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

## Abstract

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 whichmore »

- Authors:

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

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1227022

- Report Number(s):
- LLNL-JRNL-577312

Journal ID: ISSN 0021-9991

- Grant/Contract Number:
- AC52-07NA27344; FG02-05ER25710

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 242; Journal Issue: C; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

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

### Citation Formats

```
Dimits, A. M., Cohen, B. I., Caflisch, R. E., Rosin, M. S., and Ricketson, L. F. Higher-order time integration of Coulomb collisions in a plasma using Langevin equations. United States: N. p., 2013.
Web. doi:10.1016/j.jcp.2013.01.038.
```

```
Dimits, A. M., Cohen, B. I., Caflisch, R. E., Rosin, M. S., & Ricketson, L. F. Higher-order time integration of Coulomb collisions in a plasma using Langevin equations. United States. doi:10.1016/j.jcp.2013.01.038.
```

```
Dimits, A. M., Cohen, B. I., Caflisch, R. E., Rosin, M. S., and Ricketson, L. F. Fri .
"Higher-order time integration of Coulomb collisions in a plasma using Langevin equations". United States. doi:10.1016/j.jcp.2013.01.038. https://www.osti.gov/servlets/purl/1227022.
```

```
@article{osti_1227022,
```

title = {Higher-order time integration of Coulomb collisions in a plasma using Langevin equations},

author = {Dimits, A. M. and Cohen, B. I. and Caflisch, R. E. and Rosin, M. S. and Ricketson, L. F.},

abstractNote = {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(Δt1/2)] 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.},

doi = {10.1016/j.jcp.2013.01.038},

journal = {Journal of Computational Physics},

number = C,

volume = 242,

place = {United States},

year = {2013},

month = {2}

}

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