Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles

Transcription

1 Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3. Workshop Asymptotic-Preserving schemes, Porquerolles. May 23rd, INRIA Nancy - Grand Est & University of Strasbourg, IRMA. 2 INRIA Rennes - Bretagne Atlantique & ENS Cachan Bretagne, IRMAR. 3 University of Rennes I & ENS Cachan Bretagne, IRMAR.

2 Outline 1 2 3

3 Numerical simulation of plasmas Different scales (Knudsen number ε) = different models. Kinetic model: particles represented by a distribution function f (x, v, t), 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, (E electric field, B magnetic field, x position, v velocity, t time, q charge of particles, m masse of particles) in 3D = 7 variables: 3 in space, 3 in velocity and the time = heavy computations.

4 Fluid model: moment equations on physical quantities linked to f (density, mean velocity, temperature, etc.), smaller cost, but less precision. If two scales in the same simulation: two schemes with an interface? Difficult to deal with the interface! 1st obj. Construct an accurate but not too costly scheme. 2nd obj. Construct an asymptotic-preserving 4 (AP) scheme. 4 S. Jin, SIAM JSC 1999.

5 AP scheme Vlasov ε h Discretized Vlasov h,ε ε ε Limit h Discretized limit h Prop: Stability and consistency ε, particularly when ε. Aim: Standard schemes: h = O(ε) construct a scheme for which h is independent of ε.

6 Vlasov-BGK Vlasov with collisions = Vlasov-BGK equation: t f +v x f +E v f = 1 ε Q(f) coupled to Poisson equation: x E (x,t) = ρ(x,t) 1 = f (x,v,t)dv 1. (ε Knudsen number, ρ = ρ(x,t) charge density.) Q(f) BGK (Bhatnagar-Gross-Krook) collision operator: Q(f) = M(U) f, M(U) being the Maxwellian having the same first three moments than f, denoted by U.

7 ) M(U) = ρ 2πT exp ( v u 2 2T. (T = T (x,t) temperature, u = u(x,t) mean velocity.) U = 1 v v 2 2 = U = f dv = ρ ρu 1 2 (ρu2 +ρt). 1 v v 2 2 M(U) dv Remarks: M(U) N (L Q ) = Span { M,vM, v 2 M }, the null space of the linearized operator L Q of Q, f M(U) when ε : collisions move f closer to its Maxwellian.

8 Micro-macro model PIC method Projection of weights Finite volumes scheme AP property 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 +Tg = 1 ε g. Hypothesis: first three moments of g must be zero = mf := m(v)f (x,v,t) dv = U(x,t), ( where m(v) = 1,v, v2 2 ) t. True at the numerical level? If not, we have to impose it. 5 M. Lemou, L. Mieussens, SIAM JSC N. Crouseilles, M. Lemou, KRM 21.

9 Micro-macro equations PIC method Projection of weights Finite volumes scheme AP property Let Π M the orthogonal projection in L 2 (M 1 dv) onto N (L Q ) 7 : [ Π M (ϕ) = 1 (v u) (v u)ϕ ρ ϕ + T ( v u ) ( ) ] v u 2 1 ϕ M. 2T 2 T Properties: (I Π M )( t M) = Π M (g) = Π M ( t g) =. Applying (I Π M ) to Vlasov-BGK = micro equation on g t g +(I Π M )T (M +g) = 1 ε g. 7 M. Bennoune, M. Lemou, L. Mieussens, JCP 28.

10 PIC method Projection of weights Finite volumes scheme AP property Applying Π M to Vlasov-BGK = macro equation on M(U) t M +Π M T (M +g) = or by taking his first three moments F (U) = S (U) = t U + x F(U)+ x vm(v)g = S(U), ρu ρu 2 +ρt ρu ( u T) ρe ρue : source term. : Euler flux

11 Algorithm PIC method Projection of weights Finite volumes scheme AP property System { 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. 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 P. Degond, G. Dimarco, L. Pareschi, IJNMF, S. Brunner, E. Valeo, J.A. Krommes, Phys. of Plasmas, 1999

12 PIC method PIC method Projection of weights Finite volumes scheme AP property Equation t g +(I Π M )T (M +g) = 1 ε g t g +T g = (I Π M )TM +Π M T g g ε =: S g. Model: having N p 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

13 PIC method Projection of weights Finite volumes scheme AP property Mesh generation Fields computing on the mesh Initialization of positions and velocities of particles Computation of charge and current densities on the mesh (deposition) Emission of new particles Movement of particles Interpolation of fields on the particles

14 Shape functions PIC method Projection of weights Finite volumes scheme AP property Use of shape functions such as B-spline of order l B l (x) = (B B l 1 )(x), with In particular B (x) = { 1 x if x < x/2, else. B 1 (x) = 1 x Order : Nearest Grid Point. Order 1: Cloud In Cell. { 1 x / x, if x < x, else.

15 Deposition and interpolation PIC method Projection of weights Finite volumes scheme AP property Computation of the moment M of order p on the cell i: M p,i (t) = M p (x,t)b l (x i x) dx R = 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 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

16 PIC method Projection of weights Finite volumes scheme AP property 1. Initialization: particles randomly (or quasi) distributed in phase space (x, v), weights initialized to ω k () = g (x k,v k,) LxLv N p. (L x x-length of the domain, L v v-length.) 2. Solving the electric field on the mesh: x E (x,t) = ρ(x,t) 1 = f (x,v,t)dv 1 thanks to the knowledge of ρ i (t), and then interpolation on the particles E (x k,t).

17 Solving t g +T g = PIC method Projection of weights Finite volumes scheme AP property 3. Movement of particles thanks to motion equations: dx k dt (t) = v k (t) et Verlet scheme (for example): v n+1 2 k = v n k + t 2 En (x n k ) dv k dt (t) = E (x k,t). x n+1 k = xk n + tvn+1 2 k v n+1 k = v n+1 2 k + t 2 En+1( x n+1 ) k.

18 PIC method Projection of weights Finite volumes scheme AP property Solving t g = S g 4. Evolution of weights ω k (step specific to kinetic equations with source term): dω k (t) = s k (t), dt where s k (t) = S g (x k,v k ) L xl v N p is the weight associated to the source term S g = (I Π M )T M +Π M Tg g ε. In our case: dω k (t) = α k (t) ω k (t), dt ε where α k (t) is associated to (I Π M )T M +Π M Tg.

19 PIC method Projection of weights Finite volumes scheme AP property (I Π M )T M +Π M Tg is constituted of moments of g, their derivatives and M. Moments of g are computed by deposition, derived by finite differences and evaluated on (x k,v k ) by interpolation. ρ, u and T are evaluated on (x k,v k ) by interpolation. ) M = ρ exp is now easily evaluated on (x 2πT k,v k ). ( v u 2 2T To have an AP scheme, we make the stiff term ω k(t) ε ω n+1 k = ωn k + tαn k 1+ t. ε implicit:

20 Projection step PIC method Projection of weights Finite volumes scheme AP property We now have N p g n+1 (x,v,t) k=1 We want to ensure mg n+1 =. ω n+1 k δ ( x x n+1 k We compute U g := mg n+1, N p ) ( ) δ v v n+1 k. U g (x i ) = mg n+1 xi = ω k m(v k )B l (x i x k ). k=1

21 PIC method Projection of weights Finite volumes scheme AP property We seek h N (L Q ) of the form h(x,v) = λ(x) m(v)m(x,v) s.t. U g (x i ) = mh(x i,v). 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 3. j=1 For example l = (l = 1 is also computed). Let p k the weights associated to M, we compute mh x i 1 x 2 m(v k ) m(v k )p k λ i. k/ x k x i x/2

22 PIC method Projection of weights Finite volumes scheme AP property We solve the 3 3 linear system U g (x i ) = A i λ i, with A i = 1 x M,i M 1,i M 2,i M 1,i M 2,i M 3,i M 2,i M 3,i M 4,i and M j,i = 1 x k/ x k x i x/2 p kv j k. We compute the weights γ k of h, N x γ k = λ j B (x k x j ) m(v k )p k = 1 x λ j k m(v k )p k, j=1 where j k is such that x k x jk < x/2.

23 PIC method Projection of weights Finite volumes scheme AP property We correct the weights of g to obtain = 1 x k/ x k x i x/2 ω new k ω k γ k, mg n+1,new xi = 1 x ω k m(v k ) 1 x Remark: order 1 = (3N x 3N x ) system. k/ x k x i x/2 k/ x k x i x/2 ω new k m(v k ) γ k m(v k ) =.

24 Macro part PIC method Projection of weights Finite volumes scheme AP property Equation t U + x F(U) = S(U,g) (= S(U) x vm(v)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. S n i = S (U n i ) ( vmg n+1 xi+1/2 vmg n+1 xi 1/2 x where vmg n+1 xi+1/2 = 1 x k/ x k x i+1/2 x/2 ωn+1 k v k m(v k ). )

25 Analytical limit PIC method Projection of weights Finite volumes scheme AP property t g +(I Π M )T (M +g) = 1 g, ε (1) t U + x F(U)+ x vm(v)g = S(U). (2) Limit ε in (1): g = O(ε) = g = ε(i Π M )T M +O(ε 2 ), = ε(i Π M )(v x M +E v M)+O(ε 2 ), = ε(i Π M )(v x M)+O(ε 2 ), because E v M N (L Q ) and then (I Π M )(E v M) =. Injected into (2), we obtain the Navier-Stokes model t U+ x F(U) = S(U)+ε x vm(v)(i Π M )(v x M) +O(ε 2 ).

26 Numerical limit PIC method Projection of weights Finite volumes scheme AP property (1) can be discretized as ω n+1 k t ω n k ω n+1 k = = α n 1,k +αn 2,k ωn+1 kε, 1 ( ω n 1+ t/ε k tα n ) 1,k + tαn 2,k. 1 But 1+ t/ε = ε t +O(ε2 ) when ε, ωk n = O(ε) thus α n 2,k ω n+1 k = εα n 1,k +O(ε2 ). = O(ε) and We obtain the diffusion term of the Navier-Stokes model = AP property. g n+1 = ε(i Π M )T M +O(ε 2 )

27 Landau damping First test case Second test case Computational cost Initial distribution function: f (x,v,) = 1 exp( v2 )(1+αcos(kx)), x [,2π/k] 2π 2 Micro-macro initializations: 1+αcos(kx) U(x) = 1+αcos(kx) and g(x,v,t = ) =. Parameters: α =.1, k =.5. Electrical energy E(t) = E(t,x) 2 dx.

28 First test case Second test case Computational cost Convergence at the ε limit, N x = 128, N p = : log E(t) L Euler -1 MiMa, eps=.1 MiMa, eps=.1 MiMa, eps=.3 MiMa, eps= Time t

29 First test case Second test case Computational cost Heat flux of MiMa (blue) and Navier-Stokes (red) at t = 1 with N p = 1 5 for ε =.1 (left),.1 (middle) and.1 (right): eps=.1 eps=.1 eps= NS MiMa.5.4 NS MiMa.5.4 NS MiMa Heat flux Heat flux Heat flux Position x Position x Position x where heat fluxes correspond to: (3/2)ρT x T for NS and 1 ε v u 3 (M +g) for MiMa.

30 First test case Second test case Computational cost Distribution function f of PIC BGK, MiMa, g and M, at t = 2, ε = 1, N p = : f - PIC BGK f - MiMa Position x Velocity v Position x Velocity v g - MiMa M - MiMa Position x Velocity v Position x Velocity v

31 First test case Second test case Computational cost Noise reduction on ρ, ε = 1, N x = 128, N p = 5 1 5, at t =.2 (left) and t =.4 (right):.15 PIC BGK MiMa.2 PIC BGK MiMa Charge density 1-rho -.5 Charge density 1-rho Position x Position x

32 First test case Second test case Computational cost Necessary number of particles, ε = 1, N x = 128: log E(t) L PIC BGK, Npart=2 5-2 PIC BGK, Npart= 5 PIC BGK, Npart= 1 MiMa, Npart= Time t

33 First test case Second test case Computational cost Convergence in particles, ε = 1 4, N x = 128: -2 MiMa, Npart= 5 MiMa, Npart= log E(t) L Time t

34 First test case Second test case Computational cost Importance of the projection step on ρ at t = 5, ε = 1, N p = :.6 Without correction Correction of order Correction of order 1.4 Charge density 1-rho Position x

35 Test 2 First test case Second test case Computational cost Initial distribution function: f (x,v,) = v4 exp( v2 )(1+αcos(kx)), x [,2π/k] 2π 2 Micro-macro initializations: U(x) = 1+αcos(kx) 5(1+αcos(kx)) g (x,v) = 1+αcos(kx) ( v 4 2π 3 exp( v2 /2) 1 ) exp( v 2 /1). 5 Parameters: α =.5, k =.5.

36 First test case Second test case Computational cost Convergence at the ε limit, N x = 128, N p = : log E(t) L Euler MiMa, eps=.1-1 MiMa, eps=.1 MiMa, eps=.5 MiMa, eps= Time t

37 First test case Second test case Computational cost L1-norm of g, ε = 1 4 (left) and ε = 1 7 (right): 1 eps=1-4 1 eps= g(t) L1 1e-6 g(t) L1 1e-6 1e-8 1e-8 1e-1 1e-1 1e-12 1e Time t Time t

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

39 Conclusions First test case Second test case Computational cost Fluid limit recovered when ε : AP scheme. Projection step to numerically force the moments of g to zero. g when ε = 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 = calculation time is reduced. Total energy preserved (about.1% ε). And perspectives: Other collision operators. 2D (4D in phase space).

40 First test case Second test case Computational cost References - 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 (28). - S. Brunner, E. Valeo, J.A. Krommes: Collisional delta-f scheme with evolving background for transport time scale simulations, Phys. of Plasmas 12, (1999). - 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 (211). - P. Degond, G. Dimarco, L. Pareschi: The moment guided Monte Carlo method, International Journal for Numerical Methods in Fluids 67, pp (211). - S. Jin: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, 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, J. Sci. Comp. 31, pp (28). Thank you for your attention!