Multi-water-bag model and method of moments for Vlasov

Transcription

1 and method of moments for the Vlasov equation 1, Philippe Helluy 2, Nicolas Besse 3 FVCA 2011, Praha, June 6. 1 INRIA Nancy - Grand Est & University of Strasbourg - IRMA crestetto@math.unistra.fr (PhD thesis supported by DGA) 2 University of Strasbourg - IRMA 3 University Henri Poincaré Nancy I - IECN & IJL and method of moments for Vlasov

2 Summary: 1 Introduction and method of moments for Vlasov

3 Vlasov-Poisson 1D system: t f (x, v, t) + v x f (x, v, t) + q m E (x, t) v f (x, v, t) = 0 x E (x, t) = q m where n 0 is the constant density of ions. ( f (x, v, t) dv n0 (x) ), Difficulties: - in 3D = 7 variables: 3 positions, 3 velocities and time = very heavy computations, - no dissipation = the solution may become turbulent (filaments, vortices, etc.). Obectives: - reduce the dimension of the problem, - keep enough information about the solution. and method of moments for Vlasov

4 We consider f piecewise constant in v: f (x, v, t) = f (x, v, t) = k A, for v [ v k (x, t), v + k (x, t)] =1 N ( A (H =1 v + (x, t) v ) ( H where H is the Heaviside function: { 0 if u < 0, H (u) = 1 if u > 0, v )) (x, t) v, N is an integer, v ± (x, t) are velocities and A are constants, for = 1,..., N. and method of moments for Vlasov

5 3 pairs of bags model in phase space (left) and corresponding distribution function (right). Idea of the model: evolve the v ± (x, t) instead of f. Resolution of transport equations for the v ± by a finite volume method (Godunov for example): t v ± + v ± x v ± q E = 0, = 1,..., N. m and method of moments for Vlasov

6 Definition: moment of order k, in v, of a function f integrable in v: M k (x, t) = + v k f (x, v, t) dv. Moments system (by taking the 2N first moments of the Vlasov equation): t M 0 + x M 1 = 0 t M k + x M k+1 k q m EM k 1 = 0, for k = 1,..., 2N 1 t E (x, t) = q m (M 1 (0, t) M 1 (x, t)). and method of moments for Vlasov

7 Remarks: - v no longer involved = one dimension less, - N can be chosen according to the expected precision and the computation capabilities, - unknown: M 0,..., M 2N 1, and M 2N = need of a closure relation. Closure relation: we have to choose a representation of f, for example by the multi-water-bag model. Moments are then written: M k (x, t) = N =1 v +k+1 A = expression of M 2N as a function of v ±. (x, t) v k+1 (x, t), k = 0,..., 2N k + 1 and method of moments for Vlasov

8 Algorithm Introduction Moments, E and v ± (x, t) known at time n. Computation of moments and E at time n + 1 with a kinetic scheme (natural expression of fluxes): - upwind discretization for the Vlasov equation: f n+1 i fi n + 1 ( v + ( fi n f n ) i 1 + v ( fi+1 n f n )) i E n t x i v fi n = 0, - multiplication by v k and integration in v: 1 v k ( f n+1 i f n ) i dv t + 1 ( ( v +k+1 f n i f n ) i 1 + v k+1 ( fi+1 n f n ) ) i dv x E n i v k v f n i dv = 0, and method of moments for Vlasov

9 - finite volume scheme: M n+1 k,i M n k,i t - expression of the flux: + 1 ( F n x k,i Fk,i 1) n ke n i Mk 1,i n = 0, Fk,i n = F ( Mk,i n, ) N ( Mn k,i+1 = A + =1 ( N A =1 v ++ v + where u + = max (u, 0) and u = min (u, 0), v +k+1 v k+1 k + 2 v + k + 2 v +k+1 v k+1 k + 2 v k + 2 ) n i ) n i+1, and method of moments for Vlasov

10 - scheme for the electric field: E n+1 i t E n i = M n 1,i M n 1,0. Computation of v ± (x, t) at time n + 1 by: - the Newton-Raphson method, iterative and hard to initialize, - the algorithm of Gosse and Runborg when A = ±1, unstable when two bags are close to each other. and method of moments for Vlasov

11 Validation Introduction Validation of the method on Landau damping and two-stream instability, by comparing it to the Particle In Cell (PIC) method. Initial condition of the Landau damping: 1 f (x, v, t = 0) = (1 + ɛ cos (kx)) e v2 2 2π E (x, t = 0) = q m ɛ sin (kx). k Initial condition of the two-stream instability: ) 1 f (x, v, t = 0) = (1 + ɛ cos (kx)) (e 2 (v v 0) e (v+v 0) 2 2 2π E (x, t = 0) = q ɛ sin (kx). m k and method of moments for Vlasov

12 Landau damping Landau damping: k=0.5, ɛ = Electrical energy (logarithmic scale) as a function of time. and method of moments for Vlasov

13 Studied test cases Initial conditions such that: - f (x, v, t = 0) is exactly described by one or two pairs of bags, - the solution destabilizes: creation of filaments, vortices, etc. Examples at initial time: and method of moments for Vlasov

14 First test case: 1 pair of bags Phase space (v as a function of x) at times 0 and 2: particles (PIC method), v + 1 and v 1 (water-bag). and method of moments for Vlasov

15 Phase space (v as a function of x) at times 5 and 10: particles (PIC method), v + 1 and v 1 (water-bag). and method of moments for Vlasov

16 Phase space (v as a function of x) at times 20 and 50: particles (PIC method), v + 1 and v 1 (water-bag). and method of moments for Vlasov

17 Second test case: 2 pairs of bags Phase space (v as a function of x) at times 0 and 2: particles (PIC method), v + 1, v 1, v + 2 and v 2 (water-bag). and method of moments for Vlasov

18 Phase space (v as a function of x) at times 3 and 4: particles (PIC method), v + 1, v 1, v + 2 and v 2 (water-bag). and method of moments for Vlasov

19 Phase space (v as a function of x) at times 5 and 10: particles (PIC method), v + 1, v 1, v + 2 and v 2 (water-bag). and method of moments for Vlasov

20 Conclusions: - monovalued solution correctly described, - v ± (x, t) cannot be multivalued = loss of information when filaments appear, - used algorithm unstable when at least two bags are close to each other (ill-conditioned matrix). Work in progress in order to improve the representation of the filaments: - stabilize the algorithm of Gosse and Runborg when several bags are close to each other, - first idea: merge these bags. and method of moments for Vlasov

21 Phase space (v as a function of x) at times 0 and 0.1: particles (PIC method), v + 1, v 1, v + 2, v 2, v + 3 and v 3 (water-bag). and method of moments for Vlasov

22 Phase space (v as a function of x) at time 0.4: particles (PIC method), v + 1, v 1, v + 2, v 2, v + 3 and v 3 (water-bag). and method of moments for Vlasov

23 Phase space (v as a function of x) at time 0.4: - left: f obtained by the PIC method (particles), - right: f reconstructed by the method of moments/multi-water-bag. and method of moments for Vlasov

24 Perspectives: - generalize the algorithm of Gosse and Runborg when the A are different of ±1, - couple our method to a PIC method: create particles when filaments appear. Thank you for your attention! and method of moments for Vlasov

25 References Introduction - N. Besse, F. Berthelin, Y. Brenier, P. Bertrand: The multi-water-bag equations for collisionless kinetic modeling, Kinetic and Related Models 2, (2009) - R.O. Fox, F. Laurent, M. Massot: Numerical simulation of spray coalescence in an Eulerian framework: Direct quadrature method of moments and multi-fluid method, Journal of Computational Physics 227, (2008) Algorithm of Gosse and Runborg - Y. Brenier, L. Corrias: A kinetic formulation for multi-branch entropy solutions of scalar conservation laws, Annales de l Institut Henri Poincaré. Analyse Non Linéaire 15, (1998) - L. Gosse, O. Runborg: Resolution of the finite Markov moment problem, Comptes Rendus Mathématique. Académie des Sciences. Paris 12, (2005) and method of moments for Vlasov