Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles
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1 Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3. The 8th International Conference on Computational Physics, Hong Kong January, University Paul Sabatier Toulouse III, IMT. 2 INRIA Rennes - Bretagne Atlantique & ENS Cachan Bretagne, IRMAR. 3 CNRS & University of Rennes I & ENS Cachan Bretagne, IRMAR. 1 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
2 Outline 1 Problem and objectives A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
3 Numerical simulation of plasmas Different scales in plasmas, for example collisions parameterized by the Knudsen number ε different models. Kinetic models Particles represented by a distribution function f (x, v, t). Solving the Vlasov equation (with source term S (ε)) t f + v x f + q m (E + v B) vf = S (ε) coupled to Maxwell equations or Poisson equation. In 3D 7 variables: 3 in space, 3 in velocity and the time heavy computations. 3 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
4 Fluid models Moment equations on physical quantities linked to f (density, mean velocity, temperature, etc.). Smaller cost, but lost of precision. General difficulties Find a well adapted model for our system, with a good precision/cost ratio. If two scales in the same simulation: two schemes with an interface? Difficult to deal with the interface! 4 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
5 Vlasov-Poisson-BGK system Collisions between particles parameterized by the Knudsen number ε. The Vlasov-Poisson-BGK equations (on [0, L x ] R R +, L x R + ) t f + v x f + E v f = 1 ε Q (f ), x E = 1 + f dv. Periodic conditions on f and E. Zero mean condition on E and initial conditions Lx 0 E (x, t) dx = 0, t 0, f (x, v, 0) = f 0 (x, v), x [0, L x ], v R. A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
6 Q (f ) is the BGK collisions operator (Bhatnagar-Gross-Krook): Q (f ) = M (U) f, M (U) the Maxwellian having the same first three moments than f, denoted by U. We denote by N (L Q ) = Span { M, vm, v 2 M }, the kernel of the linearized operator L Q of Q. 6 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
7 Objectives Problem and objectives Construction of a stable and consistent at the ε 0 limit scheme, of constant cost with respect to ε: an asymptotic preserving (AP) scheme 4. Reduction of the numerical cost: micro-macro decomposition with particles. Especially at the ε 0 limit since collisions bring f to its Maxwellian : f M (U) 0 when ε 0. 4 Jin, SIAM JSC A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
8 Micro-macro model Problem and objectives PIC method Projection step Macro part Micro-macro decomposition 56 f = M (U) + g: t M (U)+v x M (U)+E v M (U)+ t g +v x g +E v g = 1 ε g. Transport operator T = v x +E v : t M (U) + T M (U) + t g + T g = 1 ε g. Hypothesis: first 3 moments of g are zeros: mg := m (v) g (x, v, t) dv = 0, with m (v) := ) T (1, v, v 2. 2 Verified at the numerical level? If not, we have to impose it. 5 Lemou, Mieussens, SIAM JSC Liu, Yu, CMP A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
9 Micro-macro equations PIC method Projection step Macro part Let Π M the orthogonal projection in L 2 ( M 1 dv ) on N (L Q ): Π M (ϕ) = 1 ρ + [ (v u) (v u) ϕ ϕ + T ( v u 2 1 ) ( v u 2 2T 2 T ) ] 1 ϕ M. Properties: (I Π M ) ( t M) = Π M (g) = Π M ( t g) = 0. By applying (I Π M ) to Vlasov-BGK micro equation on g t g + (I Π M ) T (M + g) = 1 ε g. 9 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
10 PIC method Projection step Macro part By applying Π M to Vlasov-BGK macro equation on M (U) t M + Π M T (M + g) = 0. And by taking its first 3 moments t U + x F (U) + x vm (v) g = S (U), with F (U) the Euler flux and S (U) a source term. Micro-macro system 7 { t g + (I Π M ) T (M + g) = 1 ε g t U + x F (U) + x vm (v) g = S (U) equivalent to Vlasov-BGK. 7 Bennoune, Lemou, Mieussens, JCP A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
11 Algorithm Problem and objectives PIC method Projection step Macro part 1. Solving the micro part by a Particle-In-Cell (PIC) method. 2. Projection step to numerically force to zero the first three moments of g (matching procedure 8 ). 3. Solving the macro part by a finite volume scheme (mesh on x), with a source term dependent on g. 1-3 coupling: similarities with the δf method 9 but here: AP scheme. 8 Degond, Dimarco, Pareschi, IJNMF, Brunner, Valeo, Krommes, Phys. of Plasmas A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
12 1. PIC method on g Problem and objectives PIC method Projection step Macro part Solving by a PIC method the equation t g + (I Π M ) T (M + g) = 1 ε g t g + T g = (I Π M ) T M + Π M T g g ε =: S g, coupled to the Poisson equation for the electric field. Model: having N p numerical particles, with position x k, velocity v k and weight ω k, 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 2 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
13 PIC algorithm Problem and objectives PIC method Projection step Macro part Mesh generation Electric field computing on the mesh Initialization of positions and velocities of particles Computation of charge density on the mesh (deposition) Interpolation of electric field on the particles Evolution of the weights (if source term) Movement of particles 13 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
14 PIC method Projection step Macro part Equation with source term splitting. 1. Solving the transport part t g + T g = 0. Displacement of particles thanks to the equations of motion dx k dt (t) = v k (t), dv k dt (t) = E (x k (t), t). Second order Verlet scheme (for example): v n+ 1 2 k = v n k + t 2 E n (x n k ) x n+1 k = xk n + tv n+ 1 2 k v n+1 k = v n+ 1 2 k + t 2 E n+1 ( x n+1 ) k. 14 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
15 PIC method Projection step Macro part 2. Solving the source part t g = (I Π M ) (T M) + Π M (T g) 1 ε g. Equation on weights of the form dω k dt (t) = α k (t) ω k (t) ε where α k is the weight associated to (I Π M ) (T M) + Π M (T g). To have an AP scheme, we make the stiff term implicit: ω n+1 k = ω n k + t αn k ωn+1 t kε. 15 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
16 2. Projection step At time n + 1, we have N p k=1 Problem and objectives PIC method Projection step Macro part g n+1 (x, v) = ω n+1 k δ ( x x n+1 ) ( ) ( k δ v v n+1 k g x, v, t n+1 ). Micro-macro structure: mg ( x, v, t n+1) = 0. A priori not true for g n+1 (x, v) correction We compute U g := mg n+1 0 on each cell x i : U g (x i ) = mg n+1 xi = ω k m (v k ). k/x k [x i,x i+1 ] 2. We seek h N (L Q )= Span { M, vm, v 2 M } h (x, v) = λ (x) m (v) M (x, v) s.t. U g (x i ) = mh (x i, v). 10 C., Crouseilles, Lemou, KRM A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
17 PIC method Projection step Macro part Solving N x linear systems (3 3) U g (x i ) = A i λ i with A i a matrix containing moments of M and λ i λ (x i ). Computation of weights γ k associated to h on each particle. 3. Correction of the weights in order to preserve the micro-macro structure ω new k ω k γ k. By construction : mg n+1,new xi = U g (x i ) mh (x i, v) = 0. Remark: correction of order 1 (by using linear shape functions) (3N x 3N x ) system. 17 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
18 3. Macro part Problem and objectives PIC method Projection step Macro part Solving the equation t U + x F (U) = S (U) x vm (v) g =: S (U, g). Finite volume method U n+1 i = Ui n t x ( ) Fi+1/2 n F i 1/2 n t S i n, with Rusanov flux Fi+1/2 n = 1 ( ( ) F U n 2 i+1 + F (U n i ) a i+1/2 (U i+1 U i ) ), where a i+1/2 = max j=i,i+1 ( abs R ( JF ( xj ))), R (JF ) being the eigenvalues of the Jacobian of F. Computation of the moments of g to evaluate S n i = S (U n i ) ( vmg n+1 xi+1/2 vmg n+1 xi 1/2 x ). 18 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
19 Landau damping Problem and objectives Landau damping Computational cost Initial distribution function: f (x, v, 0) = 1 2π exp ( v 2 2 ) (1 + α cos (kx)). Micro-macro initializations: 1 + α cos (kx) U (x) = α cos (kx) and g (x, v, t = 0) = 0. Parameters: α = 0.01, k = 0.5, L x = 2π/k. Observation: electrical energy E (t) = E (t, x) 2 dx. 19 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
20 AP property Problem and objectives Landau damping Computational cost Convergence to Euler-Poisson at the ε 0 limit? E (t), comparison with Euler-Poisson. N x = 128, N p = , different values of ε log E(t) L Time t Good convergence AP scheme. Euler MiMa, eps= MiMa, eps=0.1 MiMa, eps=0.3 MiMa, eps=1 0 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
21 Importance of the projection step Landau damping Computational cost ρ (x), with and without correction. ε = 1, N p = at t = Without correction Correction of order 0 Correction of order Charge density 1-rho Position x Correction the instabilities disappear. 1 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
22 Numerical-noise reduction Landau damping Computational cost ρ (x), comparison with a PIC method on the whole f. ε = 1, N x = 128, N p = , at t = PIC BGK MiMa Charge density 1-rho Position x Micro-macro decomposition noise due to the PIC method reduced. 2 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
23 Computational cost Problem and objectives Landau damping Computational cost E (t), comparison with a PIC method on the whole f. ε = 10 ε = 10 4 N p Time MiMa s. PIC-BGK s. N p Time MiMa s. PIC-BGK s. 23 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
24 Conclusion Problem and objectives Landau damping Computational cost Micro-macro decomposition for Vlasov-Poisson-BGK using a PIC method. Projection step to numerically force the first 3 moments of g to zero. AP scheme. Reduction of the numerical noise due to the PIC method because only on g. Reduction of the cost compared to grid methods at the ε 0 limit because few particles are sufficient. Also true when g (x, v, 0) A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
25 Landau damping Computational cost - M. Benoune, M. Lemou, L. Mieussens: Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys. 227, pp (2008). - S. Brunner, E. Valeo, J.A. Krommes: Collisional delta-f scheme with evolving background for transport time scale simulations, Phys. of Plasmas 12, (1999). - 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, Kinetic and Related Models 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, SIAM J. Sci. Comput. 21, pp (1999). - M. Lemou, L. Mieussens: A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput. 31, pp (2008). - T.-P. Liu, S.-H. Yu: Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles, Comm. in Math. Phys. 246, pp (2004). - B. Yan, S. Jin: A successive penalty-based asymptotic-preserving scheme for kinetic equations, SIAM J. Sci. Comput., to appear. Thank you for your attention! 25 A. Crestetto, N. Crouseilles, M. Lemou Micro-macro scheme for Vlasov-Poisson-BGK
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