Modèles hybrides préservant l asymptotique pour les plasmas

Transcription

1 Modèles hybrides préservant l asymptotique pour les plasmas Anaïs Crestetto 1, Nicolas Crouseilles 2, Fabrice Deluzet 1, Mohammed Lemou 3, Jacek Narski 1 et Claudia Negulescu 1. Groupe de Travail Méthodes Numériques, LJLL. 8 avril Université Paul Sabatier Toulouse 3, IMT. 2 INRIA Rennes - Bretagne Atlantique & ENS Cachan Bretagne, IRMAR. 3 CNRS & Université de Rennes I & ENS Cachan Bretagne, IRMAR. 1 A. Crestetto AP-hybrid models for plasmas

2 Plasma and applications What is a plasma? A globally neutral gaz, constituted of at least two species of charged particles: positive ions and electrons. Where does it appear? At natural state: - stars, - ionosphere, - auroras, etc. At artificial state: - nuclear fusion, - TV screens, - neon lighting, etc. 2 A. Crestetto AP-hybrid models for plasmas

3 Outline 1 Collisional plasmas: problem and objectives A. Crestetto AP-hybrid models for plasmas

4 Numerical simulation of plasmas Different scales in plasmas, for example collisions parameterized by the Knudsen number ε different kinds of 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. 4 A. Crestetto AP-hybrid models for plasmas

5 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, develop a numerical scheme efficient in each regime: - spatial coupling of two schemes, with an interface, - asymptotic-preserving (AP) scheme 4. 4 Jin, SIAM JSC A. Crestetto AP-hybrid models for plasmas

6 Asymptotic-Preserving scheme Let P ε a problem dependent on ε and P ε,h the associated numerical scheme. Pb: Standard schemes need h = O(ε)... Aim: Construct a scheme for which h does not depend on ε. Let P 0 = lim ε 0 P ε the limit problem and P 0,h the associated numerical scheme. Def: P ε,h is AP: uniform stability and consistency with P 0,h. P ε h 0 P ε,h ε 0 ε 0 P 0 P 0,h h 0 6 A. Crestetto AP-hybrid models for plasmas

7 Vlasov-BGK-Poisson system Numerical method Kinetic-fluid numerical scheme for Vlasov-BGK-Poisson using particles in collaboration with Nicolas Crouseilles 5 and Mohammed Lemou 6. 5 INRIA Rennes - Bretagne Atlantique & ENS Cachan Bretagne, IRMAR. 6 CNRS & Université de Rennes I & ENS Cachan Bretagne, IRMAR. 7 A. Crestetto AP-hybrid models for plasmas

8 Vlasov-Poisson-BGK system Vlasov-BGK-Poisson system Numerical method 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. 8 A. Crestetto AP-hybrid models for plasmas

9 Vlasov-BGK-Poisson system Numerical method 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. A. Crestetto AP-hybrid models for plasmas

10 Objectives Vlasov-BGK-Poisson system Numerical method Construction of an AP scheme, based on a micro-macro decomposition 7,8. Reduction of the numerical cost: - few points are sufficient in v at the limit since collisions bring f to its Maxwellian: f M (U) 0 when ε 0, - idea: use particles for the micro part. 7 Lemou, Mieussens, SIAM JSC Liu, Yu, CMP A. Crestetto AP-hybrid models for plasmas

11 Micro-macro model Vlasov-BGK-Poisson system Numerical method Micro-macro decomposition 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. 1 A. Crestetto AP-hybrid models for plasmas

12 Micro-macro equations Vlasov-BGK-Poisson system Numerical method 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. 12 A. Crestetto AP-hybrid models for plasmas

13 Vlasov-BGK-Poisson system Numerical method 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 9 (equivalent to Vlasov-BGK) t g +(I Π M )T (M + g) = 1 ε g, t U + x F (U)+ x vm(v) g = S (U). 9 Bennoune, Lemou, Mieussens, JCP A. Crestetto AP-hybrid models for plasmas

14 Algorithm Vlasov-BGK-Poisson system Numerical method 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 10 ). 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 11 but here: AP scheme. 10 Degond, Dimarco, Pareschi, IJNMF, Brunner, Valeo, Krommes, Phys. of Plasmas A. Crestetto AP-hybrid models for plasmas

15 1. PIC method on g Vlasov-BGK-Poisson system Numerical method 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 15 A. Crestetto AP-hybrid models for plasmas

16 Collisional plasmas: problem and objectives PIC algorithm Vlasov-BGK-Poisson system Numerical method 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 16 A. Crestetto AP-hybrid models for plasmas

17 Vlasov-BGK-Poisson system Numerical method 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 En (x n k ) x n+1 k = xk n 1 + tvn+ 2 k v n+1 k = v n+ 1 2 k + t 2 En+1( x n+1 ) k. 7 A. Crestetto AP-hybrid models for plasmas

18 Vlasov-BGK-Poisson system Numerical method 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. 8 A. Crestetto AP-hybrid models for plasmas

19 2. Projection step At time n+ 1, we have N p g n+1 (x,v) = k=1 ω n+1 k δ ( x x n+1 k Vlasov-BGK-Poisson system Numerical method Micro-macro structure: mg ( x,v,t n+1) = 0. ) ( ) ( δ v v n+1 k g x,v,t n+1 ). 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). 12 C., Crouseilles, Lemou, KRM A. Crestetto AP-hybrid models for plasmas

20 Vlasov-BGK-Poisson system Numerical method 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. 0 A. Crestetto AP-hybrid models for plasmas

21 3. Macro part Vlasov-BGK-Poisson system Numerical method 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 ( ) Fi+1/2 n x Fn i 1/2 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 ). 1 A. Crestetto AP-hybrid models for plasmas

22 Landau damping Vlasov-BGK-Poisson system Numerical method Initial distribution function: f (x,v,0) = 1 ) exp ( v2 (1+α cos(kx)). 2π 2 Micro-macro initializations: 1+αcos(kx) U(x) = 0 1+αcos(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. 2 A. Crestetto AP-hybrid models for plasmas

23 Vlasov-BGK-Poisson system Numerical method AP property? Convergence to Euler-Poisson? E (t), comparison with Euler-Poisson. N x = 128, N p = , different values of ε log E(t) L Euler -10 MiMa, eps= MiMa, eps=0.1 MiMa, eps=0.3 MiMa, eps= Time t Good convergence AP scheme. 3 A. Crestetto AP-hybrid models for plasmas

24 Importance of the projection step Vlasov-BGK-Poisson system Numerical method 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. 4 A. Crestetto AP-hybrid models for plasmas

25 Numerical-noise reduction Vlasov-BGK-Poisson system Numerical method 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. 25 A. Crestetto AP-hybrid models for plasmas

26 Computational cost Vlasov-BGK-Poisson system Numerical method E (t), comparison with a PIC method on the whole f. ε = 10 ε = 10 4 N p Time N p Time MiMa s. PIC-BGK s. MiMa s. PIC-BGK s. 26 A. Crestetto AP-hybrid models for plasmas

27 Conclusion Vlasov-BGK-Poisson system Numerical method 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) 0. Perspectives Diffusion limit. Non local collision operators. Dirichlet boundary conditions. 27 A. Crestetto AP-hybrid models for plasmas

28 Anisotropic equation for the electrical potential Coupling strategy AP-Limit coupling for the ionospheric plasma in collaboration with Fabrice Deluzet 13, Jacek Narski 14 and Claudia Negulescu CNRS & Université Paul Sabatier Toulouse 3, IMT. 14 Université Paul Sabatier Toulouse 3, IMT. 8 A. Crestetto AP-hybrid models for plasmas

29 Ionospheric plasma Anisotropic equation for the electrical potential Coupling strategy Ionosphere constituted of partially ionized plasma, submitted to a strong magnetic field. Density of neutral particles and frequency of collisions increase at the extremities of the magnetic field lines, that is at low altitude. Solar winds perturb the plasma and are responsible for irregularities or interruptions in the communication with satellites. Under some hypotheses s.t. - quasi-neutrality, - inerty of ions neglectable in front of collisions with neutral particles, - fixed magnetic field, we only consider a highly anisotropic elliptic equation for the electric potential. 9 A. Crestetto AP-hybrid models for plasmas

30 Singular perturbation problem Anisotropic equation for the electrical potential Coupling strategy Domain: Ω = Ω x Ω z = [x,x + ] [z,z + ]. Considered P-problem: (P) x (A x x u ε ) z ( Az ε(z) zu ε )+A 0 u ε = f, for (x,z) Ω x Ω z, A z(x,z ± ) ε(z ± ) z u ε (x,z ± ) = g ± (x), for x Ω x, u ε (x,z) = 0, for (x,z) Ω x Ω z, u ε the solution, A x (x,z), A z (x,z), A 0 (x,z) of the same order of magnitude, in order to have existence and uniqueness of a solution u ε for ε > A. Crestetto AP-hybrid models for plasmas

31 Numerical problem at the ε 0 limit Anisotropic equation for the electrical potential Coupling strategy ε-constant case, let formally ε tends to zero in (P): z (A z z u) = 0, for (x,z) Ω x Ω z, z u(x,z ± ) = 0, for x Ω x, u(x,z) = 0, for (x,z) Ω x Ω z. Non-unique solution: functions constant along the z-coordinate and satisfying u(x,z) = 0 on Ω x Ω z. Ill-conditioned P-model when ε 1. Remark: unique solution if Dirichlet or periodic conditions on Ω z. 1 A. Crestetto AP-hybrid models for plasmas

32 Limit model Anisotropic equation for the electrical potential Coupling strategy Notation: f (x) := 1 L z Ω z f (x,z) dz. Properties: f = 0, f z = f z, ( f x ( f x ) = f x, fg = f ḡ + f g, ) = f x, (fg) = f g f g + f g + f ḡ. (L) Integrating (P) along the z-coordinate and assuming u = u(x) (valid on the limit) gives the limit model: x ( Ax x u ) + A 0 u = f + g+(x) L z g (x) L z, for x Ω x, u(x) = 0, for x Ω x. 2 A. Crestetto AP-hybrid models for plasmas

33 AP reformulation ( AP ) (AP ) Decomposition: u(x,z) = u(x)+u (x,z). AP reformulation 15,16 : ( x Ax x u ) + A 0 u = f + g+(x) L z ( + x A x z u ) A 0 u, u(x) = 0, ( ) x (A x x u ) Az z ε(z) zu + A 0 u = f A 0 u + x (A x x u), A z(x,z ± ) ε(z ± ) z u (x,z ± ) = g ± (x), u (x,z) = 0, u = 0, Anisotropic equation for the electrical potential Coupling strategy g (x) L z for x Ω x, for x Ω x, for (x,z) Ω x Ω z, for x Ω x, for (x,z) Ω x Ω z, constraint. 15 Degond, Deluzet, Negulescu, SIAM MMS Besse, Deluzet, Negulescu, Yang, JSC A. Crestetto AP-hybrid models for plasmas

34 Coupling strategy Anisotropic equation for the electrical potential Coupling strategy Assumptions: - in a large region of the computational domain, ε 1, - the domain can be decomposed in the z-direction into two subdomains, delimited by an interface on z ι [z, z + ]: Ω z = Ω 1 z Ω2 z where Ω 2 z = [z, z ι ] and Ω 1 z = [z ι, z + ], - in Ω 2 z, u does not depend on z. Decomposition: - u Ω (x, z) = u 1 1 (x, z), z - u Ω (x, z) = u 2 2 (x). z 4 A. Crestetto AP-hybrid models for plasmas

35 Anisotropic equation for the electrical potential Coupling strategy (AP 1 ) (AP 2 ) AP-L formulation: ( AP ) (AP 1 ) (AP 2 ) with ( ) x (A x x u 1 ) Az z ε(z) zu 1 + A 0 u 1 = f A 0 u + x (A x x u), A z(x,z +) ε(z +) z u 1 (x,z +) = g + (x), u 1 (x,z) = 0, u 1 (x,z ι) = u 2 (x), Ω 1 u z 1 (x,z) dz + L2 z u 2 (x) = 0, ( ) x Ω A xdz 2 x u z 2 + Ω A 0dzu 2 z 2 for (x,z) Ω x Ω 1 z, for x Ω x, for (x,z) Ω x Ω 1 z, x Ω x, constraint, = Ω fdz g 2 + Az(x,zι) z ε(z ι) z u 1 (x,z ι) ( ) Ω A 0dzu + 2 x z Ω A xdz 2 x u, for (x,z) Ω x Ω2 z u 2 (x) = 0, for x Ω x. 5 A. Crestetto AP-hybrid models for plasmas z,

36 Finite elements discretization Anisotropic equation for the electrical potential Coupling strategy Discretization of Ω x Ω z by x i = i x, i = 0,...,N x + 1, z k = k z, k = 0,...,N z + 1. Interface put on z ι (we assume k = 1,...,N z s.t. z ι = z k ). P 1 hat functions κ 0 (z) = χ i (x) = κ k (z) = z 1 z z, z [z 0,z 1 ) 0, elsewhere x x i 1, x x [x i 1,x i ) x i+1 x, x x [x i,x i+1 ) 0, elsewhere z z k 1, z z [z k 1,z k ) z k+1 z, z z [z k, z k+1 ) 0, elsewhere and κ Nz+1 (z) =, i = 1,...,N x,, k = 1,...,N z, 36 A. Crestetto AP-hybrid models for plasmas z z Nz, z z [z Nz,z Nz+1) 0, elsewhere.

37 Anisotropic equation for the electrical potential Coupling strategy Test functions χ i, i = 1,...,N x, χ i κ k, i = 1,...,N x, Unknowns approximated by u h (x) k = ι,...,n z + 1. = N x i=1 α iχ i (x), u 1h (x,z) = N x i=1 u 2h (x) P h (x) = N x i=1 γ iχ i (x), = N x i=1 δ iχ i (x). Nz+1 k=ι β ik χ i (x)κ k (z), System A 2 + A 4 1 (C L z 11 + C 31 ) 1 (C L z 12 + C 32 ) 0 C 21 + C 01 A 11 + A 01 + A 31 0 B 11 C 22 + C 02 D A 12 + A B 21 B 22 0 α β γ δ = F u F u 1 F u A. Crestetto AP-hybrid models for plasmas

38 ε-variable test case Anisotropic equation for the electrical potential Coupling strategy Domain Ω x Ω z = [0,1] [ 3 2, 1 2]. Test case ε(z) = 1 2 (ε 2(1+tanh(rz))+ε 1 (1 tanh(rz))), with r R +, ε 1 R + and ε 2 = 1, A x (x,z) = A 0 (x,z) = L z + xz 2, A z (x,z) = L z + xz. Exact solution u(x,z) = sin ( )( ( )) 2π 2π x 1+ε(z)sin z. L x L z f, g + and g are computed by injecting the exact solution into the equations. 38 A. Crestetto AP-hybrid models for plasmas

39 Anisotropic equation for the electrical potential Coupling strategy ε as a function of z for different values of r. eps Domain b z r=10 r=20 r=30 39 A. Crestetto AP-hybrid models for plasmas

40 Scheme order Anisotropic equation for the electrical potential Coupling strategy L2-relative error as a function of x z. 0.1 Various eps test case, min(eps)=1e-25 (Relative) L2-error e-05 1e-06 1e-06 1e dx*dz r=10 r=20 r=30 Slope 2 2nd order scheme. 40 A. Crestetto AP-hybrid models for plasmas

41 AP property Anisotropic equation for the electrical potential Coupling strategy L2-relative error as a function of ε 1, comparison with AP and P schemes. (Relative) L2-error Nx=Nz=64 APL scheme AP scheme P scheme AP scheme e-25 1e-20 1e-15 1e-10 1e-05 1 min(eps) 41 A. Crestetto AP-hybrid models for plasmas

42 Computational time Anisotropic equation for the electrical potential Coupling strategy Computational time and number of nonzeros for r = 30 and ε 1 = Scheme N x, N z Time/Time P # non zeros L2-error AP-L 256, AP 256, P 256, AP-L 1024, AP 1024, P 1024, Time reduced compared to the AP scheme. 42 A. Crestetto AP-hybrid models for plasmas

43 Conclusions Anisotropic equation for the electrical potential Coupling strategy 2D AP-L coupling. AP property. Computational time reduced compared to the AP formulation! Perspectives Mathematical analysis. AP-L coupling in 3D. 43 A. Crestetto AP-hybrid models for plasmas

44 Anisotropic equation for the electrical potential Coupling strategy - 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). - C. Besse, F. Deluzet, C. Negulescu, C. Yang: Efficient numerical methods for strongly anisotropic elliptic equations, Journal of Scientific Computing (2012). - 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). - P. Degond, F. Deluzet, C. Negulescu: An Asymptotic Preserving scheme for strongly anisotropic elliptic problem, SIAM Multiscale Modeling and Simulation 8, pp (2010). - 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). Thank you for your attention! 44 A. Crestetto AP-hybrid models for plasmas