Adaptive simulation of Vlasov equations in arbitrary dimension using interpolatory hierarchical bases

Transcription

1 Adaptive simulation of Vlasov equations in arbitrary dimension using interpolatory hierarchical bases Erwan Deriaz Fusion Plasma Team Institut Jean Lamour (Nancy), CNRS / Université de Lorraine VLASIX ANR Project NUMKIN Strasbourg October 20th 2016

2 Introduction Vlasov-Poisson equations Distribution function f : R R 3 R 3! R + t, x, v 7! f (t, x, v) tf + v r xf + E(t, x) r v f = 0 E(t, x) = r x (t, x) with x (t, x) =±(4 G) (t, x) +: gravitational, -: in plasmas Z (t, x) = f (t, x, v) dv R 3 Troubles with the six-dimensional simulations: (cf [Yoshikawa 13]) In uniform grids, few points in one direction: e.g. N = 10 8 points, 6p N 20 very few structures stored in 20 points in one dimension, dissipation of small scales when they are transported.

3 Introduction Adaptive Mesh Refinement Principle: Embedded meshes with local resolutions adapted to the solution. 3D mesh generated by Gerris [Popinet 03]

4 Adaptive numerical scheme Interpolant hierarchical basis Interpolant functions Refinement filter: the scaling function satisfies a self-similarity relation x ' = X a k '(x k) 2 k2z with P k2z a k = 2, & for Interpolet a 0 = 1and8k 6= 0, a 2k = 0 '( x 2 ) '( x 2 1) a 2a 1 a 0 a 1 a 2 '(x + 2) '(x + 1) '(x) a 2 a 1 a 0 a 1 a 2 '(x 1) '(x 3) '(x 2) '(x 4)

5 Adaptive numerical scheme Interpolant hierarchical basis Interpolant functions Forth order interpolant function [Deslauriers-Dubuc 89]

6 Adaptive numerical scheme Interpolant hierarchical basis Interpolant functions Forth order interpolant function [Deslauriers-Dubuc 89]

7 Adaptive numerical scheme Interpolant hierarchical basis Interpolant functions Forth order interpolant function [Deslauriers-Dubuc 89]

8 Adaptive numerical scheme Interpolant hierarchical basis Hierarchical basis Forth order interpolant function [Deslauriers-Dubuc 89]

9 Adaptive numerical scheme The adaptive grid Point activation

10 Adaptive numerical scheme The adaptive grid Point activation

11 Adaptive numerical scheme The adaptive grid Point activation

12 Adaptive numerical scheme The adaptive grid Point activation

13 Adaptive numerical scheme The adaptive grid Point activation

14 Adaptive numerical scheme The adaptive grid Point activation

15 Adaptive numerical scheme Refinement criterion Non-linear approximation theorem We consider an expansion in the hierarchical basis in dimension d with a L 2 normalization: f (x) = X X X j,k,"2 dj/2 ' " (2 j x k) j2z then Theorem (Cohen-DeVore 01) k2z d "2{0,1} d \(0,0) If f 2 B s q(l p ), we have the wavelet estimator: kf k B s q (L p ) 2 sj 2 dj( p ) k( j,k," ) k2z d k`p. j 0 `q Then minimizing the Bp(L s p )-norm of the error (p 2 [1, +1]) leadsto discarding the coefficients j,k," such that 2 (s+ d d 2 )j p j,k," <, i.e. j,k," < 2 ( s d 2 + d p )j.

16 Numerical experiments One-dimensional electrical cases Landau damping for (x, v) 2 [ 2, 2 ] [ 6, 6], t 2 [0, 100] f 0(x, v) = cos(0.5x) p 2 exp( v 2 /2)

17 Numerical experiments One-dimensional electrical cases Landau damping L2 norm of the Electric field Time

18 Numerical experiments One-dimensional electrical cases Two streams instability for (x, v) 2 [ 2, 2 ] [ 6, 6], t 2 [0, 100] f 0(x, v) = cos(0.5x) p 2 v 2 exp( v 2 /2)

19 Numerical experiments One-dimensional electrical cases Two streams instability Number of points Time

20 Numerical experiments One-dimensional electrical cases Two streams instability Entropy S Mass Relative Value Total Energy L2 norm Time

21 Numerical experiments One-dimensional electrical cases Two streams instability Electric Potential Energy Total Energy Energy Shifted Kinetic Energy Time

22 Numerical experiments Two-dimensional case Two beams instability Fijalkow experiment [JPP 1999], for (x, y) 2 [ (u, v) 2 [ 9.43, 9.43], t 2 [0, 90] f 0(x, y, u, v) = 7 ( cos(0.3x)) exp( 4 (u2 10 3, 10 3 ],and /4 + v 2 )/2) sin 2 (u/3); projection on (x, u)

23 Numerical experiments Two-dimensional case Fijalkow Two Beams instability cut along (x, u) at y = 0, v = 0

24 Numerical experiments Two-dimensional case Fijalkow Two Beams instability 7e+07 6e+07 Number of points 5e+07 4e+07 3e+07 Number of points 2e+07 Threshold 1e+07 0e Time

25 Numerical experiments Two-dimensional case Fijalkow Two Beams instability 1.01 Entropy Mass 0.97 L2 norm Total Energy Conservations and Energy

26 Numerical experiments Two-dimensional case Fijalkow Two Beams instability Total Energy 4450 Energy Kinetic Energy Shifted Potential Energy Time Conservations and Energy

27 Numerical experiments Thee-dimensional astrophysics Merging of two halos: 3D-3V, 3D-view Plummer Model [Fujiwara 83] [Yoshikawa 13]

28 Numerical experiments Thee-dimensional astrophysics Merging of two halos projection on (z, w)

29 Numerical experiments Thee-dimensional astrophysics Merging of two halos cut along (z, w) at x = y = u = v = 0

30 Numerical experiments Thee-dimensional astrophysics Merging of two halos 4.5e e+08 Number of points 3.5e e e e+08 Number of points Threshold 1.5e e e Time

31 Numerical experiments Thee-dimensional astrophysics Merging of two halos Entropy S Relative value L2 norm Mass M Time Conservations

32 Numerical experiments Thee-dimensional astrophysics Merging of two halos Kinetic Energy Time Kinetic Energy

33 Conclusion and Perspectives Conclusion-Perspectives Finite Differences and Runge-Kutta 4 to be adapted to Vlasov-Poisson with the help of Semi-Lagrangian schemes, Partially hyperbolic bases (cf Sparse Grids [Griebel 91]) to reduce the curse of dimentionality, Switch to a uniform grid in X -space for the electrical Vlasov-Poisson (adaptivity only in velocity).

34 Other numerical technics Discountinuous Galerkin, Semi-Lagrangian and AMR Eric MADAULE Figure: Bump on tail simulation at t = 102. Distribution function & adaptive grid.

35 Dimensionality reduction technics Complete tensorisation: the Tensor-Train [Oseledets, Kormann] In TT-format a function f : R d! R is approximated by f (x 1, x 2,...,x d )= dy M k (x k ) where M k = ' k ij 1appleiappler k 1 for 1 apple k apple d are matrices of functions, r 0 = r d = 1. 1applejappler k max k (r k ) is called the rank of the Tensor-Train. In practice the functions ' k ij : R! R: k=1 are known on a fixed regular grid x1 k, x2 k,...,xn k for a given n, so they are replaced by ' k ij(xi k k ), 1applei k applen and are regularly rearranged in order to minimize (r k ) 1applekappled 1 with the help of QR decompositions.

36 Dimensionality reduction technics Hyperbolic bases: the Sparse Grids From the point of view of wavelets, it corresponds to the hyperolic decomposition: X f (x 1, x 2)=c (0,0) '(x 1)'(x 2)+ d (1,0) jk (2 j x 1 k)'(x 2) + X j 0,0applekapple2 j 1 d (0,1) jk '(x 1) (2 j x 2 k)+ j 0,0applekapple2 j 1 X j i 0,0applek i apple2 j i 1 Assuming a second order scaling function ', we can estimate d (1,1) jk (2 j 1 x 1 k 1) (2 j 2 x 2 k 2) c f, d (1,0) jk K 22 2f 2, d (1,0) jk K 22 1f 2, d (1,1) jk K (j 1+j 2 1@ 2 2f 2 with K In particular, for the coefficients d (1,1) jk, if we want to retain the ones such that d (1,1) jk " then it provides a J such that j 1 + j 2 apple J.

37 Dimensionality reduction technics Hyperbolic bases: the Sparse Grids It is particularly efficient for approximating smooth functions in high dimensionality. ' (2 ) (4 ) J=1 (8 ) ' (2 ) (4 ) (8 )