Hydrodynamic Regimes, Knudsen Layer, Numerical Schemes: Definition of Boundary Fluxes

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1 Hydrodynamic egimes, Knudsen Layer, Numerical Schemes: Definition of Boundary Flues C. Besse, S. Borghol, T. Goudon, I. Lacroi-Violet and J.-P. Dudon Abstract We propose a numerical solution to incorporate in the simulation of a system of conseration laws boundary conditions that come from a microscopic modeling in the small mean free path regime. The typical eample we discuss is the deriation of the Euler system from the BGK equation. The boundary condition relies on the analysis of boundary layers formation that accounts from the fact that the incoming kinetic flu might be far from the thermodynamic equilibrium. Key words. Hydrodynamic regimes. Knudsen layer. Finite Volume scheme. Initial-boundary alue problems for conseration laws. Eaporation-condensation problem. 00 MSC Subject Classification. 35L65 35Q35 8C80 8C40 76M 65M08 Introduction A statistical picture of a cloud of particles leads to the following PDE t F + F = QF τ satisfied by the particles distribution function F t,, 0. Here, N being a positie integer, t 0, Ω N and N are the time, space and elocity ariables respectiely. The left hand side of the equation describes the transport of particles, that is the simple motion on straight line with elocity, and the interactions process the particles are subject to is embodied in the right hand side QF, for instance interparticles collisions. The parameter τ > 0 is related to the mean free path which is the aerage distance traelled by the particles without being affected by any interaction. As τ becomes small, the distribution F tends to an equilibrium F eq that is a function which makes the collision operator anish QF eq = 0. By considering conseration laws associated to the collision dynamics, we can then derie macroscopic equations satisfied by moments with respect to the elocity ariable. A difficulty arises when the boundary conditions are not compatible with the equilibrium state. In such a case boundary layers appear, whose analysis is quite delicate. The questions we address are related to the numerical treatment of the boundary layer: considering a system of conseration laws obtained as a small mean free path limit of a kinetic model, what are the associated boundary conditions for the hydrodynamic fields? How the boundary flues can be ealuated in numerical procedures? The paper is organized as follows. First we need to set up a few definition and notation. Our framework will be the Euler system, obtained as the limit of the BGK equation but the method can Project-Team SIMPAF, INIA Lille Nord Europe esearch Centre Labo P. Painleé UM 854 CNS & Uniersité des Sciences et Technologies Lille Thales Alenia Space, Cannes La Bocca Precisely, throughout the paper we shall implicitly work with dimensionless equations so that τ is actually the Knudsen number that is the ratio of a typical macroscopic length scale of the flow oer the mean free path.

2 be etended to more intricate collision operators like the Boltzmann operator or the Landau-Fokker- Planck operator. The necessary material is recalled in Section. In particular entropy dissipation has a central role. A difficulty for hyperbolic equations set on a bounded domain relies on the fact that the number of necessary boundary conditions for the problem to be well posed usually depends on the solution itself. The entropy proides a natural way to determine the incoming flues, with a direct analogy with the microscopic picture. The discussion is strongly inspired by the analysis of the Knudsen layer for the linearized Boltzmann equation by F. Coron-F. Golse-C. Sulem [0], see also the lecture notes of F. Golse [3], and their result is the cornerstone of the definition of numerical flues we propose in this paper. Section 3 is the main part of the paper: we discuss how we can take into account the boundary layer in a Finite Volume scheme for the conseration laws. The point is precisely to define a suitable numerical flu on the boundary cells. The definition that we design relies on a decomposition of the numerical solution, according to the nature of the flow sub or supersonic combined with an approimation of the half-space problem which defines the matching condition. Finally Section 4 comments the numerical eperiments with comparison to direct simulations of the kinetic model. Hydrodynamic limits, entropy dissipation and boundary layer analysis. BGK equation and Euler system For the sake of concreteness we consider the BGK collision operator: QF = M[F ] F where M[F ] = ρ πθ ep N/ u θ and the macroscopic quantities ρ, u, θ are defined by density: ρ = F d, N elocity: ρu = F d, N temperature: ρu + Nρθ = F d. N 3 4 This non linear operator can be seen as a caricature of the Boltzmann or the Landau-Fokker-Planck collision operators that arise in gas dynamics and plasma physics, see []. It allows the deriation of rather simple formulae, but our discussion can be etended to more comple collision operators. As a matter of fact, we remark that,, are collision inariants: it means that N QF d = 0 holds. In turn, integration of 3 yields the following local conseration laws t N F d + N F d = 0. 5

3 The system is not closed since the higher order moments cannot be epressed by means of the macroscopic quantities 4. Net, it is clear that the equilibria are the Mawellian functions QF eq = 0 iff F eq = ρ πθ ep N/ u = M U, θ where U stands for the triple ρ, u, θ that parametrizes the Mawellian. Therefore, as τ 0 we epect that F, solution of 3, resembles a Mawellian, whose macroscopic parameters ρ, u, θ are still functions of t and. Inserting this information into the moment system 5 we obtain t N M U d + N M U d = 0 which is now a closed system since it can be recast as the Euler system the ector in N with components N j= ja ij. t ρ + ρu = 0, t ρu + ρu u + ρθ = 0 t ρu + Nρθ + ρu + N + ρθu = 0. 6 Howeer, when lies in a domain Ω of N, equation has to be completed with boundary conditions which prescribe the incoming distribution function. We assume that Ω is smooth and we denote ν to be the outward normal unit ector at point Ω. We set Γ out = { t,, 0, Ω N, ν > 0 } and Γ inc = { t,, 0, Ω N, ν < 0 }. The theory of traces on Γ inc/out for solutions of transport equations has been deelopped in [0, 7, 8, 46] after the seminal work [5]; we denote γ inc/out the trace operators. The boundary condition for has the following epression γ inc F t,, = Φ data t,,, for t,, Γ inc, 7 where Φ data 0 is a gien function in L Γ inc, ν d dσ dt. Here and below dσ stands for the Lebesgue measure on Ω. Of course, the problem is also completed by an initial condition F 0,, = F Init, L Ω N. 8 emark. Prescribing the incoming data as in 7 might be unsatisfactory on a physical iewpoint. More realistic boundary conditions are intended to describe the interaction of the particles with the boundary. This is a complicated issue, which requires subsequent modeling efforts, see [6, 4]. Simple and classical reflection laws are, for t,, Γ inc Specular reflection law: γ inc F t,, = α γ out F t,, νν, νγ out F t,, d Mawell diffusie law: γ inc F t,, = α ν>0 e ν e /θ w d ν<0 /θ w, Here and below for gien ectors a, b in N, a b is the N N matri with components a ib j and, gien a matri alued field A, A stands for 3

4 where 0 α is an accomodation parameter that gies the fraction of reflected particles and θ w is the temperature of the wall Ω. We refer to [36, 37] for further comments and references on boundary conditions.. Entropy The collision operator erifies a dissipation property that will be crucial to our discussion. We introduce the entropy functional HF = F lnf d. N As a consequence of the definition of the collisional inariants, we obsere that N M[F ] F lnm[f ] d = 0. Then, by using integration by parts and the boundary condition 7 we obtain d HF d + γ out F lnγ out F ν d dσ + DF d d dt Ω Ω ν>0 τ Ω N = Φ data lnφ data ν d dσ, Ω ν<0 with M[F ] DF = M[F ] F ln 0. F Thus the entropy is dissipated and we deduce the following estimate of the solution by means of the data t HF t, d + γ out F lnγ out F ν d dσ ds Ω 0 Ω ν>0 t HF Init d + Φ data lnφ data ν d dσ ds. Ω 0 Ω ν<0 This remarkable relation is referred to as the H-Theorem. In what follows, we will work with the relatie entropy between two particles distribution functions F and F [ F ] [ F F HF F := F ln F + F d = ln F ] + F d 0. F F F F N This quantity is non negatie and anishes iff F = F. It has been shown to be ery useful in kinetic theory to ealuate how far a solution F of is from a reference state F see e. g. [57] and the references therein. emark. The dissipation property etends to the case of the specular or the Mawell reflection law, and een to more general reflection laws: the dissipation due to the boundary condition essentially relies on a coneity argument and on a ersion of the Jensen lemma, referred to as the Darrozes-Guiraud lemma [] see also comments in [4]. emark.3 The entropy dissipation gies the basic a priori estimate that can be used for justifying the eistence of weak or renormalized solutions. We refer to [4] for the Boltzmann equation, to [48, 5] for the BGK equation in the whole space, while the initial-boundary-alue problem is inestigated e. g. in [37, 39, 47]. Of course, entropy dissipation is also the basis for the analysis of hydrodynamic limits, according to the program addressed in [7, 8]: we refer to [35] and to the ery complete surey [57] for detailed results and further references. N 4

5 Let us go back to the Euler system 6. We consider the mapping U = ρ, u, θ 0, N 0, U = ρ, j, E = ρ, ρu, ρu / + Nρθ/, 9 which defines the consered quantities. The Euler system recasts as t U + F U = 0, with the matri with N + raws, N columns j T j j ρ F U = N + N + N E j Nρ E j Nρ j ρ I N T. An entropy for the Euler system is a cone function η : U ηu such that there eists an entropy flu q : U qu N erifying t ηu + qu = 0 for any smooth solution of 6. This definition 3 leads to a relation between the entropy η, the entropy flu q and the flu function F ; namely we hae Uk q j = N+ i= Uk F ij Ui η. 0 Gien a reference constant state U, it is conenient to introduce the relatie entropy ηu U = ηu ηu U ηu U U. Since η is cone, this quantity is non negatie and it anishes iff U = U. As far as the solution U of 6 is smooth, we hae t ηu U + qu U = 0 where the entropy flu is defined by qu U = qu qu U ηu F U F U. A crucial remark is that the kinetic entropy defines an entropy for the Euler system. Indeed, let us set ηu = M U lnm U d = HM U N with U = ρ, u, θ associated to U by 9. Lemma.4 We hae ρ ηu = ρ ln N + lnπρ θ N/ and the function U ηu is strictly cone. The associated flu is gien by qu = [ ρ ln N ] + lnπ ρu = ηu j θ N/ ρ = M U lnm U d. N 3 We refer for more details on this notion and its role for the analysis of hyperbolic systems e. g. to [5, Section 3.4 and ff.]. 5

6 The relatie entropy is ηu U = ηu ρ Finally we hae [ ln ρ θ N/ = HM U M U 0. qu U = + N N lnπ ] + ρ + θ ρ u u + ρnθ θ N [ MU ] M U ln M U + M U d. M U The proof of this lemma is just a lengthy but straightforward computation and will be omitted here..3 Linearization We consider a gien state, with fied and constant density ρ > 0, elocity u N and temperature θ > 0; we thus set U = ρ, u, θ. The Mawellian is a solution of 3. We set M U = ρ πθ ep u N/ θ F t,, = M U + ft,, where the amplitude of the fluctuation is intended to be small. The fluctuation f satisfies t f + f = τ L U f + τ f, M U 3 where L U is the linearized BGK operator and is the remainder, with at least a quadratic estimate with respect to the amplitude of the fluctuation, defined by the deelopment In other words, we hae M[M U + M U f] M U M U f M U = L U f + f, M U. L U f = M[F ] F F M U f. M U F =MU For the BGK operator, computing the linearized operator reduces to apply the chain rule to obtain the deriatie of the composite function F ρ, j, E =,, /F d U = ρ, u, θ M U. N The first application is linear and its deriatie is simply defined by moments. Introducing the fluctuation of the macroscopic quantities ρ U = U + Ũ, Ũ = ũ, θ we are led to the linearization of the relation 9 which in turn defines the following linear relation ρ 0 T 0 ρ Ũ = j = u ρ I N 0 u ũ = P Ũ. 4 + Nθ Ẽ ρ u T Nρ / θ 6

7 Thus, the deriatie of the second application is defined by the matri P. Finally, we hae ρ u U M U = M U θ. N + u θ θ Then, it turns out that the linearized operator reduces to a projection: let f L M U d, then L U f = Πf f with and Πf := U M U M U ρ ũ θ ρ ρu + ρ ũ ρu + Nθ + ρ u ũ + Nρ θ = ρ + u ũ + θ u N. ρ θ θ θ = P ρ ũ θ = N fm U d. Clearly, Π is the orthogonal projection of L M U d to the finite dimensional set spanned by the collisional inariants {,, }. In fact the asymptotic and boundary layer analysis relies on the following properties of the linearized collision operator: L U is self-adjoint for the inner product of L M U d, KerL U = Span{,, }, anl U = KerL U for the inner product of L M U d, and the following dissipation property holds N L U f f M U d 0. These properties are obiously satisfied by the linearized BGK operator which is a mere projection. But they are fulfilled for the Boltzmann operator as well, at least for hard potentials, see [4] functional framework for the case of soft potentials is discussed in [34]. Here, for the sake of simplicity we restrict the discussion to the BGK operator. Assuming that the fluctuation remains small, we get rid of the non linear remainder term in 3. 4 We are thus concerned by the linear problem t f + f = τ L U f, f t=0 = f Init, γ inc ft,, = Ψ data t,, on Γ inc, where the boundary condition is gien, coming back to 7, by Ψ data = Φdata M U. 4 which makes sense when the amplitude of the deiation f is small compared to τ. 5 7

8 Now, as τ 0 we guess that f looks like an infinitesimal Mawellian ft,, mũt, = ρt, ρ + u ũt, + θt, u N KerL U. θ θ θ Inserting this ansatz in the following moment system analog for the linearized problem of 5 t fm U d + fm U d = 0, N / N / we are led to the linearized Euler system t ρ + ρ ũ + u ρ = 0, θ t ũ + u ũ + θ + ρ = 0, ρ t θ + u θ + N θ ũ = 0. Of course, we obtain the linear system 6 when we linearize directly the system of conseration laws 6, assuming that the perturbations and their deriaties remain small. The question is now to identify the boundary conditions to be satisfied by ρ, ũ, θ, that will depend on the kinetic incoming condition in 7..4 Slab geometry For the sake of simplicity, we aoid all difficulties associated to comple geometries and from now on we adopt the following simplified framework: the flow does not depend on the transerse ariables,..., N and the particles eole in the domain ω, +ω. We slightly change the notation by using to denote the single coordinate characterizing this slab geometry. When necessary we shall write u N as u, u N. Then the kinetic equation now reads t F + F = τ M Ut, F with ρ, u, θ associated to F by 4. The boundary condition becomes 6 t 0, ω, +ω, N, 7 γ inc F t, ω, = Φ data,l t, for > 0, γ inc F t, +ω, = Φ data, t, for < 0. 8 The linearized equation 5 can be recast in a similar fashion: we hae { t f + f = τ L U f, f t=0 = f Init, 9 endowed with γ inc ft, ω, = Ψ data,l t, for > 0, γ inc ft, +ω, = Ψ data, t, for < 0, 0 where Ψ data,j = Φ data,j /M U for j = L or j =. Let us go back to the hydrodynamic equations. In consered ariables the Euler system reads t U + F U = 0, 8

9 where now F : N+ N+. It is gien by the components F i of the multi-dimensional case. Precisely we hae j F U = j/ρ + E /N j /Nρ j j /ρ N + E /N j /Nρ. j /ρ As far as the solution is smooth, we can rewrite the equation in the following non conseratie form t U + A U U = 0, with A U = U F U = j ρ + j j Nρ ρ j Nρ j Nρ T N jj j ρ j ρ ρ I N 0 N+ N E j j Nρ ρ + jj N+ Nρ 3 N E j Nρ ρ j Nρ j j T N+ j ρ Nρ N. The linearized ersion of the Euler system is simply t Ũ + A U Ũ = 0. Equialently we can epress the system with the ariations of density, elocity and temperature Ũ = ρ, ũ, θ. We remind that they are related to the conseratie unknowns by Ũ = P Ũ, see 4. In the slab geometry, 6 becomes We thus hae and of course t ρ ũ ũ θ + u ρ 0 0 θ ρ u u I N 0 0 θ /N 0 u } {{ } :=A t Ũ + P A P Ũ = 0, P A P = A U. ρ ũ ũ θ = 0. The characteristic speeds of the system are the eigenalues of A, that is u with multiplicity N+ N, u ± c, with c = N θ the sound speed. A difficulty is related to the fact that the number of boundary conditions necessary to complete the problem depends on the number of incoming characteristics, that is the dimension of the eigenspace associated to positie eigenalues at = ω, while at = +ω we care about negatie eigenalues. This intuition 5 becomes clear by using the natural symetrization of the system proided by the relatie entropy. To this end, we need to introduce a couple of notation, the goal being to rewrite the linearized system in an equialent form t S Ũ + Q Ũ = 0, with S and Q symmetric matrices. The definition of S and Q are deduced from the epression of the entropy. Since the pioneering works of K. O. Friedrichs and P. D. La [7] and S. Goduno [9], this is the standard preliminary step for studying hyperbolic problems. In particular it has an essential role for proing the local well-posedness of the non linear Cauchy problem. For the initial boundary alue problem, the new formulation of the system gies rise to the boundary terms that need to be imposed. 5 The basic tools for the analysis of hyperbolic mied problems are described e. g. in [53, Chapter 4]. 9

10 Tedious computations yield ln U ηu U = ρ θ N/ + u u u u u + ρ θ θ θ θ + u θ u θ u θ. θ + θ Differentiating once more and ealuating the Hessian matri at U = U defines the matri + N ρ + u 4 Nθ u Nρ θ u T + u ρ θ Nρ θ S = DU ηu U = U =U u Nρ θ u + u ρ θ Nρ θ I N u Nρ θ + u ρ θ Nρ θ ut Nρ θ Nρ θ. Since η is strictly cone see Lemma.4, S is symmetric positie definite. Now, we ealuate the entropy flu. Since ηu U and qu U as well as the gradient of ηu U anish at U = U, by differentiating 0 we obtain A U T DU ηu U = D U qu U. U =U U =U We denote by Q this matri. Since it is defined as an Hessian matri, it is symmetric and we hae Q = A U T S = Q T = S T A U = S A U. 3 Considering small fluctuations around U, we can epand the entropy relation rewritten as At leading order it yields t ηu + Ũ U + qu + Ũ U = 0. t S Ũ Ũ + Q Ũ Ũ = 0. 4 Of course we can obtain the same relation by remarking that the multiplication of by S leads to t S Ũ + Q Ũ = 0, owing to 3. Then we use the symmetry of the matrices S and Q to deduce 4. The interest of the identity 4 relies on the deriation of an energy estimate as follows d dt +ω ω S Ũ Ũ d + Q Ũ Ũ Separating contributions according to their sign yields 0 +ω ω =+ω = 0. = ω t ] ] S Ũ [Q Ũ t, d + Ũ Ũ t, +ω dt [Q Ũ Ũ t, ω dt ω t ] t ] = S Ũ Init Ũ Init d + [Q Ũ Ũ t, ω dt [Q Ũ Ũ t, +ω ω 0 5 Hence we realize that uniqueness is guaranteed when the non negatie part of Q Ũ Ũ t, = ω and the non positie part of Q Ũ Ũ t, = +ω are gien. t + 0 dt. 0

11 These considerations are the key ingredients to study the well-posedness of the initial boundary alue problem for the linearized system. We refer to [3] for a crystal-clean oeriew. The following claim makes the connection between the signature σ U of the quadratic form Ũ Q Ũ Ũ = S A U Ũ Ũ, see 3, and the eigenalues of the flu matri A U : the dimension of the eigenspaces associated to positie resp. negatie eigenalues of Q corresponds to the number of positie resp. negatie eigenalues of the flu matri A U, that is, for the left hand boundary = ω, the number of incoming resp. outgoing characteristics and for the right hand boundary = +ω, the number of outgoing resp. incoming characteristics. Proposition.5 The signature σ U is n +, n, with n + +n +n 0 = N +, n 0 being the dimension of KerQ, and n +, resp. n, is the cardinal of the set of the positie, resp. negatie, eigenalues of A U. Combining this statement, which comes from basic linear algebra, to the energy estimate 5 we are led to the following well-posedness statement which makes precise the nature of required boundary conditions for soling. Proposition.6 Let E ± be a subspace of N+ such that i For any Ũ E± \ {0}, we hae ±Q Ũ Ũ > 0, ii E ± is maimal in the sense that any subspace E N+ erifying the property stated in i, is included in E ±. We set E ± = { Ũ N+, such that for any V E ± we hae Q Ũ V = 0 }. Let t Ũ data,l t and t Ũ data, t be integrable functions on 0, T for any 0 < T < such that Ũ data,l t E and Ũ data, t E +. We consider the mied problem t Ũ + A U Ũ = 0, Ũ t=0 = Ũ Init H [ ω, ω], Ũ t, ω Ũ data,l E, Ũ t, +ω Ũ data, E +. Suppose that Ũ Init = ω Ũ data,l t = 0 E and Ũ Init = +ω Ũ data, t = 0 E +. Then the mied problem has a unique solution in C 0, ; L ω, +ω. We refer to [0, 3] and [53, Chapter 4] for more details and proofs of these two propositions..5 Boundary layer As already remarked HF M U = N F lnf/m U F + M U d 0 is the relatie entropy associated to the BGK equation 3. Then we remark that, for any perturbation f and 0 < δ, Accordingly, HM U + δf M U = δ f f M U d N N f M U d + Oδ 3.

12 defines an entropy for the linearized equation and indeed solutions of 9 satisfy t f M U d + f M U d = L U f f M U d 0. N τ N N Ealuating the entropy flu on an infinitesimal Mawellian defines a quadratic form of the macroscopic quantities Q : g = mũ KerL U g M U d N satisfies QmŨ = N ρ + u ũ + θ u ρ θ θ = u ρ + ρ u ũ + ρ u ρ θ θ u 0 0 ρ = ρ u 0 θ ρ u θ 0 θ ρ θ 0 N ρ u θ N MU d ũ ũ + N ρ u θ θ + ρũ + ρ ũ θ θ ρ ρ ρ θ ũ ũ = Σ Ũ ũ ũ Ũ. θ θ In fact this quantity gies the entropy flu in the ariables ρ, ũ, θ instead of the conseratie ariables, owing to the following claim. Lemma.7 We hae Σ = P T Q P. Proof. The identity can be checked by direct inspection. Instead, let us gie some hints eplaining where the formula comes from. According to the definition of the entropy flu qu U in Lemma.4, QmŨ corresponds to the leading term in the epansion of N [ M U +Ũ ln MU +Ũ M U M U +Ũ + M U ] d. Therefore, by identification with 4 and using 4, we obtain QmŨ = Σ Ũ Ũ = Q Ũ Ũ = Q P Ũ P Ũ = P T Q P Ũ Ũ. Let us split the set of infinitesimal Mawellians according to the sign of the quadratic form Q: KerL U = Λ + Λ Λ 0, with Q Λ + is positie definite, Q Λ is negatie definite and Λ 0 = { g = mũ, Qg = 0 }. By Lemma.7, we note that g = mũ Λ ± means that Ũ = P Ũ erifies ±Q Ũ Ũ > 0 and thus Ũ belongs to E ± as arising in Proposition.6. Owing to Proposition.5 and Lemma.7, we hae dimλ + = n +, dimλ = n, dimλ 0 = n 0

13 and a correspondence is established between the quadratic form Q on KerL U and the quadratic form on N+ associated to the symmetric matri Q. According to [0], and as deeloped in [9, 3], it is conenient to introduce the following basis of KerL U : In what follows, we denote χ = u NN + + u, NN + θ θ u χ 0 = N, N + θ χ k = k u k for k {,..., N}, θ χ N+ = u NN + u. NN + θ θ I ± = { k {0,..., N + }, χ k Λ ±}, I 0 = { k {0,..., N + }, χ k Λ 0}. Note that #I + = n +, #I = n, #I 0 = n 0. The interest of this specific basis relies on the orthogonality properties and useful formula summarized in the following Lemma. Lemma.8 We hae Furthermore, we obsere that N χ k χ l M U d = ρ δ kl, χ M U d = ρ N N χ k χ l M U d = 0 if k l. N + u + N θ = ρ u + c, χ k M U d = ρ u for k {0,,..., N}, N N + χ N+ M U d = ρ u N θ = ρ u c. N The final touch consists in introducing an ansatz which takes into account the boundary layer correctors. We epand the solution of 9 as follows ft,, = mũt, + G L t, + ω/τ, + G t, ω /τ, + r τ t,, with r τ a remainder which is epected to be small as τ goes to 0. The correctors G L t, z = ω + /τ, and G t, z = ω /τ, are defined from the following half space problem { z G = L U G, for z > 0 and N, γ inc G0, = Υ data 6, for > 0, where Υ data has to be suitability defined the ariables of the half-space problem are z and ; in fact we are concerned with data parametrized by the time ariable but we omit the time dependence to simplify the notations. We seek a solution which anishes at infinity: imposing Gz, z means that the influence of the corrector becomes negligible far away from the boundary = ω or = +ω. The complete analysis of the half-space problem is due to [0] after the preliminary breakthrough of [6] and formulation of the problem in [3]; for the specific case of the linearized BGK operator the analysis appeared in a different form in [3, 38]. Precisely, we hae the following statement [0, Theorem.7.] 3

14 Theorem.9 Let V + be a subspace of KerL U satisfying i For any Ũ N+ \ {0} such that mũ V + we hae QmŨ 0, ii V + is maimal in the sense that any subspace V KerL U erifying the property stated in i is included in V +. Let Υ data L N, + M U d. Then, for any m KerL U, there eists a unique m V + and a unique solution G L 0, ; L N, M U d of 6 such that for any γ > 0 small enough we hae e γz Gz, m m L 0, ; L N, + M U d. The statement can be rephrased as follows using Theorem.9 with m = 0 Corollary.0 There eists a linear mapping that can be called the generalized Chandrasekhar functional C : L N, + M U d V + Υ data m with m the limit as z of the unique solution G L 0, ; L N, M U d of 6. This statement defines the necessary boundary condition for the macroscopic field Ũ at the boundary = ω and = +ω. Indeed, coming back to 9, the boundary layer correctors are defined as follows: G L t, z, = Gt, z, with G the solution of 6 with incoming data Υ data = Ψ data,l t, mũt, ω, G t, z, = Gt, z, ˆ with G the solution of 6 with incoming data where if =,, 3, ˆ =,, 3. Υ data = Ψ data, t, ˆ mũt,+ω ˆ, The n + necessary boundary conditions at = ω resp. the n necessary boundary conditions at = +ω are proided by the determination of the asymptotic state m associated to the incoming data. Imposing 7 means that C Ψ data,l t, = C mũt, ω, C Ψ data, t, = C mũt,+ω. 8 In fact it is maybe more intuitie to split mũt, ω into its outgoing part m = k I α kχ k and its incoming part m + = k I + I 0 α kχ k : the former is determined by the flow while the latter has to be imposed as a boundary condition to complete the Euler system. Precisely, let us define G as to be the solution of 6 with incoming data Υ data = Ψ data,l m. Then, the boundary condition for the macroscopic field is obtained by requiring m + to be the asymptotic state of Gz, as z which means C m + t,. = C Ψ data,l t,. m t,.. A similar reasoning applies for the boundary condition at = ω. We are going to use this formalism in order to define numerical flues. 4

15 3 Finite Volume scheme and treatment of the boundary conditions From now on, we only consider the case N = and we assume for the elocity ariable. The space domain reduces to the interal ω, +ω, where for the simulation we will set ω = 0.5. We wish to compare the simulation of the D BGK equation 7 endowed with the initial condition F t=0 = F Init and the gien incoming boundary condition 8 with the simulation of the Euler system t ρ + ρu = 0, t ρu + ρu + ρθ = 0, ρu + ρθ t + ρu + 3 ρθ u = 0, completed with boundary conditions defined through the boundary layer analysis. We introduce a regular subdiision ω = 0 < = ω + <... < i = ω + i <... < I+ = ω + I + = +ω. In order to sole the kinetic equation 7 we use a simple splitting scheme: gien Fi,p n, an approimation of F at time n t, position i = ω + i and elocity p, with n N, i {0,..., I + }, where I + = +ω, and p { P,..., P } Firstly, we sole the free transport equation t + F = 0 starting from F n t = F n. Of course, we treat a discrete ersion of the equation which casts as F n+/ i,p = F n i,p td i,p F n where D is a discrete ersion of the adection operator. For instance, we can work with the upwind scheme which leads to D ip F = p F i,p F i,p if p > 0, D ip F = p F i+,p F i,p if p < 0, and D i0 F = 0. The scheme is completed by using the incoming boundary condition for the fictitious points. The solution at time n + t defines the state F n+/. Secondly, we sole the collision part t F = τ M[F ] F with initial data F n t = F n+/. Actually, it reduces to a mere ODE since the macroscopic quantities do not change during this time step: since,, F d =,, F n+/ d, we hae M[F ] = M[F n+/ ] and therefore we end up with F n+ = e t/τ M[F n+/ ] + e t/τ F n+/. Actually, we make use of the Strang ersion of this algorithm which is a straightforward etension intended to reach second order accuracy with respect to time. Of course as the parameter τ becomes small the resolution of the kinetic equation becomes highly demanding in computational time since the simulation can produce releant results capturing the microscopic effects only under the constraints t, τ. Thus, when τ goes to 0 there is a clear adantage in using the hydrodynamic equations which furthermore work on a reduced set of ariables since we do not need to consider the elocity ariable. In what follows, the simulation of 7 8 by using the splitting scheme will sere to obtain a reference solution, at the price of a ery long computational time, and alidate our numerical treatment of the boundary conditions for the Euler system. 9 5

16 Now, let us eplain the scheme for the Euler system 9. To this end, let us denote, for any i {,..., I}, C i = ω + i /, ω + i + / to be the cell centered on i. Similarly, the boundary cells are defined by C 0 = ω, ω + / and C I+ = ω /, ω. Then Ui n is intended to be an approimation of C i U n t, d, the mean alue of U n t, oer the cell C i. The definition of the scheme is deduced from the integration of 9 oer the cell n t, n + t C i : U n + t, d U n t, d C i C i n+ t + F U s, ω + i + / F U s, ω + i / ds = 0 suggests n t U n+ i U n i = t Fi+/ n F i / n where Fi+/ n is an approimation of the flu on the interface = ω + i + / on the time step. This approimation is obtained as a suitable function of the U n+ k s: in the simplest case, the numerical flu on the interface = ω + i + / is determined by the unknowns in the two neighboring cells Fi+/ n = FU i n, Ui+ n see [8, 43]. In what follows we work with Goduno flues see e. g. [43, Chapter 5] for the interior cells which correspond to the cells with i {,..., I} since the scheme uses only 3 points, but of course more sophisticated schemes can be used. The question is now to define the boundary flues, namely for indices i = 0 and i = I +, in a way that accounts for the boundary layer analysis. Actually, the question is two fold. The first difficulty consists in performing the non linear boundary layer analysis and deriing the hydrodynamic boundary condition coming from 8 in the fluid regime. To our knowledge the theory is not deeloped as for the linearized problem, ecept certainly for scalar equations as discussed in [56]. Net, the underlying half-space problems are usually not affordable for the numerical simulation since their resolution is at least as difficult as the original kinetic problem. Hence, we need an additional approimation procedure. The approach we propose is based on the linearized theory described aboe. It has the adantage of offering a neat framework and a natural way to determine the number and the nature of the needed boundary conditions for the hyperbolic system 9. The first attempt would be to use directly 8 as a boundary condition: as we shall see it could be alid as far as the solution of the kinetic equation remains close to the reference state M U. But of course, using this boundary condition, which comes from a purely linear theory, is certainly questionable since it relies on the permutation of the linearization limit where the amplitude of the fluctuation anishes and the small mean free path limit hydrodynamic limit τ 0 in 3. Clearly inerting these limits is not alid and the linearized theory cannot be applied in general cases. Neertheless, we can go beyond this naie idea and we shall use the neat picture offered by the linearized theory to design numerical flues based on a local linearization, considering the alues in the closest cells to the boundaries as the reference state. Let us describe how the numerical approimation works. In what follows we eplain the construction of the numerical flues and details on the underlying formulae are gien in the Appendi. Linearized equations. Let us start by considering the linearized situation which assumes that the solution reads U = U +Ũ, where Ũ is a small perturbation of the constant state U. Therefore can be considered as an approimation of the non linear equations 9. In particular, the number of necessary boundary conditions is entirely determined by the reference state U and we shall distinguish the cases: 6

17 u 3θ > 0: the signature σ U is 3, 0 and we do need 3 incoming data at = ω, and all fields are outgoing at = +ω, u > 0 > u 3θ : the signature σ U is, and we need to prescribe incoming data at = ω, only at = +ω, u + 3θ > 0 > u : the signature σ U is, and we need to prescribe incoming data at = ω, and at = +ω, u = 3θ, u = 0 and u = 3θ correspond to degenerate cases where there eists a field which is characteristic at the boundary; the signature σ U is, 0,, or 0, respectiely which means that or or 0 data are needed at = ω respectiely 0, or or at = +ω. Let us focus on the non-degenerate cases. At the boundary, for the left hand boundary the flu that we seek has the following epression F bd = Φ data d + + m bd + γ out G0, M U d, 30 >0 / <0 / with the reference Mawellian, M U = m bd = ρ bd ρ ρ πθ + u bd u θ ep u, θ + θ bd u θ θ an infinitesimal Mawellian and G solution of the half-space problem 6 with incoming data Υ data = M U Φ data M U + m bd. These quantities are well defined by Theorem.9, howeer their eact computation is out of reach. Therefore, we seek a suitable definition of the triple ρ bd, u bd, θ bd, together with the outgoing distribution γ out G0,, for < 0, intended to approimate the actual quantities. Let us introduce a few notation, restricting our discussion to the boundary = ω the boundary = +ω is treated in a similar way, changing the sign of the elocity ariable and adapting the definition of the outgoing/incoming characteristics. Let Ū = ρ, ū, θ stand for the alue of the hydrodynamic unknowns in the boundary cell that is in C 0 for the left hand case where Ū = U 0 n and in C I+ for the right hand case where Ū = U I+ n. We set U fluc = Ū U = ρ fluc, u fluc, θ fluc to which we associate the infinitesimal Mawellian m Ufluc = ρ fluc ρ + u fluc u θ emind that we can work with a suitable basis + θ fluc u. θ θ χ = u 3 + u, 6 θ θ χ 0 = u 3, 6 θ χ = u 3 u, 6 θ θ of KerL MU, which is orthogonal for the quadratic form Q : f f M U d, 3 7

18 and we split this subspace according to the signature of the quadratic form Q: it defines the subspaces Λ ± Λ 0 being reduced to {0} in the non-degenerate cases. Now, to define m bd, we proceed as follows. Firstly, we associate to m Ufluc its projection on Λ : m = k I α k χ k, α k = m Ufluc χ k M U d. χ k M U d Secondly, we define m + as to be an infinitesimal Mawellian with flues imposed by the half-space problem 6-7, with Υ data = Ψ data,l m, where we remind that Ψ data,l = Φ data,l /M U. Of course, the resolution of the half-space problem is usually not affordable and we shall use an approimation deice. Integrating the half space problem 6 oer the elocity ariable, we obtain d dz Gz, M U d = 0. 3 By using the incoming boundary condition and 7 it follows that G0, M U d = G, M U d = 0 = Υ data M U d + γ out G0, M U d. >0 <0 Then, we make an approimation which has been already proposed by Mawell [45]. It is related to the Marshak approimation [44] introduced in radiatie transfer and it is precisely discussed for gas dynamics in [], see also [33, 30]. We suppose that the outgoing distribution coincides with the distribution at infinity: γ out G0, = G,, which, coming back to 7, leads to γ out G0, = 0. Hence we determine the coefficients of m + in the basis {χ k, k I + } by equating the incoming flues >0 Ψ data m M U d >0 m + M U d = 0. Ecept in the case #I + = dimλ + = 3 this system is oerdetermined. Therefore instead we sole the optimization problem inf m += k I + α kχ k Ψ data m M U d Λ + >0 33 m + M U d. >0 It can be recast in matri formulation min Aα b data, B + α=0 8

19 where the coefficients of the 3 3 matri A only depend on the reference state U and the coefficients of b data 3 depend on U, Ψ data and m, while B +, haing n + {0,..., 3} raws and 3 columns, is associated to the constraint m + Λ + and only depends on U see formulae in the Appendi. The solution is obtained by soling the linear system A T A B +T α A B + = T b data 34 0 λ 0 with λ I+ the Lagrange multiplier associated to the constraint α KerB +. Eentually, we define the needed Mawellian by m bd = m + + m. Clearly if the signature of Q is 3, 0 resp. 0, 3, then m = 0 since I = resp. m = m Ufluc since I + =. Haing disposed of this construction of m bd and using the Mawell approimation, we go back to 30 and we are led to the following definition of the left boundary flu to be used in the numerical scheme: F bd = >0 / Φ data d + <0 / + m + m + MU d. In this formula, m is determined by the outgoing part of the hydrodynamic flow, while the coefficients of m + are determined by the resolution of a certain linear system. emark 3. Instead of soling the minimization problem, another possibility is simply to pick as many relations as needed among the three flues identities. This is the definition proposed in [, 33]. This linearized approach produces satisfying results, but its applicability remains ery limited. Indeed, starting from a gien state, due to the incoming boundary condition, the solution of the BGK equation might be conducted far away from the reference state. In particular it might happen that, net to the boundary, the flow changes type, with modification of the number of incoming/outgoing characteristics. The fully linearized approach, that will be referred to as the global linearization, is not able to capture such phenomena and therefore produces wrong results as time becomes large see Figures and. Local linearization. Neertheless, we can adapt the ideas by reasoning locally. Knowing the numerical approimation U l = ρ l, u l, θ l for l {0,..., n}, the first step consists in defining a conenient reference state. Let 0 ν <. We find an approimation of C 0 Un + ν t, y dy by linear interpolation and we set U = U n + νu n U n. In practice, we work with ν = /. Then we consider the unknown in the boundary cell as a perturbation of the reference state: we write U0 n = U + U fluc. Now, we construct the boundary flues by repeating the strategy detailed aboe. The noticeable point is that now all the coefficients of A, b data, B + arising from 33 need to be updated at each time step. It thus requires the ealuation of seeral integrals, which has a non negligible computational cost but of course, it still remains far less costly than the computation of the kinetic equation!. By the way, we notice that the ealuation of these integrals should be performed with enough accuracy, a basic requirement being to presere the constant solutions when the initial data and boundary condition coincide with the reference state F Init = Ψ data = M the simulations presented here are obtained using the fourth order Simpson rule. Figures compare the results of a kinetic simulation to the simulation of the hydrodynamic system with both the linearized and the localized approaches. In the simulation ρ Init, u Init, θ Init = ρ, u, θ =, 0., such that σ U is,. At the boundary we choose Φ data = 0 and t = 0. as final time. As for section 4, we take hydro = 9

20 kin = /500 and t is determined by the CFL condition. Moreoer the computation of the integrals respect to are performed with regularly spaced nodes on the domain [ 6, 6]. Finally we work with τ = 0 3. It shows that the localized approach noticeably improes the hydrodynamic approimation. Figure gies the eolution in time of the eigenalues for the same test case of figure. Clearly at the beginning we hae as signature σ U =, and after a short time, this signature becomes 0,3 for the right boundary and 3,0 for the left one. This eplain why the local linearization is better: the globally linearized scheme works well for short times but it is not able to adapt to the change of type sub/supersonic induced by the boundary conditions. a Global linearization b Local linearization ρ Density u Velocity ρ Density u Velocity Pressure Temperature Pressure Temperature p θ p θ Figure : Local linearization s. global linearization in the case ρ, u, θ =, 0.,, Φ data = 0 in 7, final time t = 0.. Dashed line: kinetic simulation, solid line: hydrodynamic simulation. emark 3. It might be tempting to use as a reference state the alue U n in the first internal cell. Then, the fluctuation would be U n U0 n, which resembles U; when discontinuities are produced this quantity becomes large and thus it is not a good candidate for being seen as a fluctuation. This strategy indeed leads to unstabilities. The degenerate case u = 0. Let us eplain how the preious machinery is modified in the specific degenerate case u = 0 which yields σ U =,. The starting point consists in remarking that χ 0 = 6 3 is orthogonal to KerL U since θ 3 M ρ,0,θ θ d = 0. Therefore, since KerL U = anlu, we can uniquely define the auiliary function λ eri- 0

21 a Left boundary b ight boundary Figure : Eolution of the eigenalues on right and left boundaries for the simulation of figure. fying L U λ = χ 0, with For the BGK operator, we simply hae λ M ρ,0,θ d = 0. λ = 3 = χ 0. 6 θ Then, we obsere that the solution of the half-space problem satisfies d λ Gz, M ρ,0,θ dz d = 3 Gz, M ρ,0,θ 6 θ d = K 0 which does not depend on z since we bear in mind that 3 holds. Hence, if G is a bounded solution of the half-space problem, it imposes an additional conseration law: K 0 = 0 that is 3 Gz, M ρ,0,θ θ d = 0, 35 and λ Gz, M ρ,0,θ d = K constant independent on z. But we do not hae any knowledge of the alue of the constant K; this is the reason why we use a two steps method: the first step proides an ealuation of K, which in turn can be used to define the asymptotic state. Haing defined m as before, we construct m + = k I 0 I + α kχ k by taking into account this information. We follow the method presented in [30, 33]: it relies on a simple iteration argument, but it can be interpreted as the result of a ariational problem. It is conenient to set Gz, = Gz, + m + which thus erifies the half space problem { z G = LU G, γ inc G0, = Ψ data,l m for > 0,

22 and, as detailed in Theorem.9, m + Λ + Λ 0 is the equilibrium state at infinity lim Gz, = m + = α 0 χ 0 + α χ z that we seek. The first step consists in using the Mawell approimation to find a preliminary guess for m +. We hae Gz, Mρ,0,θ d = m + M ρ,0,θ d. We assume that the outgoing trace coincides with the distribution at infinity. The obtained system is oerdetermined since it contains 3 equations while we search for the two components of m + in the basis χ 0, χ. Hence, we pick two relations among the system and we define m + = α 0 χ 0 + α χ as to be the solution of Ψ data,l m Mρ,0,θ d = m + M ρ,0,θ d. >0 Note that it is useless to use 35 as a constraint for defining m + since 35 is satisfied by any infinitesimal Mawellian see below. For the second step, we use m + as the outgoing distribution to ealuate the two following consered quantities >0 λ Ψ data,l m Mρ,0,θ d+ <0 >0 λ m + M ρ,0,θ d = Haing these quantities at hand, we define m + = α 0 χ 0 + α χ as the solution of λ K m + M ρ,0,θ d = K. K K. As a matter of fact, we remark that this construction is consistent with 35. Indeed, on the one hand, the asymptotic state m + fulfills the constraint 3 m + M ρ,0,θ 6 θ d = χ 0 α0 χ 0 + α χ Mρ,0,θ d = α 0 χ 0 M ρ,0,θ d + α χ 0 χ M ρ,0,θ d = 0 by definition of χ 0 and χ. On the other hand at the boundary we hae >0 Ψ 3 data,l m Mρ,0,θ 6 θ d + <0 Ψ = 3 data,l m Mρ,0,θ >0 6 θ d 3 m + M ρ,0,θ >0 6 θ d + 3 m + M ρ,0,θ 6 θ d 3 m + M ρ,0,θ 6 θ d = 0 by using the definition of m +. Here we hae detailed the computations for the left hand boundary = ω. The right hand boundary = +ω is of course treated analogously, ecept that integration oer 0, + resp., 0 is replaced by integration oer, 0 resp. 0, +.

23 emark 3.3 The treatment of the boundary condition differs from those presented by S. Dellacherie [, 3]. There, the idea can be summarized as follows: Firstly we define hydrodynamic quantities ρ gh, u gh, θ gh in a ghost cell by soling >0 M ρgh,u gh,θ gh d = >0 Φ data,l d. Namely, we consider the Mawellian haing the same outgoing flues as the data. A first attempt would be to use these alues as Dirichlet conditions, but a better approach is proposed. Secondly, the flu at the interface between the ghost cell and the computational domain is determined by using a kinetic scheme, as introduced in [49, 50], see also []: we set Fbd n = >0 / M ρgh,u gh,θ gh d + <0 / M ρ n 0,u n 0,θn 0 d. Consequently, owing to the specific definition of the numerical flues, we do not hae to sole the nonlinear system in step, we simply use the relation Fbd n = Φ data,l d + M ρ n >0 / <0 0,u n 0,θn 0 d. / This definition is ery natural in the framework adopted in [] where the resolution of the Euler system is precisely based on the use of a kinetic scheme. The definition of the numerical flues can be interpreted as a releant ersion of the Enquist Osher scheme [5] and [50, Chap. 6 & 8]. As crude as it might appear this definition of boundary flues is quite performing. We will go back in a future work to further comparison in the contet of fluid/kinetic interfaces. emark 3.4 The discussion of Knudsen layers and their numerical treatment is strongly related to the question of finding releant matching condition between the kinetic equation and the hydrodynamic system in a domain decomposition approach. This issue is discussed in details, with arious numerical approaches, in [,, 3, 3, 3, 30, 33, 4, ]. The mathematical analysis of such matching condition is highly delicate, and we refer to the breakthrough of A. Vasseur concerning nonlinear scalar conseration laws [56]. We shall go back to this problem elsewhere and we will also consider the effect of a coupling with an electric potential [9]. We also point out that our approach can be improed by using a more inoled approimation of the half-space problem, while here the computations are simply based on the Mawell approimation: the iteratie procedure described in [33] looks quite appealing for this purpose. This refinement will be eplored elsewhere. eflection boundary conditions. It is releant to consider generalized boundary conditions where the incoming distribution of particles depends on the outgoing distribution: such boundary condition is intended to describe the comple interaction between the impinging particles and the boundary. The simplest case corresponds to the specular reflection law γ inc F t, = ω, = αγ out F t, = ω, for > 0, where 0 α, but it is certainly too crude to model realistic boundary phenomena. More generally, the boundary condition is defined through a non local relation γ inc F t, ω, = α k, γ out F t, ω, d = γ out F t, ω, for > 0 <0 3

24 where all the physics is embodied in the kernel k and the coefficient 0 α. We refer for the deriation of boundary kernels and comments to [4, 5,, 4, 36, 37]. It is natural to require: Non-negatiity: k, 0, Mass conseration: k, d =, eciprocity principle: >0 There eistsa Mawellian M w erifying M w = k, M w d. The specular law corresponds to the case where k is the Dirac mass δ =. A more releant eample of such a boundary condition is the Mawell law where k, = M w, M w = ρ w e /θ w, Z w = M w d. Z w πθw In other words, the particles are reflected according to the Mawellian distribution M w, proportionally to the outgoing mass flu: γ inc F t, ω, = α M w γ out F t, ω, d for > Z w <0 emark that such a boundary condition can be deried from the simple specular reflection law through an homogenization analysis [4]. The parameter α is the so called accomodation coefficient and it defines the fraction of reflected particles. Indeed, we hae 0 F t, ω, d = γ inc F t, ω, d + γ out F t, ω, d 0 <0 = α 0 >0 γ out F t, ω, d 0. In particular, when α = the mass flu anishes. The method described aboe can be adapted to treat these boundary conditions. Again, we restrict our purpose to the boundary = ω. We obtain the following boundary condition for the half-space problem M U γ inc G0, = M U + m M U + m + M U γ out G0, for > 0, 37 where we use the shorthand notation : L < 0, d L > 0, d to denote the reflection operator, which in the simulation corresponds either to the diffuse Mawell law or to the specular law. We detail in the Appendi the precise form of the linear systems to be soled for defining the boundary flu in this case. Note howeer that the specular reflection law partial or total is ecluded from the boundary layer analysis in [0], while the diffuse condition can be considered see Section and Theorem.. in [0]. In the specific case of a totally refleie boundary condition, α = in 36, it is worth pointing out that the mass flu anishes. Indeed, by combining 7 and 3 we obsere that G0, M U d = 0. Therefore, integrating 37 yields M U + m M U + m d = 0 >0 = M U + m d M U + m d. <0 4 >0

25 We remind that m splits as m + m + and we arrie at the constraint m + M U d = + m M U d to be satisfied by the asymptotic state m +. It simply means that the numerical mass flu is set to 0 and this is eactly what our definition does. emark 3.5 For the specular reflection and α =, both the mass and energy flues anish; in this ery specific case the boundary condition for the hydrodynamic system are simply the wall condition that corresponds to use a ghost cell and to reerse the elocity in the ghost cell while keeping the same density and temperature. 4 Numerical results As a preliminary remark, we point out that direct comparison between hydrodynamic and kinetic simulations should be considered with caution since they could be slightly unfair. Indeed, using the Euler system makes sense when the scaling parameter τ is ery small. Positie alues of τ naturally induces diffusion effects, and Naier-Stokes correction should be taken into account. In particular diffusion will certainly becomes sensible for large time simulation. Net, performing kinetic simulations with τ is numerically demanding since capturing layer effects requires to make use of resoled meshes, with τ. Additionally to the large size of the unknown in such conditions, the stability constraint associated to the adection term makes rapidly the simulation not affordable, especially with the quite naie scheme we are using performances can be improed by using AP scheme, as introduced in [40, 6]. Neertheless, we shall see below that the comparison is qualitatiely satisfactory, and of course, the hydrodynamic simulations run much faster than the microscopic simulations. Furthermore, we obsere that the mesh size can be safely increased in the hydrodynamic simulation without altering that much the solution. A releant test case corresponds to the simulation of eaporation-condensation phenomena. The problem is presented in great details in works of Y. Sone and K. Aoki [54, 55, ], we also refer to the surey [9]. The incoming data are gien by two Mawellians with different macroscopic quantities Φ data,l = ρl w πθ L w ep θ L w, Φ data, = ρ w πθ w ep θ w. 38 where ρ L w, θ L w can differ from ρ w, θ w. We start either from a constant state ρ Init, u Init, θ Init which can be not in equilibrium with the incoming data or from a discontinuous profile. For the kinetic equation the initial data is the Mawellian defined by these macroscopic quantities. Of course, the basic requirement is that the code preseres the equilibrium: for ρ Init, u Init, θ Init = ρ L w, 0, θ L w = ρ w, 0, θ w the solution remains constant. We check that it is indeed the case, the error being goerned by the accuracy of the computation of the integrals arising in 33. The same conclusion applies if the reference elocity does not anish. We do not present a specific figure but an eidence of the ability of the scheme in presering equilibrium can be seen in the far right hand side of Figures 3, 4 and 5: the data at = +ω coincides with the initial Mawellian and we stop the simulation at a short time so that the perturbation arising from the boundary = ω has not crossed the domain. In Figures 3, 4 and 5, we compare the results of the kinetic simulation to the hydrodynamic simulation for ρ w, u w, θ w = ρ Init, u Init, θ Init =, 0, 0.5 and for seeral alues of ρ L w, θ L w ρ Init, θ Init. The computational parameters are hydro = /500 5