Une décomposition micro-macro particulaire pour des équations de type Boltzmann-BGK en régime de diffusion

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1 Une décomposition micro-macro particulaire pour des équations de type Boltzmann-BGK en régime de diffusion Anaïs Crestetto 1, Nicolas Crouseilles 2 et Mohammed Lemou 3 Rennes, 14ème Journée de l équipe Analyse Numérique 23 mars INRIA Rennes - Bretagne Atlantique, IPSO & Université de Nantes, LMJL. 2 INRIA Rennes - Bretagne Atlantique, IPSO & Université de Rennes 1, IRMAR & ENS Rennes. 3 CNRS & Université de Rennes 1, IRMAR & INRIA Rennes - Bretagne Atlantique, IPSO & ENS Rennes. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 1

2 Outline 1 Problem and objectives A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 2

3 Introduction Our problem Objectives Particle systems: some applications Plasma physics. Plasma: gaz constituted of at least two species of charged particles (positive ions and electrons). Natural state: stars, ionosphere, aurora borealis,... Industrial state: TV screen, neon light, nuclear fusion,... ITER project Radiative transfer. Interaction between photons and matter. Examples: radiotherapy,... A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 3

4 Introduction Our problem Objectives Numerical simulation of particle systems Different scales, for example collisions parameterized by the Knudsen number ε different models. Kinetic model Particles represented by a distribution function f (x, v, t). Solving a Boltzmann or Vlasov-type equation t f +A(v,ε) x f +B(v,E,B,ε) v f = S(ε) potentially coupled with Maxwell or Poisson equations. Accurate and necessary far from thermodynamical equilibrium. In 3D = 7 variables = heavy computations. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 4

5 Introduction Our problem Objectives Fluid model Moment equations on physical quantities linked to f (density ρ, mean velocity u, temperature T, etc.). Lost of precision. Small cost and sufficient at thermodynamical equilibrium. General difficulties Find a well adapted model for the problem, with a good precision/cost ratio. If two scales in the same simulation, develop a numerical scheme efficient in each regime: spatial coupling of two schemes, with an interface, or asymptotic-preserving (AP) scheme 4. 4 Jin, SIAM JSC A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 5

6 AP scheme Problem and objectives Introduction Our problem Objectives Problem ε h 0 Discretized Problem h,ε ε 0 ε 0 Limit h 0 Discretized limit h h: space step x or time step t. Prop.: Stability and consistency ε, particularly when ε 0. Standard schemes: constraint h = O(ε). Aim: Construct a scheme for which h is independent of ε. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 6

7 Introduction Our problem Objectives Our 1st Problem ε 1D Vlasov-BGK equation, diffusion scaling t f + 1 ε v xf + 1 ε E vf = 1 ε2(ρm f) (1) x [0,L x ] R, v V = R, charge density ρ = V f dv, electric field E given by Poisson equation x E = ρ 1, M(v) = 1 2π exp ( v2 2 ), periodic conditions in x and initial conditions. Main difficulty: Knudsen number ε may be of order 1 or tend to 0 at the drift-diffusion limit t ρ x ( x ρ Eρ) = 0. (2) A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 7

8 Introduction Our problem Objectives Our 2nd Problem ε 1D radiative transport equation, diffusion scaling t f + 1 ε v xf = 1 ε2(ρm f) (3) x [0,L x ] R, v V = [ 1,1], charge density ρ = 1 2 V f dv, M(v) = 1, periodic conditions in x and initial conditions. Main difficulty: Knudsen number ε may be of order 1 or tend to 0 at the diffusion limit t ρ 1 3 xxρ = 0. (4) A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 8

9 Objectives Problem and objectives Introduction Our problem Objectives Tools Idea Construction of an AP scheme. Reduction of the numerical cost. Micro-macro decomposition 5,6 for these models 7 (in [7], grid in v for the micro part). Use particles for the micro part since few points in v are enough at the limit. 5 Lemou, Mieussens, SIAM JSC Liu, Yu, CMP Crouseilles, Lemou, KRM A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 9

10 Derivation of the micro-macro system First-order reformulation for the 2nd Problem ε (3) Micro-macro decomposition 5,7 : f = ρm + g with g the rest. N = Span{M} = {f = ρm} null space of the BGK operator Q(f) = ρm f. Π orthogonal projection in L 2( M 1 dv ) onto N: Πh := h M, h := h dv. Hypothesis: first moment of g must be zero = g = 0, since f = ρ. True at the numerical level? If not, we have to impose it. 5 M. Lemou, L. Mieussens, SIAM JSC N. Crouseilles, M. Lemou, KRM A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 10 V

11 Derivation of the micro-macro system First-order reformulation Applying Π to (3) = macro equation on ρ t ρ+ 1 ε x vg = 0. (5) Applying (I Π) to (3) = micro equation on g t g + 1 ε [v xρ+v x g x vg ] = 1 ε2g. (6) Equation (3) micro-macro system: t ρ+ 1 ε x vg = 0, t g + 1 ε F(ρ,g) = 1 (7) ε 2g, where F(ρ,g) := v x ρ+v x g x vg. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 11

12 Difficulties Problem and objectives Derivation of the micro-macro system First-order reformulation Stiff terms in the micro equation (6) on g. In previous works 5,7, stiffest term (of order 1/ε 2 ) considered implicit in time = transport term (of order 1/ε) stabilized. But here: use of particles for the micro part = splitting between the transport term and the source term, = not possible to use the same strategy. Idea? Suitable reformulation of the model. 5 M. Lemou, L. Mieussens, SIAM JSC N. Crouseilles, M. Lemou, KRM A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 12

13 Derivation of the micro-macro system First-order reformulation Strategy of Lemou 8 : 1. rewrite t g + 1 ε F(ρ, g) = 1 ε 2 g as t (e t/ε2 g) = et/ε2 F(ρ, g), ε 2. integrate in time between t n and t n+1 and multiply by e tn+1 /ε 2 : g n+1 g n t = e t/ε2 1 t 3. approximate up to terms of order O( t) by: t g = e t/ε2 1 t No more stiff terms and good properties. 8 Lemou, CRAS g n ε 1 e t/ε2 F(ρ n, g n )+O( t), t g ε 1 e t/ε2 F(ρ, g). (8) t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 13

14 Properties Problem and objectives Derivation of the micro-macro system First-order reformulation Where Consistency: ε > 0 fixed, as t goes to zero, equation (8) is consistent with the initial micro equation (6). Asymptotic behaviour: t > 0 fixed, as ε goes to zero, we get from (8) g = εv x ρ+o(ε 2 ), which injected in the macro equation (5) provides the limit model (4). (8) t g = e t/ε2 1 t g ε1 e t/ε 2 t [v x ρ+v x g x vg ], (5) t ρ+ 1 ε x vg = 0, (4) t ρ 1 3 xxρ = 0. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 14

15 Algorithm Problem and objectives PIC method Finite volumes scheme Properties Second-order in time Reformulated system t ρ+ 1 ε x vg = 0, Algorithm: t g = e t/ε2 1 t g ε 1 e t/ε2 F(ρ,g). t 1. Solving the micro part by a Particle-In-Cell (PIC) method. 2. Projection step to numerically force to zero the first moment of g (matching procedure 9 ). 3. Solving the macro part by a finite volume scheme (mesh on x), with a source term dependent on g. Remark: already used in the hydrodynamic limit P. Degond, G. Dimarco, L. Pareschi, IJNMF, A. C., N. Crouseilles, M. Lemou, KRM, 2012 A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 15

16 1. PIC method Problem and objectives PIC method Finite volumes scheme Properties Second-order in time Equation t g = e t/ε2 1 g ε 1 e t/ε2 F(ρ,g) t t t g +ε 1 e t/ε2 [v x g] t = e t/ε2 1 t g ε 1 e t/ε2 [v x ρ x vg ] =: S g. t Model: having N p particles, with position x k, velocity v k and weight ω k, k = 1,...,N p, g is approximated by N p g Np (x,v,t) = ω k (t)δ(x x k (t))δ(v v k (t)). k=1 A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 16

17 Problem and objectives Classical PIC algorithm PIC method Finite volumes scheme Properties Second-order in time Mesh generation on x for fields Fields computing on the mesh Initialization of positions, velocities and weights of particles Computation of charge and current densities on the mesh (deposition) Interpolation of fields on the particles Evolution of weights (if source term) Movement of particles (in x and v) A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 17

18 Regularization using shape functions PIC method Finite volumes scheme Properties Second-order in time Use of shape functions such as B-spline of order l B l (x) = (B 0 B l 1 )(x), with In particular B 0 (x) = { 1 x if x < x/2, 0 else. B 1 (x) = 1 x Order 0: Nearest Grid Point. Order 1: Cloud In Cell. { 1 x / x, if x < x, 0 else. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 18

19 Deposition and interpolation PIC method Finite volumes scheme Properties Second-order in time Deposition: computation of the moment of order p on the cell i: v p g i = v p g (x,v,t) dv B l (x i x) dx = R N p V ω k (t)v p k (t)b l(x i x k (t)). k=1 Interpolation: evaluation of a quantity E on particle k: N x E (x k,t) = E (x i,t)b l (x i x k (t)). i=1 A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 19

20 Solving t g +ε 1 e t/ε2 t [v x g] = 0 PIC method Finite volumes scheme Properties Second-order in time 1. Initialization: particles randomly (or quasi) distributed in phase space (x, v), weights initialized to ω k (0) = g (x k, v k, 0) LxLv N p. (L x x-length of the domain, L v v-length.) 2. Deposition ρ i (t n ). 3. Movement of particles thanks to motion equations: dx k dt (t) = ε1 e t/ε2 v k (t). t For example x n+1 k = x n k +ε(1 e t/ε2 )v k. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 20

21 PIC method Finite volumes scheme Properties Second-order in time Solving t g = S g 4. Evolution of weights ω k (step specific to kinetic equations with source term): with dω k dt (t) = S g (x k,v k ) L xl v N p S g = e t/ε2 1 g ε 1 e t/ε2 [v x ρ x vg ]. t t In practice: ω n+1 k ω n k t = e t/ε2 1 ωk n t ε1 e t/ε2 [α n k t +βn k ], with α n k = vn+1 k x ρ n (x n+1 k ) L xl v N p and βk n = x vg (x n+1 k ) L xl v. N p A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 21

22 2. Projection step Problem and objectives PIC method Finite volumes scheme Properties Second-order in time We now have N p g n+1 (x,v) k=1 ω n+1 k δ ( x x n+1 k ) ( ) δ v v n+1 k. Nothing ensures g n+1 = 0 at the numerical level. We have to impose it. How? By applying a discrete approximation of (I Π) to each weight ω k. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 22

23 3. Macro part Problem and objectives PIC method Finite volumes scheme Properties Second-order in time Equation t ρ+ 1 ε x vg = 0. Finite volume method ρ n+1 i = ρ n i t ε vg n+1 i+1 vg n+1 i 1. 2 x A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 23

24 Numerical limit Problem and objectives PIC method Finite volumes scheme Properties Second-order in time Micro equation is discretized as v xρ x vg ω n+1 k = e t/ε2 ωk n {}}{{}}{ ε(1 e t/ε2 ) α n k + βk n. When ε 0, βk n = O(ε) thus ωn+1 k = εα n k +O(ε2 ) and vg n+1 i = ε v 2 n i }{{} x ρ n i +O(ε 2 ). 1/3 Np Injecting in the macro equation ρ n+1 i = ρ n i t ε x vg n+1 i gives ρ n+1 i = ρ n i + t 3 xxρ n i, = we recover a discretization of the limit equation (4). A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 24

25 "Maneuver" Problem and objectives PIC method Finite volumes scheme Properties Second-order in time Use this idea to implicit the diffusion term. v xρ x vg Write ω n+1 k = e t/ε2 ωk n ε(1 {}}{{}}{ e t/ε2 ) α n k + βk n. Let h n i := e t/ε2 g n i ε(1 e t/ε2 ) x vg and approximate vg n+1 i = ε(1 e t/ε2 ) 1 3 xρ n i + h n i. Inject it in the macro equation and take the diffusion term implicit ρ n+1 i = ρ n i + t(1 e t/ε2 ) 1 3 xxρ n+1 i t ε xhi n. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 25

26 AP property Problem and objectives PIC method Finite volumes scheme Properties Second-order in time For fixed ε > 0, the scheme is a first-order (in time) approximation of the reformulated micro-macro system, for fixed t > 0, the scheme degenerates into an implicit first-order (in time) scheme of the diffusion equation (4) = AP property. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 26

27 PIC method Finite volumes scheme Properties Second-order in time Second-order scheme in time: new reformulation When integrating in time t (e t/ε2 g) = et/ε2 ε F(ρ,g), use a midpoint method for the right-hand side g n+1 = e t/ε2 g n te t/2ε2 F ε ( ρ n+1/2,g n+1/2) +O ( t 3). Make appear a discrete time derivative g n+1 g n = e t/ε2 1 ( g n e t/2ε2 F ρ n+1/2,g n+1/2) +O ( t 2). t t ε Perform Taylor expansions at t n+1/2 ( t g n+1/2 = e t/ε2 1 g n+1/2 t ) t 2 tg n+1/2 ( e t/2ε2 F ρ n+1/2,g n+1/2) +O ( t 2). ε A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 27

28 PIC method Finite volumes scheme Properties Second-order in time New second-order micro-macro system: t ρ+ 1 ε x vg = 0, t g = 2 t e t/ε2 1 e t/ε2 + 1 g 2 e t/2ε2 ε e t/ε2 + 1 [v xρ+v x g x vg ]. Time scheme: RK2 g n+1/2 =g n + e t/ε2 1 e t/ε2 + 1 gn t ε ρ n+1/2 =ρ n t 2ε x vg n+1/2, g n+1 =g n + 2 e t/ε2 1 e t/ε2 + 1 ρ n+1 =ρ n t ε x vg n+1/2. Prediction step on t/2: e t/2ε2 e t/ε2 + 1 F (ρn,g n ), Correction step on t: g 2 t ε e t/2ε2 ( e t/ε2 + 1 F ρ n+1/2,g n+1/2), A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 28

29 PIC method Finite volumes scheme Properties Second-order in time ρ n+1 i Choice of g in order to have a second-order scheme in time and the right asymptotic limit: g = gn +g n+1 2. Do not "maneuver", but correct the macro equation: = ρ n i t ε x vg n+1/2 i + t(1 e t/ε2 ) 21 3 xx( ρn+1 i 2 +ρ n i ). Same PIC/FV discretization in space as for the first-order scheme. Use only B-splines of order l 1 for depositions and interpolations. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 29

30 Properties Problem and objectives PIC method Finite volumes scheme Properties Second-order in time For fixed ε > 0, the scheme is a second-order (in time) approximation of the reformulated micro-macro system, for fixed t > 0, the scheme degenerates into an implicit second-order (in time) scheme of the diffusion equation (4) = AP property. The scheme can be (and has been) extended to the Vlasov-BGK-Poisson case. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 30

31 ETR case Problem and objectives ETR case Vlasov-BGK-Poisson cases Initial distribution function: f (x,v,t = 0) = 1+cos ( 2π ( x + 1 )), x [0,1],v [ 1,1]. 2 Micro-macro initializations: ( ( ρ(x,t = 0) = 1+cos 2π x + 1 )) 2 and g(x,v,t = 0) = 0. Density ρ(x,t) = f(x,v,t)dv. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 31

32 Asymptotic behaviour ETR case Vlasov-BGK-Poisson cases T = 0.1, N x = 64, N p = 10 4, t = 10 3 (left), T = 0.1, N x = 64, ε = 10 6, t = 10 2 (right). Density ρ ETR, AP property Limit ε=10-6 ε=10-2 ε=0.25 ε= x Density ρ ETR, limit Limit ε=10-6, Np=10 4 ε=10-6, Np= x A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 32

33 Convergence Problem and objectives ETR case Vlasov-BGK-Poisson cases T = 0.1, N x = 16, N p = 100. Error on ρ in L norm ETR, convergence (1) 10-5 ε= ε= ε=0.1 ε= Slope t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 33

34 Landau damping Problem and objectives ETR case Vlasov-BGK-Poisson cases Initial distribution function: f (x,v,t = 0) = 1 2π exp( v2 2 )(1+α cos(kx)), x [0, 2π k Micro-macro initializations: ρ(x,t = 0) = 1+α cos(kx) and g(x,v,t = 0) = 0. Parameters: α = 0.05, k = 0.5. Electrical energy E(t) = E(t,x) 2 dx. ],v R. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 34

35 ETR case Vlasov-BGK-Poisson cases Kinetic regime, N x = 128, N p = 10 5, t = 0.1. log(e) Landau damping, ε=10-12 MiMa-Part-2-14 MiMa-Part-1 Moment G. -16 Full PIC MiMa-Grid εt log(e) Landau damping, ε=1 MiMa-Part-2 MiMa-Part-1 Moment G. Full PIC MiMa-Grid t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 35

36 ETR case Vlasov-BGK-Poisson cases Intermediate regime, N x = 256, N p = 10 5, t = Landau damping, ε=0.5 0 log(e) MiMa-Part-2 MiMa-Part-1 Moment G. Full PIC MiMa-Grid t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 36

37 ETR case Vlasov-BGK-Poisson cases Limit regime, N x = 128, N p = 10 4, t = (left), N x = 128, N p = 100, t = 0.01 (right). 0 Landau damping, ε=0.1 0 Landau damping, ε= log(e) MiMa-Part-2 MiMa-Part-1 Moment G. MiMa-Grid Limit log(e) MiMa-Part-2 MiMa-Part-1 Moment G. MiMa-Grid Limit t t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 37

38 Two stream instability ETR case Vlasov-BGK-Poisson cases Initial distribution function: f (x,v,t = 0) = v2 2π exp( v2 2 )(1+α cos(kx)), x [0, 2π k Micro-macro initializations: ρ(x,t = 0) = 1+α cos(kx) g (x,v,t = 0) = 1 ( v 2 1 ) ) exp ( v2 (1+α cos(kx)). 2π 2 Parameters: α = 0.05, k = 0.5. ],v R. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 38

39 ETR case Vlasov-BGK-Poisson cases Kinetic regime, N x = 128, N p = 10 6, t = 0.1 (left), N x = 128, N p = 10 5, t = 0.1 (right). log(e) TSI, ε=10-12 MiMa-Part-2-14 MiMa-Part-1 Moment G. -16 Full PIC MiMa-Grid εt log(e) TSI, ε=1 MiMa-Part-2 MiMa-Part-1 Moment G. Full PIC MiMa-Grid t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 39

40 ETR case Vlasov-BGK-Poisson cases Intermediate regime, N x = 256, N p = 10 5, t = TSI, ε=0.5 0 log(e) MiMa-Part-2 MiMa-Part-1 Moment G. Full PIC MiMa-Grid t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 40

41 ETR case Vlasov-BGK-Poisson cases Limit regime, N x = 128, N p = 10 4, t = (left), N x = 128, N p = 100, t = 0.01 (right). 0 TSI, ε=0.1 0 TSI, ε= log(e) MiMa-Part-2 MiMa-Part-1 Moment G. MiMa-Grid Limit log(e) MiMa-Part-2 MiMa-Part-1 Moment G. MiMa-Grid Limit t t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 41

42 Convergence Problem and objectives ETR case Vlasov-BGK-Poisson cases T = 0.1, N x = 16, N p = 100. Error on ρ in L norm Landau damping, convergence ε=1 ε=0.5 ε=0.1 ε=10-6 Slope t Error on ρ in L norm TSI, convergence ε=1 ε=0.5 ε=0.1 ε=10-6 Slope t A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 42

43 Conclusions Problem and objectives ETR case Vlasov-BGK-Poisson cases Diffusion (resp. drift-diffusion) limit recovered when ε 0. AP scheme. g 0 when ε 0 = few particles are sufficient at the limit, whereas grid methods have a constant cost, whatever the value of ε. Noise due to PIC method reduced (because only on g) = at equivalent results, fewer particles are necessary. Computational cost reduced at the limit. Second-order in time. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 43

44 Future works Problem and objectives ETR case Vlasov-BGK-Poisson cases Monte-Carlo method for adapting the number of particles automatically. Models where ε = ε(x). Extension to a Vlasov-BGK-Maxwell model: 1D in x / 2D in v.... A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 44

45 References Problem and objectives ETR case Vlasov-BGK-Poisson cases [4] S. Jin: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, J. Sci. Comput. 21, pp (1999). [5] M. Lemou, L. Mieussens: A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, J. Sci. Comp. 31, pp (2008). [6] T.-P. Liu, S.-H. Yu: Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles, Comm. Math. Phys. 246 pp (2004). [7] N. Crouseilles, M. Lemou: An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, KRM 4, pp (2011). [8] M. Lemou:, Relaxed micro-macro schemes for kinetic equations, Comptes Rendus Mathématique 348, pp , (2010). [9] P. Degond, G. Dimarco, L. Pareschi: The moment guided Monte Carlo method, International Journal for Numerical Methods in Fluids 67, pp (2011). [10] A. C., N. Crouseilles, M. Lemou: Micro-macro decomposition for Vlasov-BGK equation using particles, Kinetic and Related Models 5, pp , (2012). Thank you for your attention! A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 45