### Abstract

We present a new, for plasma physics, highly efficient multilevel Monte Carlo numerical method for simulating Coulomb collisions. The method separates and optimally minimizes the finite-timestep and finite-sampling errors inherent in the Langevin representation of the Landau-Fokker-Planck equation. It does so by combining multiple solutions to the underlying equations with varying numbers of timesteps. For a desired level of accuracy ε, the computational cost of the method is O(ε-2) or O(ε-2(lnε)2), depending on the underlying discretization, Milstein or Euler-Maruyama respectively. This is to be contrasted with a cost of O(ε-3) for direct simulation Monte Carlo or binary collision methods. We successfully demonstrate the method with a classic beam diffusion test case in 2D, making use of the Lévy area approximation for the correlated Milstein cross terms, and generating a computational saving of a factor of 100 for ε = ^{10 -5}. We discuss the importance of the method for problems in which collisions constitute the computational rate limiting step, and its limitations.

Original language | English (US) |
---|---|

Pages (from-to) | 140-157 |

Number of pages | 18 |

Journal | Journal of Computational Physics |

Volume | 274 |

DOIs | |

State | Published - Oct 1 2014 |

### Fingerprint

### Keywords

- Coulomb collisions
- Monte Carlo
- Multilevel Monte Carlo
- Particle in cell
- Plasma

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)
- Computer Science Applications

### Cite this

*Journal of Computational Physics*,

*274*, 140-157. https://doi.org/10.1016/j.jcp.2014.05.030

**Multilevel Monte Carlo simulation of Coulomb collisions.** / Rosin, M. S.; Ricketson, L. F.; Dimits, A. M.; Caflisch, Russel; Cohen, B. I.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 274, pp. 140-157. https://doi.org/10.1016/j.jcp.2014.05.030

}

TY - JOUR

T1 - Multilevel Monte Carlo simulation of Coulomb collisions

AU - Rosin, M. S.

AU - Ricketson, L. F.

AU - Dimits, A. M.

AU - Caflisch, Russel

AU - Cohen, B. I.

PY - 2014/10/1

Y1 - 2014/10/1

N2 - We present a new, for plasma physics, highly efficient multilevel Monte Carlo numerical method for simulating Coulomb collisions. The method separates and optimally minimizes the finite-timestep and finite-sampling errors inherent in the Langevin representation of the Landau-Fokker-Planck equation. It does so by combining multiple solutions to the underlying equations with varying numbers of timesteps. For a desired level of accuracy ε, the computational cost of the method is O(ε-2) or O(ε-2(lnε)2), depending on the underlying discretization, Milstein or Euler-Maruyama respectively. This is to be contrasted with a cost of O(ε-3) for direct simulation Monte Carlo or binary collision methods. We successfully demonstrate the method with a classic beam diffusion test case in 2D, making use of the Lévy area approximation for the correlated Milstein cross terms, and generating a computational saving of a factor of 100 for ε = 10 -5. We discuss the importance of the method for problems in which collisions constitute the computational rate limiting step, and its limitations.

AB - We present a new, for plasma physics, highly efficient multilevel Monte Carlo numerical method for simulating Coulomb collisions. The method separates and optimally minimizes the finite-timestep and finite-sampling errors inherent in the Langevin representation of the Landau-Fokker-Planck equation. It does so by combining multiple solutions to the underlying equations with varying numbers of timesteps. For a desired level of accuracy ε, the computational cost of the method is O(ε-2) or O(ε-2(lnε)2), depending on the underlying discretization, Milstein or Euler-Maruyama respectively. This is to be contrasted with a cost of O(ε-3) for direct simulation Monte Carlo or binary collision methods. We successfully demonstrate the method with a classic beam diffusion test case in 2D, making use of the Lévy area approximation for the correlated Milstein cross terms, and generating a computational saving of a factor of 100 for ε = 10 -5. We discuss the importance of the method for problems in which collisions constitute the computational rate limiting step, and its limitations.

KW - Coulomb collisions

KW - Monte Carlo

KW - Multilevel Monte Carlo

KW - Particle in cell

KW - Plasma

UR - http://www.scopus.com/inward/record.url?scp=84902603177&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902603177&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2014.05.030

DO - 10.1016/j.jcp.2014.05.030

M3 - Article

VL - 274

SP - 140

EP - 157

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -