Overview of Accelerated Simulation Methods for Plasma Kinetics

Transcription

1 Overview of Accelerated Simulation Methods for Plasma Kinetics R.E. Caflisch 1 In collaboration with: J.L. Cambier 2, B.I. Cohen 3, A.M. Dimits 3, L.F. Ricketson 1,4, M.S. Rosin 1,5, B. Yann 1 1 UCLA Math Dept 2 AFRL 3 LLNL 4 Courant Institute 5 Pratt Institute AFRL Program Review January 2, 215 Work performed under funding from AFOSR and DOE.

2 Introduction

3 Kinetic Theory Phase space particle number density f ( x, v, t) Boltzmann Equation: C(f, f ) is: t f + v x f + a v f = C(f, f ) (1) Boltzmann collision operator for rarefied gas dynamics (RGD) Landau-Fokker-Planck operator for Coulomb collisions Operator for excitation/deexcitation, ionization/recombination for collisional/radiative (CR) kinetics This talk omits spatial variation and mean fields.

4 Binary Collision Method Collisions are simulated directly as in DSMC, with Distribution function is represented by set of discrete particles For short range interactions (RGD and CR kinetics), each particle has ν t collisions in time step t for collision rate ν. Collision partners and parameters are randomly chosen. For long range interactions (Coulomb), each numerical collision is an aggregate of grazing collisions over time step. Every particle collides once in every time step Collisions are aggregates depending on t

5 Computational Obstacles for Monte Carlo Simulation Computational complexity for high collisionality (near continuum limit) Hybrid method for RGD and Coulomb collisions Negative particle method Many time steps Multi-Level Monte Carlo (MLMC) Multiscale features CR kinetics

6 Hybrid Schemes for Binary Collision Methods

7 Hybrid Scheme Combine fluid and particle simulation methods 1 : Treat as fluid Treat as particles Separate f into Maxwellian and non-maxwellian components: f = m + f p Treat m as fluid solves Euler equations Simulate f p by Monte Carlo algorithm Interaction of m and f p : sample particles from m and collide them with particles from f p. 1 Caflisch et. al, Multiscale Model. Simul. 7 (28)

8 Thermalization/dethermalization Hybrid scheme is efficient because collisions between m and m need not be simulated. Efficiency increased by thermalization Collisions drive particles into equilibrium Move particles from f p to m when they have collided enough (thermalization) Move sampled particles from m into f p if the collision is strong enough (dethermalization) (De)Thermalization criterion using entropy 2 Alternative criterion based on scattering angle 3 Generalization to spatial inhomogeneities is nontrivial 2 Ricketson et. al, J. Comp. Phys., Dimits et. al., private communication

9 Bump-on-Tail f f f f f f.2 t=.2 t = 1.2t FP.2 t = 2.4t FP v x v x v x t = 3.6t FP t = 6t FP t = 11t FP v x v x v x

10 Scheme Comparison.6 Bump on Tail Test Problem.55.5 Relative L 1 Error Entropy Scheme.25 Entropy + Dethermalization Scattering Angle Scheme Scattering Angle + Dethermalization Improvement Factor vs. PIC

11 Negative Particles

12 Hybrid methods need negative particles f The hybrid method presented above, assumes that f m, This may be inefficient if there is a defect in the Maxwellian v

13 Hybrid methods need negative particles The hybrid method presented above, assumes that f m, This may be inefficient if there is a defect in the Maxwellian. If f = m + f p (with f p > so that f p can represented by particles) then m will be small.

14 Hybrid methods need negative particles The hybrid method presented above, assumes that f m, This may be inefficient if there is a defect in the Maxwellian. Better: f = m + f p f n with f p, f n. Represent the defect by negative particles!

15 Meaning of Negative Particles A negative particle w (in f m ) cancels a particle in m (or in f p ) which should not be present. A collision P N between a positive particle v + and a negative particle w cancels a collision P P between v + and a positive particle w + that should not have occurred. The P P collision removes v + and w + and adds v + and w +: P-P: v +, w + v +, w + So the P N collision should add a v +, remove w (equivalent to adding w + ) and add negative particles to cancel v + and w + P-N: v +, w 2v +, v, w This can be derived from the Boltzmann equation (Hadjiconstantinou 25).

16 General rules for collisions with negative particles P-P: v +, w + v +, w + P-N: Particle number increases! v +, w 2v +, v, w N-N: v, w 2v, 2w, v +, w + P-M: m, v + m, w, v +, w + N-M: m, v m, w +, v, w. Since every particle collides in every time step, it s much worse for Coulomb collisions! Bokai Yan will present a new method for overcoming this particle increase.

17 Multilevel Monte Carlo (MLMC) for Langevin Method

18 Langevin Formulation Linear LFP equation for f (v, t) is equivalent to stochastic differential equations (SDEs) for v(t) dv i = F i dt + D ij dw j, (2) where f is probability density of v and i, j are component indices W = W (t) is Brownian motion in velocity dw is white noise in velocity Direct extension to spatial dependence Valid for nonlinear LFP, if F and D are updated as needed

19 Discretization of SDEs Euler-Maruyama discretization in time: v i,n+1 = v i,n + F i,n t + D ij,n W j,n, (3) W n = W n+1 W n (4) in which v i,n = v i (t n ) and F n = F(v n ) Computational cost vs. error ε: Statistical error is O(N 1/2 ) t error is O( t), since W = O( t) and random Optimal choice is ε = N 1/2 = t Cost = N t 1 = ε 3

20 MLMC scheme combines multiple solutions with varying MLMC Basics Like multi-grid method. Expectations with N l samples LX Convergent sum, L!1 when using Milstein method ˆv NL L = E[v ]+ E[(v l v l 1 )] l=1 Introduce time step levels, t l = T 2 l, for l =,..., L Statistical error converges with t, N l t Grid l λ 1. Giles in Monte Carlo and Quasi-Monte Carlo Method, Springer-Verlag, (26) 7 Use computation at coarse level as a control variate for computation at finer level Optimally combine the different levels Motivated by mutigrid methods

21 MLMC Scaling For RMSE < ε, the complexity now scales like 4 { ( O ε Cost = 2 (log ε) 2) for Euler-Maruyama O ( ε 2) for Milstein (5) Notes: MLMC-Euler-Maruyama scales better than standard MC MLMC-Milstein is even better O ( ε 2) scaling is possible without Milstein, using antithetic sampling method 5 4 Giles, Operations Research, 56(3):67, 28 5 Giles & Szpruch, arxiv: , 212

22 A Sample Plasma Problem 1 Direct MLMC Euler MLMC Milstein (lnε) 2 K ε ε Rosin, Ricketson, et. al., J Comp Phys 214

23 Direct Simulation of CR Kinetics

24 CR Kinetics Kinetic simulation of electron-impact excitation/deexcitation and ionization/recombination. 6 Plasma representation Discrete particles for electron energy distribution f (E) Continuum densities ρ k for atoms at k m 1 levels of excitation and ρ i for ions (Bohr model) Cross-sections Semi-classical cross-sections for excitation and ionization. Detailed balance yields cross-sections for deexcitation and recombination. Closely related to work of Hai Le. 6 Yan et. al., J Comp Phys to appear

25 Boltzmann Equation Electrons Atomic level k Ions t f (E) = Q e IR + Qe ED t ρ k = Q k IR + Qk ED t ρ i = Q i IR Q is the collision operator for ionization/recombination (IR) and excitation/deexcitation (ED).

26 Entropy Inequality Entropy S is S = kn(log(n/g) 1) N = number density of particles G = degeneracy of particles k = Boltzmann constant Boltzmann H-theorem, for H=-S, is dh dt with equality only for equilibrium New(?) explicit formula for S in terms of distribution functions. Multiple non-maxwellian equilibria for excitation/deexcitation! H theorem is always valid.

27 Multiscale and Singular Features Multiscale and singular features are a bottleneck in Monte Carlo simulation Many processes m states (m 1 levels + ion) interactions between any two states k, k (including k = i) m(m 1)/2 different types of interactions Wide range of interaction rates interaction k (k 1) has rate O(k 6 )! Singularity in recombination rate for recombining electron energies E 1 and E 2 ( ) 1 r rec = O E1 E 2

28 Numerical Results Evolution for ionization/recombination only t =. ps 8 x atom distribution actual equilibrium 4 x electron distribution actual equilibrium t =.5 ps t =.2 ps 4 x x Levels Levels Levels Energy / I H 15 x Energy / I H 2 x Energy / I H Maxwellian equilibrium as expected Similar results if excitation/deexcitation is included

29 Numerical Results Evolution for ionization/recombination only 1 entropy time / ps Decrease in H as expected

30 Numerical Results Evolution for excitation/deexcitation only, with 2 atomic levels t =. ps 8 x atom distribution actual equilibrium 2 x electron distribution actual equilibrium t =.2 ps 8 x x Levels 1 2 Levels Energy / I H 15 x Energy / I H 15 x 145 t =.52 ps Levels Energy / I H Equilibrium is clearly non-maxwellian For 1 levels, equilibrium very close to Maxwellian

31 Numerical Results Evolution for excitation/deexcitation only, with 2 atomic levels 1 Entropy time / ps Decrease in H as expected, in spite of non-maxwellian

32 Summary: Accelerated Methods for Monte Carlo Simulation Hybrid method for RGD and Coulomb collisions Negative particle method Multi-Level Monte Carlo (MLMC) CR kinetics