High Order Semi-Lagrangian WENO scheme for Vlasov Equations

Transcription

1 High Order WENO scheme for Equations Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Andrew Christlieb Supported by AFOSR. Computational Mathematics Seminar, UC Boulder 1 / 39

2 Outline -Poisson system Typical numerical approach The proposed semi- (SL) WENO method High order accuracy for smooth structure and non-oscillatory resolution for sharp interface Mass conservation Allows for large time step results 2 / 39

3 VP system The collisionless plasma may be described by the well-known -Poisson (VP) system, f t + v xf + E(t, x) v f = 0, (1) E(t, x) = x φ(t, x), x φ(t, x) = ρ(t, x). (2) f (t, x, v): the probability of finding a particle with velocity v at position x at time t. ρ(t, x) = f (t, x, v)dv - 1: charge density 3 / 39

4 Numerical approach: vs. Eulerian : tracking a finite number of macro-particles. e.g., PIC (Particle In Cell) dx dt = v, dv dt = E (3) Eulerian: fixed numerical mesh e.g., finite difference WENO, semi-, finite volume, finite element, spectral method. 4 / 39

5 Numerical Challenge : suffers from the statistical noise (O(1/ N)). Eulerian: suffers the curse of dimensionality for a multi-d (up to 6) problem. Our approach: A very high order semi- approach with relatively coarse numerical mesh in both space and time. 5 / 39

6 The SL WENO method Key Component Strang splitting Conservative form of semi- (SL) scheme WENO reconstruction. 6 / 39

7 Strang splitting f t + v xf + E(t, x) v f = 0, Nonlinear eqation a sequence of linear s. 1-D in x and 1-D in v: f t + v f x = 0, (4) f t + E(t, x) f v = 0. (5) 7 / 39

8 Strang Splitting for solving the 1 Evolve (4) for t 2, 2 Compute E from the Poisson s, 3 Evolve (5) for t, 4 Evolve (4) for t 2. 8 / 39

9 Linear transport 1 Shifting of solution profile f t + v f x = 0, 2 Mass conservation (if the boundary condition is periodic) d f (x, t)dx dt = 0 9 / 39

10 SL method for linear transport f t + v f x = 0, Numerical scheme: evolution solution from t n to t n+1 1 reconstruction: determine accuracy quality of the scheme. 2 shifting: by v t Let ξ 0 = v t x, f n+1 i = f (x i, t n+1 ) = f (x i ξ 0 x, t n ) 10 / 39

11 Third order example When ξ 0 [0, 1 2 ] t n+1 t n x 0 x i 2 x i 1 x i x i+1 x i+2 x N ξ 0 [0, 1 2 ] 11 / 39

12 Third order example When ξ 0 [0, 1 2 ] When ξ 0 [ 1 2, 0] t n+1 t n x 0 x i 2 x i 1 x i x i+1 x i+2 x N ξ 0 [0, 1 2 ] ξ 0 [ 1 2, 0] 11 / 39

13 When ξ 0 [0, 1 2 ] f n+1 i = fi n + ( 1 6 f i 2 n + fi 1 n 1 2 f i n 1 3 f i+1)ξ n 0 + ( 1 2 f i 1 n fi n f i+1)ξ n ( 1 6 f i 2 n 1 2 f i 1 n f i n 1 6 f i+1)ξ n 0, 3 (6) 12 / 39

14 Matrix Vector From f n+1 i with matrix A L 3 = = = f n i + ξ 0 (f n i 2, f n i 1, f n i, f n i+1) A L 3 (1, ξ 0, ξ 2 0), (7) 1/6 0 1/6 1 1/2 1/2 1/2 1 1/2 1/3 1/2 1/6 1/6 0 1/6 5/6 1/2 1/3 1/3 1/2 1/ /6 0 1/6 5/6 1/2 1/3 1/3 1/2 1/6. 13 / 39

15 Conservative Form f n+1 i = fi n ξ 0 ((fi 1, n fi n, fi+1) n C3 L (1, ξ 0, ξ0) 2 (fi 2, n fi 1, n fi n ) C3 L (1, ξ 0, ξ0) 2 ) with C3 L = / 39

16 Conservative Form (cont.) Numerical flux: f n+1 i = f n i ξ 0 (ˆf n i+1/2 ˆf n i 1/2 ) ˆf n i+1/2 = (f n i 1, f n i, f n i+1) C L 3 (1, ξ 0, ξ 2 0) = (fi 1, n fi n, fi+1) n C3 L (:, 1) +(fi 1, n fi n, fi+1) n C3 L (:, 2)ξ 0 +(fi 1, n fi n, fi+1) n C3 L (:, 3)ξ / 39

17 WENO reconstructions on fluxes Apply WENO reconstruction on (f n i 1, f n i, f n i+1) C L 3 (:, 1) = 1 6 f i 1 n f i n f i+1 n = γ 1 ( 1 2 f i 1 n f i n ) + γ 2 ( 1 2 f i n f i+1) n with linear weights γ 1 = 1 3 and γ 2 = / 39

18 WENO reconstructions (cont.) Idea of WENO: adjust the linear weighting γ i to a nonlinear weighting w i, such that w i is very close to γ i, in the region of smooth structures, w i weights little on a non-smooth sub-stencil. 17 / 39

19 Nonlinear weights w i Smoothness indicator β i : β 1 = (f n i 1 f n i ) 2, Nonlinear weights w i : β 2 = (f n i f n i+1) 2. w 1 = γ 1 /(ɛ + β 1 ) 2, w 2 = γ 2 /(ɛ + β 2 ) 2 Normalized nonlinear weights w i : w 1 = w 1 /( w 1 + w 2 ), w 2 = w 2 /( w 1 + w 2 ) 18 / 39

20 of algorithm 1 Compute the numerical fluxes ˆf n i+1/2 compute the nonlinear weights w 1, w 2. compute numerical fluxes ˆf i+1/2 n = w 1 ( 1 2 f i 1 n f i n ) + w 2 ( 1 2 f i n f i+1) n +(fi 1, n fi n, fi+1) n C3 L (:, 2)ξ 0 +(fi 1, n fi n, fi+1) n C3 L (:, 3)ξ0 2 2 Update the solution f n+1 i f n+1 i = f n i ξ 0 (ˆf n i+1/2 ˆf n i 1/2 ) 19 / 39

21 Mass Conservation The scheme conserves the total mass, if the periodic boundary conditions are imposed. Proof. i f n+1 i = i = i = i ( ) fi n ξ 0 (ˆf n ˆf n ) i+ 1 i ξ 0 (ˆf n f n i f n i. i i+ 1 2 ˆf n ) i / 39

22 Comparison Compare with the SL method compare with the existing scheme that the integrated mass: very high spatial accuracy in multi-dimensional problem. compare with the existing scheme that the point values: mass conservation. Compare with other Eulerian approaches allows for extra large numerical time step integration, commits no error in time for one dimensional problem, but is subject to the splitting error O( t 2 ). 21 / 39

23 Extensions The SL WENO scheme has been extended to 5 th, 7 th, 9 th order in accuracy. The scheme is planned to extended to solve the -Maxwell system. The scheme is planned to couple with the adaptive mesh refinement (AMR) framework to realize spatial adaptivity. 22 / 39

24 Numerical s 23 / 39

25 u t + u x = 0 Table: Order of accuracy with u(x, t = 0) = sin(x) at T = 20. CFL = rd order 5th order 9th order mesh error order error order error order E E E E E E E E E E E E E E E E E E / 39

26 u t + u x = 0 Figure: 320 grid points, CFL = 0.6, T = / 39

27 Linear transport: u t + u x + u y = 0 Table: Order of accuracy with u(x, y, t = 0) = sin(x + y) at T = 20. CFL = rd order 5th order 9th order mesh error order error order error order E E E E E E E E E E E E E / 39

28 Linear transport: u t + u x + u y = 0 Figure: The numerical mesh is CFL = 0.6 at T = / 39

29 Rigid body rotation: u t yu x + xu y = 0 Figure: The numerical mesh is CFL = 1.2 at T = 2π. 28 / 39

30 Landau damping Consider the VP system with initial condition, f (x, v, t = 0) = 1 (1 + αcos(kx))exp( v 2 ), (8) 2π 2 29 / 39

31 Weak Landau damping: α = 0.01, k = 2 Figure: Time evolution of L 2 norm of electric field. 30 / 39

32 Weak Landau damping (cont.) Figure: Time evolution of L 1 (upper left) and L 2 (upper right) norm, discrete kinetic energy (lower left), entropy (lower right). 31 / 39

33 Strong Landau damping: α = 0.5, k = 2 Figure: Time evolution of the L 2 norm of the electric field using third, fifth, seventh and ninth order reconstruction. 32 / 39

34 Strong Landau damping (cont.) Figure: Phase space plots of strong Landau damping at T = 30 using third, fifth, seventh and ninth order reconstruction. The numerical mesh is / 39

35 Two-stream instability Consider the symmetric two stream instability, the VP system with initial condition f (x, v, t = 0) = 2 7 2π (1 + 5v 2 )(1 + α((cos(2kx) +cos(3kx))/1.2 + cos(kx))exp( v 2 2 ), with α = 0.01, k = 0.5 and the length of domain in x direction L = 2π k. 34 / 39

36 Figure: Two stream instability T = 53. The numerical mesh is / 39

37 Figure: Third order semi- WENO method. Two stream instability T = 53. The numerical mesh is (left) and (right). The computational cost is 4 and 16 times as the first plot in Figure / 39

38 Figure: Time development of numerical L 2 norm (left) and entropy (right) of two stream instability. 37 / 39

39 Equations: the VP system ology: Strang-split semi- WENO method (3rd, 5th, 7th and 9th order) s: High order schemes is more efficient than low order ones. It better preserves the energy, entropy of the system. 38 / 39

40 THANK YOU! 39 / 39