Particle simulation of Coulomb collisions: Comparing the methods of Takizuka & Abe and Nanbu

Abstract

The interactions of charged particles in a plasma are in a plasma is governed by the long-range Coulomb collision. We compare two widely used Monte Carlo models for Coulomb collisions. One was developed by Takizuka and Abe in 1977, the other was developed by Nanbu in 1997. We perform deterministic and stochastic error analysis with respect to particle number and time step. The two models produce similar stochastic errors, but Nanbu's model gives smaller time step errors. Error comparisons between these two methods are presented.

Time-Step Considerations in Particle Simulation Algorithms for Coulomb Collisions in Plasmas

“…We also find that when a higher-order Milstein correction [11], [12] to the Langevin equations model is included, there is no significant change in the results of the collisional relaxation process for the specific example considered. Results using the binary collision algorithm for the same collisional relaxation test problem have been reported by Wang et al [13]. In the Wang et al study, it was found that the mixing of statistical errors with time-step effects made it difficult to obtain unambiguous and clear scalings of the errors with respect to the time step used.…”

“…A lack of a clear scaling with time step is also observed for the binary collision algorithm reported in the work of Wang et al [13]. In [13], the binary collison models of [1] and [2] are employed, and convergence with respect to particle number and time step are investigated with the additional feature that ensemble averages are computed over many realizations of the simulations to reduce the statistical variance.…”

“…The scaling of the errors with respect to Δt is less clear, but some cases show a rough error scaling approximately as Δt 1/2 in Fig. 5 of [13]. In Fig.…”

“…In the work of Wang et al, [13] the convergence properties of the Takizuka and Abe and the Nanbu binary collision operators with respect to particle number and time step were studied. It was found that the Nanbu collision algorithm achieved a factorof-two improvement in relative accuracy over the Takizuka and Abe basic algorithm for the same time step.…”

“…In [13], the binary collison models of [1] and [2] are employed, and convergence with respect to particle number and time step are investigated with the additional feature that ensemble averages are computed over many realizations of the simulations to reduce the statistical variance. For fixed time step, the simulations in [13] show a convergence with respect to increasing the number of particles or the number of realizations, i.e., the variance decreases as 1/N , where N is the product of the number of particles and realizations. The scaling of the errors with respect to Δt is less clear, but some cases show a rough error scaling approximately as Δt 1/2 in Fig.…”