (2.5), we assume that the functionR satisfies(1.2),(1.5)and choose a >0,β >0 such that(2.6) is verified andhsatisfies(2.7). Then, for all n ∈ N, we have:

T V(wⁿ⁺¹) + T V(zⁿ⁺¹) ≤ T V(wⁿ) + T V(zⁿ).

The proof is straightforward applying (iii) of Lemma 3.1 to initial conditions

w_j^n+1/2, z_j^n+1/2 and

w^n+1/2_j+1 , z_j+1^n+1/2 .

4. Trend to equilibrium (Proof of Theorem 2.2)

For a sequenceu= (u_j)_j∈Z we set

u1 :=

j∈Z

Δx|u_j|.

In this section we ﬁrst focus on the asymptotic behavior of the numerical solution to (2.12)–(2.14) whenεgoes to zero or when times goes to inﬁnity. Then, we prove that this numerical solution converges to a consistent approximation of the conservation laws (1.3) whenεgoes to zero.

4.1. Asymptotic behavior

In this subsection, we drop the subscripts ε for sake of clarity and estimate the deviation to the local equilibriumδⁿ=vⁿ−A(uⁿ).

Proposition 4.1. Assume that u₀, v₀ are uniformly bounded with respect to εin BV(T). For any a₀>0 and N₀ given by (2.5), we assume that the functionR satisfies(1.2),(1.5)and choose a >0,β >0 such that(2.6) is verified and hsatisfies(2.7). Then the deviation from the equilibrium, δ =v−A(u) satisfies for alln ∈ N

and all ε >0 ⎧

⎨

⎩

δ^n+1/2

1 ≤ δⁿ₁ +C Δt, δⁿ₁ ≤ e^−β⁰^tⁿ^/εδ⁰

1 + C ε. (4.1)

whereC >0 is a constant only depending on the parametersa,β₀ and the BV norm of the initial data.

Moreover, if ε < Δtthen we get

δⁿ₁ ≤ e^−β⁰^tⁿ^/εδ⁰

1 + C_aΔte^−β⁰^Δt/ε. (4.2)

Proof. We set forj∈Z,n∈Nthe sequence of deviations from the equilibrium:

δ_jⁿ=vⁿ_j −A uⁿ_j

.

We ﬁrst consider the transport step (2.12) of the numerical scheme: for allj ∈ Z, we apply a Taylor expansion toA, then there existsξ_jⁿ such thatξ_jⁿ ≤ V(N₀, a₀) and

δ^n+1/2_j =δ_jⁿ− Δt 2Δx

a

uⁿ_j+1−uⁿ_j−1

−√ a

v_j+1ⁿ −2v_jⁿ+v_j−1ⁿ

− Δt

2ΔxA(ξ_jⁿ)

v_j+1ⁿ −vⁿ_j−1

−√a

uⁿ_j+1−2uⁿ_j +uⁿ_j−1 .

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Thanks to the uniformBV estimate, proved in Proposition3.4, the sub-characteristic conditionA(ξ_jⁿ) < √a and the TVD property of the numerical ﬂuxes we get the ﬁrst estimate (4.1), by multiplying byΔxand summing overj∈Z:

δ^n+1/2₁ ≤ δⁿ₁+Δt C_a

T V(v⁰) +√a T V(u⁰)

, (4.3)

whereC_a>0 is a constant only depending ona.

Then, we consider the second step of the numerical scheme (2.14). On the one hand, sinceuⁿ⁺¹=u^n+1/2, it yields

δⁿ⁺¹_j = δ_j^n+1/2

1 +β Δt ε

e^{−β Δt/ε} − Δt

u^n+1/2_j , v^n+1/2_j .

On the other hand, applying a Taylor expansion, since R(u, A(u)) = 0 we get that there exists η such that

|η| ≤ √

a V(N₀, a₀) and:

R

u^n+1/2_j , v_j^n+1/2

= ∂_vR

u^n+1/2_j , η

δ_j^n+1/2. Hence, we have

δⁿ⁺¹_j = δ_j^n+1/2

1 +

1−∂_vR(u^n+1/2_j , η) β

βΔt ε

e^{−β Δt/ε}. Therefore under the assumption (1.5), we set for alls≥0

g(s) =

1 +

1−β₀ β

s

e^−s,

for which we easily show that for alls∈R⁺, we have that e^−s ≤ g(s) ≤ e^−β⁰^s/β. Hence, takings=βΔt/ε δⁿ⁺¹

1 ≤ e^−β⁰^Δt/εδ^n+1/2₁. (4.4)

Finally, gathering (4.3) and (4.4), we obtain that there exists a constantC₁ >0 depending only ona,T V(u⁰) andT V(v⁰) such thatδⁿ⁺¹

1 ≤ e^−β⁰^Δt/ε [δⁿ₁ +C₁Δt]. By induction, we easily get δⁿ₁ ≤ e^−β⁰^tⁿ^/ε δ⁰

1 + C_aΔt e^−β⁰^Δt/ε

1−e^−β⁰^Δt/ε. (4.5)

To conclude we only observe thatxe^−x≤1−e^−x, for anyx≥0, then it gives the second estimate of (4.1): there exists a constantC >0, only depending ona,β₀,T V(u⁰) andT V(v⁰) such thatδⁿ₁ ≤ e^−β⁰^tⁿ^/εδ⁰

1+ C ε.

Moreover, whenε < Δt, we again start from the estimate (4.5) and note that 1/(1−e^−β⁰^Δt/ε)≤1/(1−e^−β⁰).

Thus, there exists another constantC >0, only depending ona,β₀, T V(u⁰) andT V(v⁰) such that δⁿ₁≤e^−β⁰^tⁿ^/ε δ⁰

1 +C Δte^−β⁰^Δt/ε,

which gives (4.2).

4.2. Proof of Theorem 2.2

We are now ready to perform the asymptotic analysis of the numerical scheme (2.12)–(2.14) when εgoes to zero. We keep now the subscriptεfor the solution to the relaxation system.

Let us consider the numerical solution (u^ε_h, v_h^ε) to the scheme (2.12)–(2.14) written in the form (3.2)–(3.3) with

w^ε_h = −v_h^ε −√a u^ε_h and z_h^ε = +v^ε_h − √a u^ε_h,