Relative entropy for the finite volume approximation of strong solutions to systems of conservation laws

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1 Relative entropy for the finite volume approximation of strong solutions to systems of conservation laws Clément Cancès, Hélène Mathis, Nicolas Seguin To cite this version: Clément Cancès, Hélène Mathis, Nicolas Seguin. Relative entropy for the finite volume approximation of strong solutions to systems of conservation laws. 23. HAL Id: hal Submitted on 8 Mar 23 v, last revised 5 Jan 26 v3 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 RELATIVE ENTROPY FOR THE FINITE VOLUME APPROXIMATION OF STRONG SOLUTIONS TO SYSTEMS OF CONSERVATION LAWS CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN Abstract. We study in this paper the finite volume approximation of strong solutions to systems of conservation laws. We derive error estimates in the multidimensional case, using the relative entropy between the strong solution and its finite volume approximation. The error terms are carefully studied, leading to a classical h /2 estimate under a BV assumption on the numerical approximation. Contents. Systems of conservation laws.. Strong, weak, and entropy weak solutions.2. Relative entropy 2.3. Organization of the paper 4 2. Entropy satisfying finite volume schemes Definition of the numerical scheme Stability estimates and main result 6 3. Continuous weak and entropy formulations for the discrete solution 8 4. Error estimate using the relative entropy Relative entropy for approximate solutions End of the proof of Theorem Concluding remarks 2 References 2. Systems of conservation laws.. Strong, weak, and entropy weak solutions. We consider a system of m conservation laws d t ux, t + α f α ux, t =. The system is set on the whole space x R d, and for any time t [, T ], T >. We assume that there exists a convex bounded subset of R m, denoted by Ω and called set of the admissible states such that 2 ux, t Ω, x, t R d [, T ]. The system is complemented with the initial condition 3 ux, = u x Ω, x R d. We assume for all α {,..., d} the functions f α : R m R m to belong to C 2 Ω; R m, and be such that u f α are diagonalizable with real eigenvalues, where

3 2 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN u φ denotes the matrix of the differential of φ : u φu. Moreover, we assume that there exists L f > such that 4 spec 2 uf α u + u f α u T [ L f, L f ], α {,..., d}, u Ω. The system is endowed with a uniformly convex entropy η C 2 Ω; R such that there exists M β > so that 5 spec 2 uηu [β; M], u Ω, and the corresponding entropy flux ξ C 2 Ω; R d satisfies for all α {,..., d} 6 u ξ α = u η T u f α. The existence of the entropy flux ξ amounts to assume the integrability condition 7 2 uη u f α = u f α T 2 uη. Despite it is well-known that even for smooth initial data u, the solutions of 3 may develop discontinuities after a finite time, our study is restricted to the approximation of smooth solutions u C R d R + ; Ω of 3. Such solutions are called strong solutions and they automatically satisfy the conservation of the entropy 8 t ηu + d α ξ α u = in R d R +. We refer for example to [Daf] for more details. Assuming that u L R d ; Ω, a function u L R d R + ; Ω is said to be a weak solution to 3 if, for all φ Cc R d R + ; R n, one has d 9 u t φ dxdt + u φ, dx + f α u α φ dxdt =. R d R + R d R d R + Moreover, u is said to be an entropy weak solution to 3 if u is a weak solution, i.e., u satisfies 9, and if, for all ψ Cc R d R + ; R +, it satisfies d ηu t ψ dxdt + ηu ψ, dx + ξ α u α ψ dxdt. R d R + R d R d R +.2. Relative entropy. In the scalar case, the comparison of two entropy weak solutions lies on the ruzhkov s paper [ru7] which has been extended to the comparison of an entropy weak solution with an approximate solution by uznetsov [uz76]. Several improvements can be found in [CCL94, Vil94, EGGH98, BP98, CH99, EGH]. In the case of systems of conservation laws, these techniques no longer work. Basically, the family of entropy entropy flux pair η, ξ is not sufficiently rich to control the difference between two solutions. Nevertheless, let us assume that one of these solutions is a strong solution u in the following definition and introduce: Definition. Relative entropy. Let u, v Ω, we define the relative entropy of v w.r.t. u by Hv, u = ηv ηu u ηu T v u, while the corresponding relative entropy flux Q : R n R n R d is defined by Q α v, u = ξ α v ξ α u u ηu T f α v f α u, α {,..., d}.

4 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 3 The notion of relative entropy for systems of conservation laws goes back to the early works of DiPerna and Dafermos see [DiP79], [Daf79] and the condensed presentation in [Daf]. It is also a powerful tool which has been extensively used for the study of hydrodynamic limits of kinetic equations see the first works [Yau9] and [BGL93], but also [SR9] for more recent results. In the context of systems of conservation laws, it is easy to check that, given a strong solution u and an entropy weak solution v with respective initial data u and v, one has d d t Hv, u + α Q α v, u α u T Z α v, u in the weak sense, where 2 Z α v, u = 2 uηu f α v f α u u f α u T v u. On the other hand, it follows from the definition of H that 3 Hv, u = θ which, together with 5, leads to v u T 2 uηu + γv uv u dγdθ, M 4 2 v u2 Hv, u β 2 v u2, u, v Ω. If u is assumed to be a strong solution, its first and second derivatives are bounded and by a classical localization procedure à la ruzhkov and a Gronwall lemma, one obtains a L 2 loc stability estimate for any r > 5 vx, T ux, T 2 dx CT, u v x u x 2 dx x <r x <r+l f T where the dependence of C on u reflect the needs of smoothness on u C blows up when u becomes discontinuous. This inequality, rigorously proved in [Daf], provides a weak strong uniqueness result and similar but more sophisticated ideas have been recently applied to other fluid systems [FN2]. Remark.. For general conservation laws, the relative entropy is not symmetric, i.e, Hu, v Hv, u and Qu, v Qv, u. In the very particular case of Friedrichs systems, i.e. when there exist symmetric matrices A α R m m α {,..., d} such that f α u = A α u, then u u 2 is an entropy and the corresponding entropy flux ξ is ξ α u = u T A α u, α {,..., d}. It is then easy to check that Hv, u = v u 2 = Hu, v, Q α v, u = v u T A α v u = Q α v, u, Z α v, u =, for all u, v R m. As a consequence, inequality becomes d t Hv, u + α Q α v, u even if u is a weak solution since H and Q are symmetric functions. This allows to make use of the doubling variable technique [ru7] to compare u to v, recovering the classical uniqueness result for Friedrichs systems [Fri54]. Our aim is to replace the entropy weak solution v in by an approximate solution provided by finite volume schemes on unstructured meshes. Following the formalism introduced in [EGGH98], this makes appear in bounded Radon measures which can be controlled, leading to error estimates in h /2 between a

5 4 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN strong solution and its finite volume approximation, h being the characteristic size of the cells of the mesh. Remark.2. In the context of Friedrichs systems, such error estimates can be obtained, even in the weak setting, for the sames reasons as those given in Remark., see [VV3, RJ5]. Remark.3. In [Tza5], Tzavaras studies the comparison of solutions of a hyperbolic system with relaxation with solutions of the associated equilibrium system of conservation laws. He also makes use of the relative entropy but, since he only focus on strong solutions for both systems, he does noeed to place himself in a weak setting with measure terms. Very similar questions have been adressed in [BV5, BTV9] for the convergence of kinetic equations towards the system of gas dynamics. Here again, only strong solutions of the Euler equations are considered. To finish the bibliographical review, let us mention the work by Leger and Vasseur [LV] where the reference solution may include some particular discontinuities..3. Organization of the paper. We first introduce in Section 2 the family of numerical schemes we consider for approximating the solution to the problem 3. Some natural assumptions are made on the scheme, but also on the numerical solution on which we assume some reasonable stability property. In order to compare the discrete solution denoted by u h in the sequel of the paper with the strong solution u, we write continuous weak and entropy formulations in Section 3, so that we can adapt the uniqueness proof proposed in [Daf]. Nevertheless, the discrete solution u h is obviously not a weak entropy solution. Therefore, some error terms coming from the discretization have to be taken into account in the formulation, which take the form of positive locally bounded Radon measures, following [EGGH98]. A large part of Section 3 consists in making these measures explicit and to bound them with quantities tending towards with the discretization size. In Section 4, we make use of the weak and entropy weak formulations for the discrete solution and of their corresponding error measures to derive an error estimate. The distance between the strong solution u and the discrete solution u h is quantified thanks to the relative entropy Hu h, u introduced in Definition.. 2. Entropy satisfying finite volume schemes 2.. Definition of the numerical scheme. Let T be a mesh of R d such that R d is the union of the closure of the elements of T. We denote h = sup{diam, T } <. For all T, we denote by its d dimensional Lebesgue measure, and by N the set of its neighboring cells. For L N, the common interface between and L is denoted by σ L and σ L is its d Lebesgue measure. We assume that there exists a > such that 6 ah d and := σ L hd a, T. L N The uniormal vector to σ L from to L is denoted n L. Let t > be the time step and we set = n t, n N. Remark 2.. In order to avoid some additional heavy notations, we have chosen to deal with an uniform time discretization and a space discretization that does not depend on time. Nevertheless, it is possible, by following the path described in [O], to adapt our study to the case of time depending space discretizations and to non-uniform time discretizations. This would be mandatory for considering

6 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 5 a dynamic mesh adaptation procedure based on the a posteriori numerical error estimators that can be derived from our study. For all, L T 2, L N, we consider the numerical flux G L C 2 Ω Ω; R m : u, v G L u, v which satisfies the following conditions: 7 8 G L u, v = G L v, u, u, v Ω 2, G L u, u = fu n L, u Ω. We assume that there exists λ > such that, for all λ > λ, for all T, and for all L N, 9 u λ [G Lu, v fu n L ] Ω, u, v Ω 2. We also assume there exists a numerical entropy flux ξ L which is conservative 2 ξ L u, v = ξ L v, u, u, v Ω 2, and such that, for all λ λ >, the following entropy inequality by interface is satisfied: 2 ξ L u, v ξu n L λ η u λ[ GL u, v fu n L ] ηu. We additionally assume that the following CFL condition holds t 22 λ σ L, T. L N In particular, the regularity of the mesh 6 implies that 22 holds if 23 t a2 λ h. We have now introduced all the necessary material to define the numerical scheme we will consider in what follows. Definition 2. Finite volume scheme. Let us introduce N T = max{n N, n T/ t + } and define the discrete unknowns u n, T and n {,..., N T }, by 24 u n+ t un + together with the initial condition 25 u = L N σ L G L u n, u n L = u xdx, T, under assumptions 7 2 on the numerical flux G L and under the CFL condition 23. The approximate solution u h L R d R + provided by the finite volume scheme is defined by 26 u h x, t = u n, for x, t < +, T, n {,..., N T }. An important property of such a numerical scheme is the preservation of the set of admissible states. Lemma 2.2. Assume that 3, 9 and 22 holds, then, for all T, for all n {,..., N T }, u n belong to Ω. Proof. Recall first that for all T, u is defined by 25. Since, thanks to 2, u takes its values in the convex set Ω, then so does its spatial mean values on the cells T. This ensures that u belongs to Ω for all T.

7 6 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN Now, let us notice that assumption 8 implies 27 σ L G L u, u =, u R m, T. L N Assume now that for all T, u n Ω. Using 24 and 27, we know that for all T and for all n N, one has 28 u n+ = σ L u n t G L u n, u n L fu n n L. L N Thanks to 9 and 22, u n+ is a convex combination of elements of the convex set Ω, then it also belongs to Ω. Thanks to 2, we can derive entropy consistency properties on the numerical scheme. Proposition 2.3. Let u h be defined by and thus, satisfying assumptions 7 2. Then the numerical entropy flux ξ L is consistent with ξ, i.e. 29 ξ L u, u = ξu n L, u Ω. Moreover, under the CFL condition 23, u h satisfies the discrete entropy inequality m 3 ηu n+ t ηun + ξ L u n, u n L. L N Proof. First we focus on the consistance property 29. We write the inequality 2 for ξ L u, u L and ξ L u L, u : ξ L u, u L ξu n L λ ηu λ [G Lu, u L fu n L ] ηu, ξ L u L, u ξu L n L λ ηu L λ [G Lu L, u fu L n L ] ηu L. Setting u = u L = u, then using the conservation property 2 and the consistency relation 8 lead to the consistency of the numerical entropy flux 29. Applying η to 28 and using its convexity yields 3 ηu n+ L N σ L η u n t G L u n, u n L fu n n L Now, it follows from the CFL condition 23, the interface entropy condition 2 and 32 σ L ξ L u, u =, u Ω that 3 holds. L N 2.2. Stability estimates and main result. In order to carry out an error estimate, we have to assume some regularity on the approximate solution u h. In practice, we assume from now on that there exists C BV > that may depend at most on r, T, u and on G L such that 33 N T t n=,l E r σ L G L u n, u n L fu n n L C BV. Remark 2.2. Assumption 33 requires several comments..

8 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 7 Since we consider expliciumerical schemes, the stability estimate 33 requires a CFL condition to be satisfied in order to prevent oscillations. One can reasonably assess that 34 t C stab h for some C stab > which may depend on the same quantities as C BV. We then have to choose t small enough so that both 34 and 23 are satisfied. 2 Assuming 33 is weaker than assuming that u h BV loc R + R d m. Indeed, consider the simple linear transport case where fu = u e for e =, T in R 2 with a grid whose edges are along e and e 2 =, T, and define G,L as the upstream flux. Then 33 only requires u h to have a bounded variation in the direction of e. We refer to [Des4, Lemma 5] for an example where u h is not uniformly bounded in BV loc R + R 2 but where 33 still holds. 3 In the scalar case m =, it is possible to recover estimate 33 on cartesian grids even for nonlinear flux functions f : R R d cf. e.g. [CH99, EGGH98, GR9]. On unstructured grids, despite 33 seems to remain true in the numerical experiments cf. [O], only a weaker inequality can be proved, namely N T t σ L G L u n, u n L fu n n L C wbv n=,l E h r for some C wbv >. This estimate, called weak BV estimate in [EGGH98, CH99], relies on the quantification of the numerical diffusion introduced by the scheme [Vil94, CCL94]. Using the BV assumption 33, one may derive entropy BV estimates and time BV estimates. Lemma 2.4. Assume that 7 2 and 33 hold, then one has 35 N T t n=,l E r σ L ξ L u n, u n L ξu n L u η L ΩC BV. Proof. This is a direct consequence of 2 in which the Lipschitz continuity of η has been taken into account. Lemma 2.5 time BV estimates. Under Assumption 33, one has N T n= N T n= T r u n+ un C BV, T r ηu n+ ηun u η L ΩC BV. Proof. Thanks to the definition 24 of the scheme and thanks to the divergence free property 27, for all T and all n N, one has un = t σ L G L u n, u n L fu n n L, so that u n+ u n+ un t L N L N σ L G L u n, u n L fu n n L.

9 8 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN Summing over T r and n {,..., N T } and using 33 provides 36. Inequality 37 then follows from the Lipschitz continuity of η. Let us now state our main result. Theorem 2.6. Assume that u W, R d and that the solution u of the Cauchy problem 3 belongs to W, [, T ] R d. Assume also that 3, 6, 7 2, 23, and identities 6 7 hold. Let u h defined by 24 26, and assume that the stability estimate 33 holds, then, for all r > and T > there exist C depending only on T, r, Ω, a, λ, u, G L, η and f, such that T B,r+L f T t u u h 2 dxdt Ch. 3. Continuous weak and entropy formulations for the discrete solution In order to obtain the error estimate of Theorem 2.6, we aim at using the relative entropy of u h w.r.t. u. Since u h is only an approximate solution, it does not exactly satisfies neither the weak formulation 9 nor the entropy weak formulation. Some numerical error terms appear in these formulations, and thus also appear the inequality of the relative entropy 38 d d t Hu h, u + α Q α u h, u α u T Z α u h, u + numerical error terms. As usual, these terms may be described by Radon measures see [BP98, EGGH98, CH99, O] in the scalar case and [RJ5] in the case of Friedrichs systems. Note that for nonlinear systems of conservation laws, a function which satisfies the entropy inequality is noecessarily a weak solution 9. This leads us to consider the measures not only in the entropy inequality of u h but also in the weak formulation of u h. Let us first begin with the entropy formulation and the related measures. For X = R d or X = R d R +, we denote MX the set of locally bounded Radon measures on X, i.e., MX = C c X. If µ MX we set µ, ϕ = X ϕdµ, ϕ C c X. Definition 3.. For ψ C c R d, ϕ C c R d R +, we define µ MR d and µ MR d R + by µ, ψ = ηu x ηu h x, ψxdx, R d µ, ϕ = µ T, ϕ + t σ L ξ L u n, u n L ξ L u n, u n µ L, ϕ n=,l E r + t σ L ξ L u n, u n L ξ L u n L, u n L µ L, ϕ, n=,l E r

10 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 9 where t n+ µ T, ϕ = ηu n+ ηun ϕx, tdxdt, n= T r µ L, ϕ = mmσ L t 2 t n+ t n+ h + tϕγ + θx γ, s + θt sdθdxdtdγds, σ L µ L, ϕ = mlmσ L t 2 t n+ t n+ L σ L h + tϕγ + θx γ, s + θt sdθdxdtdγds. The measures µ and µ describe the error of approximation in the entropy formulation satisfied by u h. Indeed, we have: Lemma 3.2. Assume that 3, 6, 7 2, 23, and identities 6 7 hold. Let u h defined by 24 26, and assume that the stability estimate 33 holds, then, for all r > and T > there exist C µ >, depending only on u, u η L Ω := sup u Ω u ηu and r, and C µ >, depending only on T, r, a, λ, u, G L and η such that, for all h < r, 39 4 µ B, r C µ h, µb, r [, T ] C µ h. Proof. The measure µ is the measure of density ηu x ηu h x, with respect to the Lebesgue measure. For all r > we have lim µ B, r =. h For smooth u : R d R m, the measure µ satisfies µ B, r h u η L Ω B,r+h u dx. For r > and T > the measure µ T satisfies T N T µ T B, r [, T ] = ηu n+ ηun [tn,+ ]dxdt. B,r n= T r Then, using the time BV estimate 37, N T µ T B, r [, T ] t ηu n+ ηun t u η L ΩC BV. n= T r It follows from 23 that 4 µ T B, r [, T ] C µt h, where C µt := a2 u η L Ω λ C BV. The measures µ L and µ L satisfy: µ L R d R + h + t, µ L R d R + h + t.

11 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN Therefore, µb, r [, T ] N T C µt h + h + t t n= N T +h + t t n=,l E r σ L ξ L u n, u n L ξ L u n, u n,l E r σ L ξ L u n, u n L ξ L u n L, u n L. Hence, using Lemma 2.4, the CFL condition 23 and the bound 4 provides the following bound: µb, r [, T ] C µ h, where C µ = C µt a2 λ u η L ΩC BV. Proposition 3.3. Assume 3, 6, 7 2, 23 and identities 6 7. Let u h defined by 24 26, and assume that the stability estimate 33 holds. Let µ and µ be the measures introduced in Definition 3., then, for all ϕ Cc R d R + ; R +, one has 42 d ηu h t ϕx, t + ξ α u h α ϕx, tdxdt R d R + + ηu xϕx, dx ϕ + t ϕ dµx, t ϕx, dµ x. R d R d R + R d Proof. Let ϕ Cc R d R + ; R +. Let T > and r > such that supp ϕ t n+ B, r [, T. Let us multiply 3 by ϕx, tdxdt and sum over the control volumes T r and n N T. It yields 43 T + T 2 where 44 T = and 45 T 2 = N T n= N T n= T r T r t ηun+ t n+ t n+ ηun ϕx, tdxdt L N ϕx, tdxdt σ L ξ L u n, u n L. The term T corresponds to the discrete time derivative of ηu h and T 2 to the discrete space derivative of ξu h. The proof relies on the comparison firstly between T and T and secondly between T 2 and T 2, where T and T 2 denote respectively the temporal and spatial term in 42: T = ηu h t ϕx, tdxdt R d R + ηu xϕx, dx, R d 46 T 2 = d ξ α u h α ϕx, tdxdt. R d R +

12 RELATIVE ENTROPY FOR FINITE VOLUME METHODS Let us first focus on T. Following its definition 26, the approximate solution u h is piecewise constant, then so does ηu h. This yields T = = n= N T n= N T n= N T = ηu n T r ηu n T r ηu n+ T r t n+ t ϕdxdt ηu xϕx, dx R d ϕx, + ϕx, dx ηu xϕx, dx R d ηun ϕx, + dx ηu x ηu h x, ϕx, dx R d N T = ηu n+ ηun t n+ t n= T r ηu x ηu h x, ϕx, dx. R d Then we consider the quantity T T : ϕx, + dxdt N T n+ ηu T T ηun t n+ ϕx, t ϕx, + dxdt t n= T r + ηu x ηu h x, ϕx, dx R d N T t n+ ηu n+ ηun t ϕ dxdt n= T r + ηu x ηu h x, ϕx, dx. R d Then, accounting from Definition 3., the inequality reads: 47 T T t ϕ dµ T x, t + R d R + ϕx, dµ x. R d We now consider the terms T 2 and T 2. Performing a discrete integration by parts by reorganizing the sum, and using the properties 32 and 2 lead to 48 T 2 = T 2, + T 2,2, with N T σ L T 2, = n=,l E r N T σ L T 2,2 = L n=,l E r t n+ t n+ Gathering terms of T 2 by edges yields L ϕx, tξ L u n, u n L ξ L u n, u n dxdt, ϕx, tξ L u n L, u n ξ L u n L, u n Ldxdt. T 2 = T 2, + T 2,2,

13 2 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN where, thanks to 2, we have set N T t n+ T 2, = ξ L u n, u n L ξu n n L ϕγ, tdγdt, σ L T 2,2 = n=,l E r N T t n+ n=,l E r Let us now compare the terms T 2, and T 2, : σ L ξ L u n L, u n ξu n L n L ϕγ, tdγdt. N T T 2, T 2, = t σ L ξ L u n, u n L ξu n n L n=,l E r t n+ ϕx, tdxdt t t n+ ϕγ, sdγds. σ L t σ L In order to handle the mean value of ϕ over the cell and its mean value over the edge σ L, we write t n+ ϕx, tdxdt t n+ t n+ = ϕx, tdγdsdxdt, σ L t σ L t n+ ϕγ, tdγdt σ L σ L t n+ t n+ = ϕγ, sdγdsdxdt. σ L t σ L Thus one has N T T 2, T 2, = t n=,l E r σ L ξ L u n, u n L ξ L u n, u n σ L t 2 t n+ t n+ σ L ϕx, t ϕγ, sdγdsdxdt. Writing a Taylor s expansion of ϕ with the integral form of the remainder provides for all x, γ, t, s σ L [, + [ 2 49 ϕx, t ϕγ, s h + t ϕ + t ϕ γ + θx γ, s + θt sdθ. Then using Definition 3. of the measure µ L we obtain the following estimate: 5 N T T 2, T 2, t σ L ξ L u n, u n L ξ L u n, u n µ L, ϕ + t ϕ. n=,l E r Similarly, one obtains that 5 N T T 2,2 T 2,2 t σ L ξ L u n L, u n ξ L u n L, u n L µ L, ϕ + t ϕ, n=,l E r

14 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 3 where the measure µ L MR d R + is given by Definition 3.. Bearing in mind the definition of µ MR d R + given in Definition 3., inequalities 43, 47, 48, 5 and 5, one has T T 2 ϕ + t ϕ dµx, t R d R + ϕx, dµ x R d that concludes the proof. The same kind of tools can be used to deduce how close is u h w.r.t. to a weak solution. For that purpose we define the following measures. Definition 3.4. For ψ C c R d and ϕ C c R d R +, we set µ, ψ = u x u h x, ψxdx, R d µ, ϕ = µ T, ϕ + t σ L G L u n, u n L G L u n, u n µ L, ϕ n=,l E r + t σ L G L u n, u n L G L u n L, u n L µ L, ϕ, n=,l E r where µ T, ϕ = u n+ n= T r µ L, ϕ = σ L t 2 t n+ t n+ un t n+ µ L, ϕ = L σ L t 2 t n+ L t n+ σ L σ L ϕx, tdxdt, h + tϕγ + θx γ, s + θt sdθdxdtdγds, h + tϕγ + θx γ, s + θt sdθdxdtdγds. Remark 3.. It follows from the definitions of the measures µ and µ that they can be extended in a unique way into continuous linear forms defined on the set { E := ϕ L R d R + ; R suppϕ is compact, and ϕ MR d R + d}. Indeed, any ϕ E admits traces on σ L, so that the quantities µ L, ϕ, µ L, ϕ, µ L, ϕ and µ L, ϕ are well defined. Moreover, one has µ, ϕ ϕ L µ{ϕ }, µ, ϕ ϕ L µ{ϕ }, ϕ E. We now state a lemma and a proposition whose proofs are left to the reader, since they are similar to the proofs of Lemma 3.2 and Proposition 3.3 respectively as one uses the estimates 33 and 36 instead of 35 and 37. Lemma 3.5. Assume that 3, 6, 7 2, 23, and identities 6 7 hold. Let u h defined by 24 26, and assume that the stability estimate 33 holds, then, for all r > and T > there exist C µ >, depending only on u and r, and C µ >, depending only on T, r, a, λ, u, G L such that, for all h < r, 52 µ B, r C µ h, µb, r [, T ] C µ h.

15 4 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN The quantities C µ and C µ are related to C µ and C µ by the relations C µ = u η L ΩC µ, C µ = u η L ΩC µ, where C µ and C µ have been explicitly calculated during the proof of Lemma 3.2. We are now in position to provide the approximate weak formulation satisfied by u h. Proposition 3.6. Assume 3, 6, 7 2, 23 and identities 6 7. Let u h defined by 24 26, and assume that the stability estimate 33 holds. Let µ and µ be the measures introduced in Definition 3., then, for all ϕ Cc R d R + ; R m, one has 53 d u h T t ϕx, t + f α u h T α ϕx, tdxdt + u x T ϕx, dx R d R + R d ϕ + t ϕ dµx, t + ϕx, dµ x. R d R + R d Let us stress that ϕ is a vector-valued function. This means that ϕ = max αϕ. α {,...,d} Nevertheless, the proof of Proposition 3.6 follows the same guidelines as the proof of Proposition 3.3 and is left to the reader. 4. Error estimate using the relative entropy Thanks to the measures defined above and Propositions 3.3 and 3.6, we are now in position to provide the precise meaning of inequality 38 satisfied by the relative entropy Hu h, u and conclude the proof of Theorem Relative entropy for approximate solutions. Proposition 4.. Assume 3, 6, 7 2, 23 and identities 6 7. Let u h defined by 24 26, and assume that the stability estimate 33 holds. Let µ and µ be the measures introduced in Definition 3., and let µ and µ be the measures introduced in Definition 3.4 then, for all ϕ Cc R d R + ; R +, one has d Hu h, u t ϕx, t + Q α u h, u α ϕx, t dxdt 54 R d R + ϕ + t ϕ dµx, t ϕx, dµ x R d R + R d [ϕ u ηu] + t [ϕ u ηu] dµx, t R d R + d [ϕ u ηu]x, dµ x + ϕ α u T Z α u h, udxdt, R d R d R + where Z α u h, u = 2 uηuf α u h f α u u f α u T u h u. Proof. Let ϕ be any nonnegative Lipschitz continuous test function with compact support in R d [, T ]. Since u is a classical solution of 3, it satisfies d ηu t ϕx, t + ξ α u α ϕx, tdxdt + ηu ϕx, dx =. R d R + R d

16 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 5 Subtracting this identity to 42 yields 55 d ηu h ηu t ϕx, t + ξ α u h ξ α u α ϕx, tdxdt R d R + ϕ + t ϕ dµx, t ϕx, dµ x. R d R + R d We now exhibit the relative entropy-relative entropy flux pair in the inequality 55 and obtain d Hu h, u t ϕ + Q α u h, u α ϕ dxdt 56 R d R + ϕ + t ϕ dµx, t ϕx, dµ x R d R + R d d u ηu T u h u t ϕ + f α u h f α u α ϕ dxdt. R d R + Since u is a strong solution of 3, it satisfies the following weak identity, ψ C c R d R + ; R m 57 R d R + u t ψx, t + d f α u α ψx, tdxdt + u xψx, dx =. R d Then we combine 57 and 53 using the Lipschitz continuous vector field [ϕ u ηu] as test function to get 58 [ ] d u ηu T u h u t ϕ + f α u h f α u α ϕ dxdt R d R + [ϕ u ηu] + t [ϕ u ηu] dµx, t [ϕ u ηu]x, dµ x R d R + R d d + u h uϕ t u ηu + f α u h f α uϕ α u ηudxdt. R d R + Moreover identity 7 together with gives 59 t u ηu = t u T 2 uηu d = α f α u T 2 uηu = = d α u T u f α u T 2 uηu d α u T 2 uηu u f α u. Injecting 58 and 59 into 56 leads to conclusion. Lemma 4.2. There exists C Z depending only on f, η and Ω such that, for all α {,..., d}, 6 Z α u h, u C Z u h u 2.

17 6 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN Proof. For M : Ω R m m and Υ : Ω LR m ; R m m, we set M, = sup Mu, u Ω Υ,2 = sup u Ω sup Υu v 2, v R m, v = where 2 and denote the usual matrix 2- and -norms respectively. Using the Taylor expansion of f α around u, we get that f α u h f α u u f α u T u h u 2 2 u f,2 α u h u 2, then, estimate 6 holds for C Z = 2 2 uη, 2 uf,2 α. We now prove the following lemma on the finite speed of propagation, which is usually assumed to be true in the literature [DiP79, Tza5, Daf]. Actually, this result is the reason why we makes assumption 4. Lemma 4.3. For all s L f, one has 6 shu h, u + d x α x Q αu h, u. Proof. Denote by w h := u h u, then it follows from the characterization 3 of the relative entropy H that 62 H = θ w h T 2 uηu + γw h w h dγdθ. Denoting by A γ the symmetric definite positive matrix 2 uηu + γw h, and by, Aγ the scalar product on R n defined by the relation 62 can be rewritten 63 H = u, v Aγ = u T A γ v, u, v R n, θ w h, w h Aγ dγdθ. On the other hand, it follows from the definition 6 of the entropy flux ξ that, for all α {,..., d}, Q α = = = θ θ The Rayleigh quotient u ηu + θw h u ηu T u f α u + θw h T w h dθ w h T A γ u f α u + θw h T w h dγdθ w h, u f α u + θw h T w h Aγ dγdθ. w h, u f α u + θw h T w h Aγ = w h, u f α u + θw h w h Aγ = w h, u f α u + θw h T + u f α u + θw h w h Aγ 2 satisfies, thanks to 4, w h, u f α u + θw h T w h Aγ Lf w h, w h Aγ.

18 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 7 Therefore, we obtain that 64 Q α θ L f w h, u f α u + θw h T w h Aγ dγdθ θ w h, w h Aγ dγdθ = L f H. The fact that 6 holds is a straightforward consequence of End of the proof of Theorem 2.6. We now have all the preliminary tools required for comparing u h to u via the relative entropy Hu h, u. Let δ, T be a parameter to be fixed later, and, for k N, we define the nonincreasing Lipschitz continuous function θ k : R + [, ] by θ k t = min, max, k + δ t δ, t. Let us also introduce the Lipschitz continuous function ψ : R d R + [, ] defined by ψx, t = min, max, x r L f T t +, where L f is so that 4 holds. The function ϕ k : x, t R d R + θ k tψx, t [, ] can be considered as a test function in 54. Indeed, denoting by I δ k = [kδ, k + δ], C r,t t = {x, t x [r + L f T t, r + L f T t + ]}, one has t ϕ k x, t = δ I δ k tψx, t L f θ k t Cr,T tx, ϕ k x, t = x x θ kt Cr,T tx, so that both t ϕ k and ϕ k belong to the set E defined in Remark 3.. Then taking ϕ k as test function in 54 yields where δ Ik δ R d Hψdxdt + T R = T θ k t C r,t t R d [,T ] θ k t R d L f H + d α u T Z α u h, u ψdxdt d Q α x α x ϕ k x, t + t ϕ k x, t dµx, t, dxdt + R + R 2 + R 3 + R 4, R 2 = ψx, dµ x, R d R 3 = u ηu ϕ k x, t + t ϕ k x, t dµx, t + R d [,T ] R d [,T ] R 4 = R d ψx, u ηu dµ x. ϕ k x, t 2 uηux, t t u + u dµx, t,

19 8 CLÉMENT CANCÈS, HÉLÈNE MATHIS, NICOLAS SEGUIN Thanks to Lemma 4.3, one has 65 T d Hψdxdt + θ k t α u T Z α u h, u ψdxdt R + R 2 + R 3 + R 4. δ R d R d I δ k In view of the definition of ϕ k, one has suppϕ k B, r + L f T t + {t}, This leads to t [,k+δ] ϕ k =, ϕ k, t ϕ k δ + L f. R δ + L f + µsupp ϕ k supp t ϕ k. Thanks to Lemma 3.2, we obtain that there exists Cµ k depending on k, r, T, δ, L f, a, λ, u, G L and η such that 66 R Cµ k δ + L f + h. It follows from similar arguments that there exists C k µ depending on k η, u, r, L f, T and δ such that 67 R 2 C k µ h, and, thanks to Lemma 3.5, we obtain that there exists C k µ depending on k, r, u, L f, T and δ such that 68 R 4 C k µ u ηu h. Similarly, there exists Cµ k depending on k, T, r, L f, a, λ, u, G L and δ such that 69 R 3 Cµ k u ηu δ + L f uη, t u + u h. By using Lemma 4.2, and using that θ k t, we obtain T d θ k t α u T Z α u h, u ψdxdt 7 R d C Z u R d [,k+δ] Since the entropy η is supposed to be β-convex, we have 7 Hx, t β 2 uh x, t ux, t 2. u h x, t ux, t 2 ψx, tdxdt. Putting 66 7 together with 65 provides that β 2δ C Z u u h u 2 ψ dxdt Ik δ R d 72 C Z u u h u 2 ψ dxdt + C k h, R d [,kδ] where C k =Cµ k δ + L f + + Cµ k + Cµ k u ηu + Cµ k u ηu δ + L f uη, t u + u.

20 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 9 Choose now δ = T p +, { where p = min p N T p + } β 2C Z u + 2 note thaeither δ nor p depend on h, so that 72 turns to k 73 e k ω e i + C k h, where e k = I δ k i= R d u h u 2 ψ dxdt, ω = C Z u. Denoting by e = e,..., e p T, and c = C,..., C p T, we deduce from 73 that... Me hc, where M = ω ω... ω It is easy to verify that M i,j = if j > i, if j = i, ω + ω i j if j < i, and then, since M, that i,j k 74 e k C k + ω + ω k j C j h. Define v =,..., T R p +, then Since we obtain that 75 p k= M T v j= e k = e v h M c v = h c M T v. k p k= p k = + ω + ω i = + ω p k+ ω, e k = h p k= i= C k + ω p k+ ω. Noticing that ψx, t = if x B, r + L f T t, and that ψx, t for all x, t R d, T, ensure that 76 T B,r st u u h 2 dxdt T We conclude the proof by using using 75 in 76. R d u u h 2 ψx, tdxdt = p k= e k.

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22 RELATIVE ENTROPY FOR FINITE VOLUME METHODS 2 [LV] [RJ5] [SR9] [Tza5] [Vil94] [VV3] [Yau9] N. Leger and A. Vasseur. Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-bv perturbations. Arch. Ration. Mech. Anal., 2:27 32, 2. C. Rohde and V. Jovanović. Finite-volume schemes for friedrichs systems in multiple space dimensions: a priori and a posteriori error estimates. Numer. Meth. P.D.E., 2:4 3, 25. L. Saint-Raymond. Hydrodynamic limits: some improvements of the relative entropy method. Ann. Inst. H. Poincaré Anal. Non Linéaire, 263:75 744, 29. A. E. Tzavaras. Relative entropy in hyperbolic relaxation. Commun. Math. Sci., 32:9 32, 25. J.-P. Vila. Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes. RAIRO Modl. Math. Anal. Numr., 283: , 994. J.-P. Vila and P. Villedieu. Convergence of an explicit finite volume scheme for first order symmetric systems. Numer. Math., 943:573 62, 23. H.-T. Yau. Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys., 22:63 8, 99. Clément Cancès UPMC Univ Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-755, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-755, Paris, France cances@ann.jussieu.fr Hélène Mathis Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2, Rue de la Houssinière, Nantes Cedex 3, France helene.mathis@univ-nantes.fr Nicolas Seguin INRIA Rocquencourt, BP 5, F-7853, Le Chesnay Cedex, France UPMC Univ Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-755, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-755, Paris, France nicolas.seguin@upmc.fr