An asymptotic-preserving micro-macro scheme for Vlasov-BGK-like equations in the diffusion scaling

Transcription

1 An asymptotic-preserving micro-macro scheme for Vlasov-BGK-like equations in the diffusion scaling Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3 Saint-Malo 13 December Université de Nantes, LMJL & INRIA Rennes - Bretagne Atlantique, IPSO. 2 INRIA Rennes - Bretagne Atlantique, IPSO & Université de Rennes 1, IRMAR & ENS Rennes. 3 CNRS & Université de Rennes 1, IRMAR & ENS Rennes. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 1/39

2 Outline 1 Problem and objectives A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 2/39

3 Introduction Our problem Objectives Numerical simulation of particles 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 Vlasov-type equation t f +A(v,ε) x f +B(v,E,B,ε) v f = S(ε) coupled to Maxwell or Poisson equations. Accurate and necessary far from thermodynamical equilibrium. In 3D = 7 variables = heavy computations. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 3/39

4 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 AP micro-macro scheme for Vlasov-BGK-like eq. 4/39

5 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 AP micro-macro scheme for Vlasov-BGK-like eq. 5/39

6 Introduction Our problem Objectives Our 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 R, charge density ρ = 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 AP micro-macro scheme for Vlasov-BGK-like eq. 6/39

7 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 (with a 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 AP micro-macro scheme for Vlasov-BGK-like eq. 7/39

8 Problem and objectives First formulation of the micro-macro system Reformulation 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 AP micro-macro scheme for Vlasov-BGK-like eq. 8/39

9 First formulation of the micro-macro system 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 [vm xρ+v x g x vg M vmeρ+e v g] = 1 ε 2g. (4) Equation (1) micro-macro system: t ρ+ 1 ε x vg = 0, t g + 1 ε F(ρ,g,E) = 1 (5) ε 2g, where F(ρ,g,E) := vm x ρ+v x g x vg M vmeρ+e v g. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 9/39

10 Difficulties Problem and objectives First formulation of the micro-macro system Reformulation Stiff terms in the micro equation (4) on g. In [Lemou, Mieussens, SIAM JSC 2008] and in [Crouseilles, Lemou, KRM 2011], 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. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 10/39

11 First formulation of the micro-macro system Reformulation Strategy of Lemou 8 : 1. rewrite t g + 1 ε F(ρ, g, E) = 1 ε 2 g as t (e t/ε2 g) = et/ε2 F(ρ, g, E), 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 2 ) 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, E n ), t g ε 1 e t/ε2 F(ρ, g, E). (6) t A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 11/39

12 Properties Problem and objectives First formulation of the micro-macro system 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 = εvm( x ρ Eρ)+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 [vm x ρ+v x g x vg M vmeρ+e v g], (3) t ρ+ 1 ε x vg = 0, (2) t ρ x ( x ρ Eρ) = 0. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 12/39

13 Algorithm Problem and objectives PIC method Projection of weights Finite volumes scheme Properties 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,E). 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 AP micro-macro scheme for Vlasov-BGK-like eq. 13/39

14 1. PIC method Problem and objectives PIC method Projection of weights Finite volumes scheme Properties Equation t g = e t/ε2 1 t t g +ε 1 e t/ε2 [v x g + E v g] t = e t/ε2 1 t g ε 1 e t/ε2 F(ρ,g,E) t g ε 1 e t/ε2 [vm x ρ x vg M vmeρ] =: 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 AP micro-macro scheme for Vlasov-BGK-like eq. 14/39

15 Problem and objectives PIC method Projection of weights Finite volumes scheme Properties 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 AP micro-macro scheme for Vlasov-BGK-like eq. 15/39

16 Regularization using shape functions PIC method Projection of weights Finite volumes scheme Properties 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 AP micro-macro scheme for Vlasov-BGK-like eq. 16/39

17 Deposition and interpolation PIC method Projection of weights Finite volumes scheme Properties 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 R ω k (t)v p k (t)b l(x i x k (t)). k=1 Interpolation: evaluation of the electric field 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 AP micro-macro scheme for Vlasov-BGK-like eq. 17/39

18 PIC method Projection of weights 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. Deposition ρ i (t n ). 3. Solving the Poisson equation x E (x,t) = ρ(x,t) 1 on the mesh: E i (t n ). E i+1 (t n ) E i (t n ) x = ρ i (t n ) 1 (for example) 4. Interpolation on the particles E (x k,t n ). A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 18/39

19 PIC method Projection of weights Finite volumes scheme Properties Solving t g +ε 1 e t/ε2 t [v x g + E v g] = 0 5. Movement of particles thanks to motion equations: dx k dt (t) = ε1 e t/ε2 v k (t) and dv k t dt (t) = ε1 e t/ε2 E (x k,t). t Verlet scheme (for example): v n+ 1 2 k = v n k ε(1 e t/ε2 )E n (x n k ) x n+1 k = xk n +ε(1 e t/ε2 )v n+ 1 2 k v n+1 k = v n+ 1 2 k ε(1 e t/ε2 )E n+1( x n+1 k ). A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 19/39

20 PIC method Projection of weights Finite volumes scheme Properties Solving t g = S g 6. 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 [vm x ρ x vg M vmeρ]. t t In practice: ω n+1 k with ω n k t = e t/ε2 1 ωk n t ε1 e t/ε2 [α n k t +βn k ], α n k = vn+1 k M( x ρ n (x n+1 k ) E n (x n+1 k )ρ n (x n+1 k and βk n = x vg (x n+1 k ) L xl v. N p )) L xl v N p A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 20/39

21 2. Projection step Problem and objectives PIC method Projection of weights 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 We want to ensure g n+1 = 0. How? By correcting the weights, cell by cell. We compute g n+1 0 on each cell C i N p g n+1 i = ω k B l (x i x k ). k=1 ) ( ) δ v v n+1 k. We seek h N (Q) of the form h(x,v) = λ(x) M(v) s.t. g n+1 i = h i. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 21/39

22 PIC method Projection of weights Finite volumes scheme Properties We expand λ(x) on the basis of B-splines of degree l N x λ(x) = λ j B l (x x j ), λ j R. j=1 For example l = 0 (l = 1 is also computed). Let p k the weights associated to M, we compute h i 1 x 2λ i k/ x k x i x/2 p k. g n+1 i = h i gives λ i = x 2 g n+1 i k/ x k x i x/2 p k. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 22/39

23 PIC method Projection of weights Finite volumes scheme Properties We compute the weights γ k of h N x γ k = λ j B 0 (x k x j )p k = j=1 for each particle k in the cell C i. We correct the weights of g to obtain ω new k ω k γ k, g n+1,new i = 0. x g n+1 i k/ x k x i x/2 p k p k, Remark: order 1 = solve a tridiagonal system to obtain λ i. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 23/39

24 3. Macro part Problem and objectives PIC method Projection of weights 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 AP micro-macro scheme for Vlasov-BGK-like eq. 24/39

25 Numerical limit Problem and objectives PIC method Projection of weights Finite volumes scheme Properties Micro equation is discretized as k = e t/ε2 ωk n ε(1 e t/ε2 ) ω n+1 x vg M {}}{{}}{ α n k + βk n. vm xρ vmeρ When ε 0, β n k = O(ε) thus ωn+1 k = εα n k +O(ε2 ) and vg n+1 i = ε v 2 M n i ( }{{} x ρ n i Ei n ρ n i )+O(ε2 ). =1 Injecting in the macro equation ρ n+1 i gives ρ n+1 i = ρ n i t ε x vg n+1 i = ρ n i + t x ( x ρ n i E n i ρ n i ), = we recover a discretization of the limit equation (2). A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 25/39

26 "Maneuver" Problem and objectives PIC method Projection of weights Finite volumes scheme Properties Use this idea to implicit the diffusion term. Write k = e t/ε2 ωk n ε(1 e t/ε2 ) ω n+1 vm xρ vmeρ {}}{ α n k + x vg M {}}{ βk n. Let h n i := e t/ε2 g n i ε(1 e t/ε2 ) x vg M and approximate vg n+1 i = ε(1 e t/ε2 )( x ρ n i E n i ρ 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 ) x ( x ρ n+1 i Ei n ρ n i ) t ε xhi n. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 26/39

27 AP property Problem and objectives PIC method Projection of weights 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 drift-diffusion equation (2) = AP property. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 27/39

28 PIC method Projection of weights Finite volumes scheme Properties Second-order scheme in time: few words ρ n+1 i Reformulation: when integrating in time t (e t/ε2 g) = et/ε2 ε F(ρ, g, E), use a midpoint method for the right-hand side. Time scheme: RK2. Use only B-splines of order l 1 for depositions and interpolations. Do not "maneuver", but "cheat": = ρ n i t ε x vg n+1/2 i + t(1 e t/ε2 ) 2 x ( x ( ρn+1 i 2 +ρ n i ) E n i ρ n i ). A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 28/39

29 Landau damping Problem and objectives First test case Second test case Initial distribution function: f (x,v,0) = 1 exp( v2 )(1+α cos(kx)), x [0,2π/k] 2π 2 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. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 29/39

30 First test case Second test case 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 AP micro-macro scheme for Vlasov-BGK-like eq. 30/39

31 First test case Second test case 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 AP micro-macro scheme for Vlasov-BGK-like eq. 31/39

32 First test case Second test case 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 AP micro-macro scheme for Vlasov-BGK-like eq. 32/39

33 Two stream instability First test case Second test case Initial distribution function: f (x,v,0) = v2 exp( v2 )(1+α cos(kx)), x [0,2π/k] 2π 2 Micro-macro initializations: ρ(x,t) = 1+α cos(kx) g 0 (x,v) = 1 ( v 2 1 ) ) exp ( v2 (1+αcos(kx)). 2π 2 Parameters: α = 0.05, k = 0.5. A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 33/39

34 First test case Second test case 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 AP micro-macro scheme for Vlasov-BGK-like eq. 34/39

35 First test case Second test case 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 AP micro-macro scheme for Vlasov-BGK-like eq. 35/39

36 First test case Second test case 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 AP micro-macro scheme for Vlasov-BGK-like eq. 36/39

37 Conclusions Problem and objectives First test case Second test case 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. Error on ρ in L2 and L norms Landau damping, convergence ε=1, L ε=1, L2 ε=0.5, L ε=0.5, L2 ε=0.1, L ε=0.1, L2 ε=10-6, L ε=10-6, L2 Slope t Error on ρ in L2 and L norms TSI, convergence ε=1, L ε=1, L2 ε=0.5, L ε=0.5, L2 ε=0.1, L ε=0.1, L2 ε=10-6, L ε=10-6, L2 Slope t A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 37/39

38 Future works Problem and objectives First test case Second test case Monte-Carlo method for adapting the number of particles automatically. Dirichlet boundary conditions.... A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 38/39

39 References Problem and objectives First test case Second test case - A. C., N. Crouseilles, M. Lemou: Micro-macro decomposition for Vlasov-BGK equation using particles, Kinetic and Related Models 5, pp , (2012). - 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). - P. Degond, G. Dimarco, L. Pareschi: The moment guided Monte Carlo method, International Journal for Numerical Methods in Fluids 67, pp (2011). - S. Jin: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, J. Sci. Comput. 21, pp (1999). - T.-P. Liu, S.-H. Yu: Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles, Comm. Math. Phys. 246 pp (2004). - M. Lemou:, Relaxed micro-macro schemes for kinetic equations, Comptes Rendus Mathématique 348, pp , (2010). - 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). Thank you for your attention! A. Crestetto, N. Crouseilles, M. Lemou AP micro-macro scheme for Vlasov-BGK-like eq. 39/39

40 First test case Second test case Noise reduction on ρ, ε = 1, N x = 128, N p = , at t = 0.2 (left) and t = 0.4 (right): PIC BGK MiMa 0.02 PIC BGK MiMa Charge density 1-rho Charge density 1-rho Position x Position x

41 First test case Second test case Necessary number of particles, ε = 10, N x = 128: log E(t) L PIC BGK, Npart= PIC BGK, Npart= PIC BGK, Npart= MiMa, Npart= Time t

42 First test case Second test case Importance of the projection step on ρ at t = 5, ε = 1, N p = : Without correction Correction of order 0 Correction of order Charge density 1-rho Position x

43 Computational cost Problem and objectives First test case Second test case Landau damping, ε = 0.1: MiMa N p = s. PIC-BGK N p = s log E(t) L PIC BGK, Npart=1*10 6 MiMa, Npart=1* Time t