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 La Tremblade, Congrès SMAI juin 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 1 Problem and objectives Introduction Our problem Objectives 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 1D Problem ε 1D radiative transport equation, diffusion scaling t f + 1 ε v xf = 1 ε2(ρm f) (1) 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. (2) A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 7
8 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 this model 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 8
9 Derivation of the micro-macro system First-order reformulation 1 Problem and objectives 2 Derivation of the micro-macro system First-order reformulation 3 4 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 Problem ε (1) 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 (1) = macro equation on ρ t ρ+ 1 ε x vg = 0. (3) Applying (I Π) to (1) = micro equation on g t g + 1 ε [v xρ+v x g x vg ] = 1 ε2g. (4) Equation (1) micro-macro system: t ρ+ 1 ε x vg = 0, t g + 1 ε F(ρ,g) = 1 (5) 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 (4) 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). (6) 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 (6) is consistent with the initial micro equation (4). Asymptotic behaviour: t > 0 fixed, as ε goes to zero, we get from (6) g = εv x ρ+o(ε 2 ), which injected in the macro equation (3) provides the limit model (2). (6) t g = e t/ε2 1 t g ε1 e t/ε 2 t [v x ρ+v x g x vg ], (3) t ρ+ 1 ε x vg = 0, (2) t ρ 1 3 xxρ = 0. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 14
15 PIC method Finite volumes scheme Properties 1 Problem and objectives 2 3 PIC method Finite volumes scheme Properties 4 A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 15
16 Algorithm Problem and objectives Reformulated system t ρ+ 1 ε x vg = 0, PIC method Finite volumes scheme Properties t g = e t/ε2 1 g ε 1 e t/ε2 [v x ρ+v x g x vg ]. t t Algorithm: 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 16
17 1. PIC method where Equation Problem and objectives t g = e t/ε2 1 t S g := e t/ε2 1 t PIC method Finite volumes scheme Properties g ε 1 e t/ε2 [v x ρ+v x g x vg ] t t g +ε 1 e t/ε2 [v x g] = S g t g ε 1 e t/ε2 [v x ρ x vg ]. 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 17
18 Steps of the PIC method PIC method Finite volumes scheme Properties Initializations: Mesh in space x i = i x for the macroscopic quantities such as ρ(t,x) ρ i (t) and vg (t,x) vg i (t). Initialization of particles independently of the mesh. At each time step t n : Transport part: solving t g +ε 1 e t/ε2 t [v x g] = 0. Source part: solving t g = S g. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 18
19 Solving t g +ε 1 e t/ε2 t [v x g] = 0 PIC method Finite volumes scheme Properties 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. 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 19
20 PIC method Finite volumes scheme Properties Solving t g = S g 3. 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 20
21 2. Projection step Problem and objectives PIC method Finite volumes scheme Properties 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 21
22 3. Macro part Problem and objectives PIC method Finite volumes scheme Properties 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 22
23 Numerical limit Problem and objectives PIC method Finite volumes scheme Properties 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 (2). A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 23
24 "Maneuver" Problem and objectives PIC method Finite volumes scheme Properties 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 24
25 AP property Problem and objectives PIC method Finite volumes scheme Properties 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 (2) = AP property. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 25
26 ETR case Vlasov-BGK-Poisson cases 1 Problem and objectives ETR case Vlasov-BGK-Poisson cases A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 26
27 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 27
28 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 28
29 ETR case Vlasov-BGK-Poisson cases Our 2nd Problem ε 1D Vlasov-BGK equation, diffusion scaling t f + 1 ε v xf + 1 ε E vf = 1 ε2(ρm f) (7) 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. (8) A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 29
30 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 30
31 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 31
32 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 32
33 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 33
34 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 34
35 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 35
36 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 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 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 37
38 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. A. Crestetto, N. Crouseilles, M. Lemou Schéma AP micro-macro pour Boltzmann-BGK 38
39 Moreover Problem and objectives ETR case Vlasov-BGK-Poisson cases The presented scheme is first-order in time. We have also developed a second-order in time scheme (in a submitted paper). Errors on ρ as a function of t: Error on ρ in L norm ETR, convergence (1) ε=1 ε=0.5 ε=0.1 ε=10-6 Slope t Error on ρ in L norm Landau damping, convergence ε=1 ε=0.5 ε=0.1 ε=10-6 Slope t Parameters: T = 0.1, N x = 16, N p = 100. 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 39
40 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 40
41 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 41
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