A particle-in-cell method with adaptive phase-space remapping for kinetic plasmas

Transcription

1 A particle-in-cell method with adaptive phase-space remapping for kinetic plasmas Bei Wang 1 Greg Miller 2 Phil Colella 3 1 Princeton Institute of Computational Science and Engineering Princeton University 2 Department of Chemical Engineering and Materials Science University of California at Davis 3 Applied Numerical Algorithm Group Lawrence Berkeley National Laboratory Mathematical and Computer Science Approaches to High Energy Density Physics Workshop, IPAM May 10, 2011

2 Outline Background Vlasov equation Numerical methods for the Vlasov equation Particle Methods The particle in cell method What is particle noise? Noise Reduction: Phase-space Remapping Conservative, high order and positive remapping Mesh refinement Numerical results

3 The Vlasov-Poisson equation We consider a collisionless electrostatic plasma with two species, electron and ion, described by the Vlasov-Poisson equation Vlasov equation Poisson equation f e t + v xf e + F v f e = 0 ρ = 1 q e R n f e dv F = q e m e E φ = ρ E = φ f e (x, v, t): the electron distribution function in phase space F(x, t): Lorentz force (electrostatic case) E(x, t): the self-consistent electrostatic field

4 Lagrangian description of the system The distribution function f is conserved along the characteristics. At time s and t f (X(t), V(t), t) = f (X(t 0 ), V(t 0 ), t 0 ) where (X(t), V(t)) is the characteristics of the Vlasov equation: dx dt dv dt = V(t) = F(t) where X(t 0 ) = x 0 and V(t 0 ) = v 0

5 Review of numerical methods: grid methods 1. Grid-based methods include spectral methods (Knorr 63, Armstrong 69, Flimas-Farrell 94), semi-lagrangian methods (Cheng-Knorr 76, Sonnendrucker 99, Nakamura-Yabe 99), finite volume methods (Fijalkow 99, Filbet 01, Colella 11) and finite element methods (Zaki 88, Kilimas-Farrell 94). 2. They have drawn much attention in the past decade thanks to increasing processing power. Advantage: Smooth representation of f Disadvantage: High dimensions (up to 6) high computational cost (specifically memory)

6 Review of numerical methods: particle methods Particle methods,e.g.,the PIC method,are widely used and are preferred for high dimensions. Advantages: Naturally adaptive, since particles only occupy spaces where the distribution function is not zero Simpler to implement, in particular in high dimensions Disadvantages: particle noise difficulties to get precise results in some cases, for example, in simulating the problems with large dynamic ranges

7 Particle methods In particle methods, we approximate the distribution function by a collection of finite-size particles f (x, v, t) k q k δ εx (x X k (t))δ εv (v Ṽk(t)) q k = f (x i, v i, 0)h x h v where X k (0) = x i, Ṽk (t) = v i. δ εx (y)dy = u1 u0 δ εx (y) = 1 ε x u( y ε x ) u(y) At each time step, particles are transported along trajectories described by the equation of motion y where X k (0) = x k and Ṽ k (0) = v k. dq k dt d X k dt dṽk dt = 0 = Ṽ k (t) = F k (t)

8 The PIC method Charge assignment: ρ(x j, t n ) = k q k u 1 ( x j X k (t n ) ) ε x ε x where j is the grid index. Field solver: i.e., FFTs or multigrid methods φ j 1 2φ j + φ j+1 ε x 2 = ρ j Force interpolation: E j = φ j 1 φ j+1 2ε x Ẽ( X k, t n ) = j E j u 1 ( x j X k (t n ) ε x )

9 What is particle noise? Particle noise: the numerical error introduced when evaluating the moments of the distribution function using particles in phase space and the particle disorder induced by the numerical error Error analysis: Monte Carlo estimate (Aydemir 93) error σ N where σ is the standard deviation depends on the particle sampling and the distribution function. The approach in vortex methods: consistency error + stability error (Cottet-Raviart 84): error consistency error (exp(at) 1) where a is a physical parameter.

10 Error Analysis of the PIC method for the VP system The charge density error is ρ(x, t) ρ(x, t) = ρ(x, t) q k δ εx (x X k (t)) k ρ(x, t) ρ(y, t)δ εx (x y)dy R }{{} moment error:e m(x,t) ε 2 x + ρ(y, t)δ εx (x y)dy q k δ εx (x X k (t)) R k }{{} discretization error:e d (x,t) ε x 2 ( hx εx )2 + q k δ εx (x X k (t)) q k δ εx (x X k (t)) k k }{{} stability error:e s (x,t) εx 1 max k X k X k ( ( E(x, t) Ẽ(x, t) ε 2 2 hx x + ε x ε x ) 2 + max X k (t) X k (t) k )

11 By following Cottet and Raviart (84): and t (Ṽ k V k )(t) = 0 t = 0 t = 0 t ( X k X k )(t) = (Ṽk V k )(t )dt 0 (Ẽ( X k, t ) E(X k, t )) dt ( (Ẽ E)( X ( k, t ) + E( X )) k, t ) E(X k, t ) dt ( (Ẽ E)( X k, t ) + E ) x ( X k X k )(t ) dt

12 By using a variation on Gronwall s inequality, we get max ( X k(t) X k (t) + V k (t) Ṽ k (t) + E Ẽ(, t) ) k L C 1 εx 2 + ε 2 x ( hx 2 ( ) ) + ε 2 x + ε 2 x ( hx ) 2 (exp(at) 1) ε x ε x }{{}}{{} e c (x,t) e s (x,t) where a is max(1, E x (, t) L ), but not εx and hx. Requirements for Convergence Particle overlapping: hx ε x 1 Particle regularization: control the exponential-like term ref: Cottet-Raviart 84 and Wang,Miller,Colella 11

13 Options if we want to reduce particle noise Perturbative methods, such as the δf method (Kotschenreuther 88, Dimits-Lee 93, Parker-Lee 93): discretize the perturbation with respect to a (local) Maxwellian in velocity space using particles reduce σ ( error σ N ) Our approach: remapping: remap the distorted charge distribution on regularized grid(s) in phase space and then create a new set of particle charges from the grids with regularized distribution reduce exponential term

14 Conservative remapping on phase space (x, v) Particle charges are remapped (interpolated) to a grid in phase space q i = q k u( x k x ix h x )u( v k v iv h v ) u1 u0 u(y) y Total charge and momentum are conserved if the interpolation function u satisfies i Z u( x x i h ) = 1 i Z x iu( x x i h ) = x

15 High order interpolation A modified B-spline by Monaghan (85) 1 5 y y y 1 u 2 (y) = 1 2 (2 y )2 (1 y ) 1 < y 2 0 otherwise u2 0.6 u(y) y u 2 can approximate a quadratic function exactly (error is O(h 3 )). Moreover, its first and second order derivatives are continuous.

16 Positivity for high order interpolation functions Algorithm: Global mass redistribution based on flux corrected transport (Zalesak 78) Treat interpolation as advection f n+1 i = k 1 q k u 1 2 ( h x h v f n+1 i x i x k h = fi n + F ) f n i = k 1 q k u 0 ( x i x k h x h v h ) u1 u2 u0 y x Obtain the flux by solving a Poisson equation F = f n+1 i f n i Define a low order flux F lo and a high order flux F hi as being using a low order u1 and a high order u2 interpolation. Correct the low interpolation value by using the idea of FCT

17 Positivity for high order interpolation functions Algorithm: Local mass redistribution by Chern-Colella (87) We redistribute the undershoot of cell i δf i = min(0, f n i ) to neighboring cells in proportion to their capacity ξ Neighboring Cells ξ i+l = max(0, f n i+l ) The distribution function is conserved, which fixes the constant of proportionality l i f n+1 i+l = f i+l n + ξ i+l neighbors l 0 ξ i+l δf i Cell with Negative Charge

18 Mesh refinement Motivation: Remapping creates too many small strength particles at the edge of the distribution function. Algorithm: Interpolate as in uniform grid first, then transfer the charge from invalid cells to valid cells Ω l c Ω l c Ω l+1 c Ω l+1 c Particle is at the coarser level side Particle is at the finer level side The valid deposit cells: filled circles The invalid cell: open circles

19 Introduce collision term 1. Remapping provides an opportunity to integrate collision models with a grid-based solver. 2. Example: simplified Fokker-Planck equation suggested by Rathmann and Denavit (75) ( f )c = v [νvf + D v (νf )] t where (ν, D) are constants. Discretize it using a finite volume discretization and a second order L 0 stable, implicit scheme Solve the matrix system using multigrid method Couple it with the Vlasov equation with operator splitting f t + v f x E f v = ( f t )c R R+C R N N+1/2 N+1 t PIC

20 1D Vlasov-Poisson: linear Landau damping The initial distribution of linear Landau damping is f 0 (x, v) = 1 2π exp( v 2 /2)(1 + α cos(kx)) where α = 0.01, k = 0.5, L = 2π/k. The evolution of the amplitude of the electric field: exponential decay with rate γ = γ= no remapping, h x =L/32, h v =h max / γ= with remapping, h x =L/32, h v =h max / e t w/o remapping particle number: 960 CPU times: 3.99 seconds 1e t w/ remapping particle number: 960-1,539 CPU times: 5.21 seconds

21 1D Vlasov-Poisson: linear Landau damping 0.1 γ= no remapping, h x =L/128, h v =h max / γ= with remapping, h x =L/32, h v =h max / e-05 1e t w/o remapping particle number: 60,416 CPU times: seconds 1e t w/ remapping particle number: 1,539 CPU times: 5.21 seconds

22 1D Vlasov-Poisson: the two stream instability The initial distribution of the two stream instability is f 0 (x, v) = 1 2π v 2 exp( v 2 /2)(1 + α cos(kx)) where α = 0.05, k = 0.5, L = 2π/k. The evolution of f(x,v,t) w/o remapping, particle number: 18,360 w/ remapping, particle number: 18,360-24,242

23 1D Vlasov-Poisson: the two stream instability Comparison of f (x, v, t) at the same instant of time t = 20 w/o remapping max c = vs. max e = 0.3 w/ remapping max c = vs. max e = 0.3

24 1D Vlasov-Poisson: the two stream instability E field Errors and convergence rates error error e-05 1e-06 h x =L/64, h v =v max /128 h x =L/128, h v =v max /256 h x =L/256, h v =v max / t 1e-05 1e-06 h x =L/64, h v =v max /128 h x =L/128, h v =v max /256 h x =L/256, h v =v max / t convergence rate h x =L/128, h v =v max /256 h x =L/256, h v =v max /512 convergence rate h x =L/128, h v =v max /256 h x =L/256, h v =v max / t t w/o remapping w/ remapping

25 1D Vlasov-Poisson-Fokker-Planck: the two stream instability The evolution of f(x,v,t) Simulation w/o (left) and w/ (right) collision at t=30

26 2D Vlasov-Poisson:linear Landau damping The initial distribution of linear Landau damping in 2D is f 0 (x, y, v x, v y ) = 1 2π exp( (v 2 x + v 2 y )/2)(1 + α cos(kx) cos(ky)) where α = 0.05, k = 0.5, L = 2π/k. The evolution of the electric energy ξ e: exponential decay with constant rate γ log(ξ e ) γ without remapping, h x =L/32, h vx =h max /16 log(ξ e ) γ with remapping, h x =L/32, h vx =h max / w/o remapping, particle number: 1,228,800 t w/ remapping, particle number: 1,228,800-1,835,008 t

27 2D Vlasov-Poisson: the two stream instability The initial distribution for the two stream instability in 2D is f 0 (x, y, v x, v y ) = 1 12π exp( (v 2 x + v 2 y )/2)(1 + α cos(kx x))(1 + 5v 2 x ) where α = 0.05, k x = 0.5, L = 2π/k. Comparison of projected distribution function in (x, v x ) at time t = 20 w/o remapping, particle number: 14,408,192 w/ remapping particle number: 14,408,192-22,184,217

28 Code implementation The parallel and multidimensional Vlasov-Poisson solver is implemented using Chombo framework, a C/C++ and FORTRAN library for the solutions of partial differential equations on a hierarchy of block-structured grid with finite difference methods developed in the ANAG group of LBNL. The physical domain is decomposed into patches Each patch is assigned to a processor Particles are assigned to a processor according to their physical position Particles are transferred between patches using MPI Continuous work in ANAG: scalable multidimensional particle in cell code with remapping

29 Future Work Adaptivity on creating a hierarchy of grids for remapping Apply the method to magnetic fusion plasmas, e.g., gyrokinetic particle in cell method Parallel scalability GPU acceleration of remapping

30 Thank you! Questions?