Micro-macro methods for Boltzmann-BGK-like equations in the diffusion scaling

Transcription

1 Micro-macro methods for Boltzmann-BGK-like equations in the diffusion scaling Anaïs Crestetto 1, Nicolas Crouseilles 2, Giacomo Dimarco 3 et Mohammed Lemou 4 Saint-Malo, 14 décembre Université de Nantes, LMJL & INRIA Rennes - Bretagne Atlantique, IPSO. 2 INRIA Rennes - Bretagne Atlantique, IPSO & Université de Rennes 1, IRMAR & ENS Rennes. 3 Université de Ferrara, Department of Mathematics and Computer Science. 4 CNRS & Université de Rennes 1, IRMAR & INRIA Rennes - Bretagne Atlantique, IPSO & ENS Rennes. A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 1

2 Outline 1 Problem and objectives 2 3 A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 2

3 Introduction Our problem Objectives 1 Problem and objectives Introduction Our problem Objectives 2 3 A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 3

4 Introduction Our problem Objectives Numerical simulation of particle systems We are interested in the numerical simulation of kinetic Problems ε, different scales: collisions parameterized by the Knudsen number ε, the development of Asymptotic Preserving (AP) schemes 5, the reduction of the cost at the limit ε 0. 5 Jin, SIAM JSC A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 4

5 Introduction Our problem Objectives Our 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, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 5

6 Objectives (1) Introduction Our problem Objectives Tools Idea Construction of an AP scheme. Reduction of the numerical cost at the limit ε 0. Micro-macro decomposition 6,7 for this model. Previous work with a grid in v for the micro part 8, cost was constant w.r.t. ε. Use particles for the micro part since few information in v is necessary at the limit. 6 Lemou, Mieussens, SIAM JSC Liu, Yu, CMP Crouseilles, Lemou, KRM A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 6

7 Objectives (2) Introduction Our problem Objectives Two points of view: Particle-In-Cell method with weights depending on time, Monte-Carlo techniques 9,10. Some upgrades: scheme of order 2 in time, add an electric field: Vlasov-BGK-Poisson system in 1Dx, 1Dv, space and/or time dependent ε, time-diminishing property. 9 P. Degond, G. Dimarco, L. Pareschi, IJNMF P. Degond, G. Dimarco, JCP A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 7

8 Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme 1 Problem and objectives 2 Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme 3 A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 8

9 Micro-macro decomposition Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme Micro-macro decomposition 11,12 : 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. 11 M. Lemou, L. Mieussens, SIAM JSC N. Crouseilles, M. Lemou, KRM A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 9 V

10 Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme 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 [vm xρ+v x g x vg M] = 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) := vm x ρ+v x g x vg M. A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 10

11 Difficulties Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme Stiff terms in the micro equation (4) on g. In previous works 13,14, 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. 13 M. Lemou, L. Mieussens, SIAM JSC N. Crouseilles, M. Lemou, KRM A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 11

12 Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme Strategy of Lemou 15 : 1. rewrite (4) 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 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 No more stiff terms and consistent with the initial micro equation (4). 15 Lemou, CRAS A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 12

13 Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme 1. PIC method with evolution of weights where Equation t g = e t/ε2 1 t S g := e t/ε2 1 t g ε 1 e t/ε2 t t g +ε 1 e t/ε2 [v x g] = S g t [vm x ρ+v x g x vg M] g ε 1 e t/ε2 [vm x ρ x vg M]. 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 (t,x,v) = ω k (t)δ(x x k (t))δ(v v k (t)). k=1 A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 13

14 Solving t g +ε 1 e t/ε2 t [v x g] = 0 Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme 1. Initialization: mesh in space X i = i x for macroscopic quantities such as ρ(t, x) ρ i (t) and vg (t, x) vg i (t), particles uniformly distributed in phase space (x, v), independently of the mesh, weights initialized to ω k (0) = g (0, x k, v k ) LxLv N p. (L x x-length of the domain, L v v-length.) 2. Movement of particles thanks to motion equations: For example dv k dt (t) = 0 and dx k dt (t) = ε1 e t/ε2 v k. t x n+1 k = x n k +ε(1 e t/ε2 )v k. A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 14

15 Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme Solving t g = S g 3. Deposition and interpolation for communications between macro mesh and particles. 4. Evolution of weights ω k : dω k dt (t) = S g (x k,v k ) L xl v N p with S g = e t/ε2 1 g ε 1 e t/ε2 [vm x ρ x vg M]. 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 = v km x ρ n (x n+1 k ) L xl v N p and βk n = x vg (x n+1 k )M L xl v. N p A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 15

16 2. Projection step Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme We now have N p g n+1 (x,v) ω n+1 k δ ( x x n+1 ) k δ(v vk ). k=1 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. In the case of no regularization (Dirac masses): k I i := { k / x k [X i 1/2,X i+1/2 ] } g i, ω k ω k x p k, k I i p k where p k := ρ(x k )M LxLv N p Maxwellian. is the weight associated to the A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 16

17 3. Macro part Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme Equation t ρ+ 1 ε x vg = 0. First proposition: discretization of Finite Volumes type 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, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 17

18 Correction of the macro discretization Micro equation is discretized as ω n+1 k = e t/ε2 ωk n ε(1 e t/ε2 ) Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme vm xρ {}}{ α n k + x vg M {}}{ βk n. Take the moment on cell i: i, and let h n i := e t/ε2 vg n i ε(1 e t/ε2 ) x v 2 g M so that vg n+1 i = ε(1 e t/ε2 ) v 2 M n i }{{} x ρ n i + hi n. 1/3 Np Inject it in the macro equation ρ n+1 i = ρ n i t ε x vg n+1 i, and take the diffusion term implicit new macro discretization: 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, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 18

19 Numerical limit Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme When ε 0, x vg M = O(ε) thus h n i = O(ε 2 ) and vg n+1 i = ε 1 3 xρ n i +O(ε 2 ). 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, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 19

20 AP property Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme 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, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 20

21 What about upgrades? Derivation of the micro-macro system First-order reformulation Particle-In-Cell / FV scheme Scheme of second order in time. Consider an electric field. Space and/or time dependent ε + time-diminishing property: number of particles fixed initially. A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 21

22 New discretization Monte-Carlo technique Numerical results 1 Problem and objectives 2 3 New discretization Monte-Carlo technique Numerical results A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 22

23 New discretization Monte-Carlo technique Numerical results Our Problem ε 1D radiative transport equation, diffusion scaling x [0,L x ] R, v R, charge density ρ = V ( f ) dv, M(v) = 1 2π exp, t f + 1 ε v xf = 1 ε2(ρm f) (7) 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 diffusion limit t ρ xx ρ = 0. (8) A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 23

24 Discretization of the micro equation New discretization Monte-Carlo technique Numerical results Model: considering at each time step N n p particles, with position x n k, velocity v k and constant weight ω k, k = 1,...,N n p, g is approximated by 16 N n p g N n p (t n,x,v) = ω k δ(x xk n )δ(v v k). k=1 Previously, moments of g evolved through the weights of N p (= cst) particles, now it evolves through the number of particles, each one having a constant weight. Solve the transport part as previously by computing x n+1 k = x n k +ε(1 e t/ε2 )v k. 16 N. Crouseilles, G. Dimarco, M. Lemou, KRM A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 24

25 New discretization Monte-Carlo technique Numerical results Monte-Carlo procedure for the source part The source part t g = e t/ε2 1 t is discretized in time with g ε 1 e t/ε2 [vm x ρ x v g M] t g n+1 = e t/ε2 g n (1 e t/ε2 )ε(vm x ρ n x ( v g n )M) where g n denotes the value of the function g after the transport part. Introduce p = e t/ε2 and write g n+1 = p g n (1 p)ε(vm x ρ n x ( v g n )M). A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 25

26 New discretization Monte-Carlo technique Numerical results From a given set of particles (xk n,vn k ) k=1,...,np n at tn : choose pnp n particles randomly and keep them unchanged, discard the others, sample a number of particles from (1 p)p(t n,x,v) := (1 p)ε(vm x ρ n x ( v g n )M). A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 26

27 Initialization New discretization Monte-Carlo technique Numerical results Choose the characteristic weight m p or the characteric number of particles N p necessary to sample the full distribution function f, and link them with m p = 1 xmax f(t = 0,x,v)dvdx. N p x min Now, we want to sample g(t = 0, x, v), that has no sign. We impose ω k {m p, m p }. For velocities, we impose v k {v j, j = 0,...,N v 1} k = 1,...,N n p, where v j = v min + j v, j = 0,...,N v 1. R A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 27

28 New discretization Monte-Carlo technique Numerical results The number of initial positive (resp. negative) particles having the velocity v k = v j in the cell C i = [x i,x i+1 ] is given by N 0,± i,j that is an approximation of 1 m p = ± x v m p g ± (t = 0,x i,v j ), xi+1 vj+1 x i v j ±g ± (t = 0,x,v)dvdx, with g ± = g± g 2 the positive (resp. negative) part of g. Positions of these N 0,± i,j particles are taken uniformly in C i = [x i,x i+1 ]. A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 28

29 From t n to t n+1 New discretization Monte-Carlo technique Numerical results Transport part with motion equation. Source part (1): keep qni n particles in each cell C i, discard the others. Source part (2): create a number (1 p)m n,± i,j of positive (resp. negative) particles of velocity v j, where M n,± i,j = ± 1 m p x vp n,± (x i,v j ), and P n,± (x i,v j ) is the positive (resp. negative) part of the function P(t n,x,v) evaluated in x i and v j. Source part (3): positions of these new particles are uniformly sampled in C i. A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 29

30 AP property New discretization Monte-Carlo technique Numerical results Discretization of macro part: ρ n+1 i = ρ n i t e t/ε2 v gn i+1 v g n i 1 ε 2 x + t(1 e t/ε2 ) ρ i+1 2ρ i +ρ i 1 x 2. Explicit scheme if ρ i = ρ n i. Implicit scheme if ρ i = ρ n+1 i. ε 0: right asymptotic limit, = AP property. A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 30

31 What about upgrades? New discretization Monte-Carlo technique Numerical results Scheme of second order in time. Consider an electric field. Space and/or time dependent ε + time-diminishing property: number of particles evolves in time and diminishes with ε. A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 31

32 Density and momentum for ε = 1 New discretization Monte-Carlo technique Numerical results Initially, N 0 p = for MiMa-MC and PIC MC. Density ρ ε=1, T=2 0.8 PIC MC 0.7 Reference MiMa MC x First momentum <vf> ε=1, T=2 PIC MC Reference MiMa MC x A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 32

33 Density and momentum for ε = 0.2 New discretization Monte-Carlo technique Numerical results Density ρ ε=0.2, T=2 0.8 PIC MC 0.7 Reference MiMa MC x First momentum <vf> ε=0.2, T=2 PIC MC Reference MiMa MC x A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 33

34 AP property New discretization Monte-Carlo technique Numerical results Comparison with a discretization of the diffusion limit. First momentum <vf> AP property, T=2 0.8 Limit ε= ε=0.01 ε= x First momentum <vf> AP property, T=2 Limit ε=0.2 ε=0.01 ε= x A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 34

35 Time-diminishing property New discretization Monte-Carlo technique Numerical results Total number of particles is plotted as a function of time for ε = 1, ε = 0.5, ε = 0.2 and ε = 0.1 on the left, ε = 10 2, ε = 10 3 and ε = 10 4 on the right. Total number of particles ε=1, 0.5, 0.2 and 0.1 ε=1 ε=0.5 ε=0.2 ε= t Total number of particles ε=0.01, and ε=0.01 ε=0.001 ε= t A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 35

36 New discretization Monte-Carlo technique Numerical results Thank you for your attention! A. Crestetto, N. Crouseilles, G. Dimarco, M. Lemou Micro-macro methods for Boltzmann-BGK-like eq. 36