Monte Carlo method with negative particles

Transcription

1 Monte Carlo method with negative particles Bokai Yan Joint work with Russel Caflisch Department of Mathematics, UCLA Bokai Yan (UCLA) Monte Carlo method with negative particles 1/ 2

2 The long range Coulomb collisions in plasma can be modeled by Landau-Fokker-Planck (LFP) equation f t = Q LFP(f,f), with binary collision term Q LFP (f,f)= 1 ( u 3 (u 2 δ ij u i u j ) ) f (w)f (v) dw. 4 v i R 3 v j w j Problem in DSMC As f m, most computation is spent on the collision between particles sampled from m. Q LFP (m, m)=. The major part of collisions has no net effect. Highly inefficient! Bokai Yan (UCLA) Monte Carlo method with negative particles 2/ 2

3 Hybrid methods with negative particles Apply splitting f (v)=m(v)+f d (v), Equilibrium m(v): evolved according to a fluid equation cheap In spatial homogeneous case, m(v) can be constant. Deviation f d (v): represented by particles expensive To minimize the deviation part, we allow f d (v)<. Write f (v)=m(v)+f p (v) f n (v), with f p (v), f n (v). We introduce negative particles to represent f n. f p and f n are represented by P and N particles. Perform P-P, P-N, N-N, P-M and N-M collisions. Bokai Yan (UCLA) Monte Carlo method with negative particles 3/ 2

4 Negative particle methods for rarefied gas (Hadjiconstantinou 5) One negative particle means the number of particle is 1. An N particle cancels a P particle with the same velocity w + + w = particle. A P-N collision cancels a regular P-P collision P-P: v +, w + v +, w +, P-N: v +, w 2v +, v, w. This can be derived from the Boltzmann equation. Bokai Yan (UCLA) Monte Carlo method with negative particles 4/ 2

5 Particle number increases! In rarefied gas (charge free), short range collision # collisions in one time step =O ( t) The particle number grows in the physical scale (N p + N n ) = (1+c t) (N p + N n ) t+ t. t In Coulomb gas (charged), long range collision # collisions in one time step = N The particle number grows in the numerical scale in Coulomb collisions! ( (N p + N n ) = 1+ N ) m+ 2N n (N p + N n ) t+ t. N m + N p N n t Bokai Yan (UCLA) Monte Carlo method with negative particles 5/ 2

6 A new negative particle method for general binary collisions Bokai Yan (UCLA) Monte Carlo method with negative particles 6/ 2

7 Key idea # 1: Combine collisions The Boltzmann equation t f= Q(f, f ) is reformulated t f= Q(f, f )=Q(f, f p ) Q(f, f n )+Q(f, m) = Q(f, f p ) Q(f, f n )+Q(f p f n, m)+q(m, m), } {{ } = and split: t m=q(m, m)=, t f p = Q(f, f p )+(Q(f p f n, m)) +, t f n = Q(f, f n )+(Q(f p f n, m)). Bokai Yan (UCLA) Monte Carlo method with negative particles 7/ 2

8 Key idea # 1: Combine collisions The Boltzmann equation t f= Q(f, f ) is reformulated t f= Q(f, f )=Q(f, f p ) Q(f, f n )+Q(f, m) = Q(f, f p ) Q(f, f n )+Q(f p f n, m)+q(m, m), } {{ } = and split: t m=q(m, m)=, t f p = Q(f, f p )+(Q(f p f n, m)) +, t f n = Q(f, f n )+(Q(f p f n, m)). P - P N - P M - P P - N N - N M - N M - M N - M P - M Bokai Yan (UCLA) Monte Carlo method with negative particles 8/ 2

9 Key idea # 1: Combine collisions Apply a forward Euler method in time. A Monte Carlo method: f p (t+ t)=f p + tq(f, f p ) + t(q(f p f n, m)) +. } {{ }} {{ } ւ ց regular collisions between f and f p, N p not change source term, N p increases by O ( t(n p + N n ) ) The particle number grows in the physical scale for any binary collisions (N p + N n ) = (1+c t) (N p + N n ) t+ t. t Bokai Yan (UCLA) Monte Carlo method with negative particles 9/ 2

10 Step 1, collisions between f and f p f p (t+ t)=f p + tq(f, f p )+ t(q(f p f n, m)) +. Sample a particle from f and collide with a P particle. } {{ } How? f= m+f p f n. Need to recover the distributions f p and f n from P and N particles computationally expensive and inaccurate. Bokai Yan (UCLA) Monte Carlo method with negative particles 1/ 2

11 Key idea # 2: Approximate f by F particles We introduce F particles give a solution to the original equation t f= Q(f, f ). Initially sampled from f (v, t=) directly. Then perform regular DSMC method. To sample a particle from f, just randomly pick one sample from F particles. One only needs #(F particles) N p + N n. Hence F particles give a coarse approximation of f. P and N particles are finer approximation of f m. Bokai Yan (UCLA) Monte Carlo method with negative particles 11/ 2

12 The new method A new Monte Carlo method with negative particles t f= Q( f, f ), t m=, t f p = Q(f, f p )+(Q(f p f n, m)) +, t f n = Q(f, f n )+(Q(f p f n, m)). Bokai Yan (UCLA) Monte Carlo method with negative particles 12/ 2

13 The new method A new Monte Carlo method with negative particles t f= Q( f, f ), t m=, t f p = Q( f, f p )+(Q(f p f n, m)) +, t f n = Q( f, f n )+(Q(f p f n, m)). f : coarse solution. Simulated by F particles. f= m+f p f n : finer solution, the desired result. Simulated by P and N particles. Bokai Yan (UCLA) Monte Carlo method with negative particles 12/ 2

14 To summarize Combine collisions to reduce the total number of collisions. Use F particles to perform the combined collisions. Bokai Yan (UCLA) Monte Carlo method with negative particles 13/ 2

15 An extra step: Particle Resampling v v 1 Global interpolation f p f n 4.5 v 2 2 Resample v Bokai Yan (UCLA) Monte Carlo method with negative particles 14/ 2

16 Control of particle number Particle Resampling Evolution of Particle Numbers 5 x 14 # of particles 1/2 N f 4 N p N n time Particle resampling is accurate but expensive. But it is only needed whenever N f (N p + N n ) is violated. After resampling, only need to keep a subset of the F particles. Bokai Yan (UCLA) Monte Carlo method with negative particles 15/ 2

17 Bump on Tail problem t =. t = t = t = t = t = Negative TA method Regular TA method Figure: The snaps of time evolution of marginal distribution f (vx, v y, v z ) dv y dv z in Bump-on-Tail problem. Bokai Yan (UCLA) Monte Carlo method with negative particles 16/ 2

18 Bump on Tail problem t = t = t = t = t = t = m f p f n Figure: The snaps of time evolution of the components m, f p and f n in Bump-on-Tail problem. Bokai Yan (UCLA) Monte Carlo method with negative particles 17/ 2

19 Convergence test L 1 error 1 2 slope = 1/2 T =.4, fine solution T =.4, coarse solution T = 2., fine solution T = 2., coarse solution T = 9., fine solution Initial N p Figure: The convergence rate for fine solution f and coarse solution f at different times. Test on Rosenbluth s problem. Bokai Yan (UCLA) Monte Carlo method with negative particles 18/ 2

20 Efficiency test t = 5/16; Negative TA t = 5/16; Regular TA t = 5/4; Negative TA t = 5/4; Regular TA t = 5; Negative TA t = 5; Regular TA error cputime Figure: The efficiency test on Rosenbluth s problem. Bokai Yan (UCLA) Monte Carlo method with negative particles 19/ 2

21 Future work Intro Old New Numerics Multi-component plasma. Evolve Maxwellian part m to further improve the efficiency. General non-linear operators. Spatial inhomogeneous. Design a hybrid method which uses very few particles in the fluid regime. Bokai Yan (UCLA) Monte Carlo method with negative particles 2/ 2