Higher-order time integration of Coulomb collisions in a plasma using Langevin equations

Journal Article · Fri Feb 08 00:00:00 EST 2013 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2013.01.038· OSTI ID:1227022

Dimits, A. M. ^[1]; Cohen, B. I. ^[1]; Caflisch, R. E. ^[2]; Rosin, M. S. ^[2]; Ricketson, L. F. ^[2]

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1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2. Univ. of California, Los Angeles, CA (United States). Dept. of Mathematics

The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt^1/2)] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering if and only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration.

• Research Organization: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC52-07NA27344; FG02-05ER25710
• OSTI ID: 1227022
• Report Number(s): LLNL-JRNL-577312
• Journal Information: Journal of Computational Physics, Vol. 242, Issue C; ISSN 0021-9991
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
70 PLASMA PHYSICS AND FUSION
Monte-Carlo methods
Milstein method
Collision processes
Plasmas
Collisions
Computer applications