Multilevel Monte Carlo simulation of Coulomb collisions

M. S. Rosin, L. F. Ricketson, A. M. Dimits, R. E. Caflisch, B. I. Cohen

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)140-157
Number of pages18
JournalJournal of Computational Physics
StatePublished - Oct 1 2014


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

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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