Multilevel Monte Carlo simulation of Coulomb collisions

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)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)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=10−5ε=10−5. We discuss the importance of the method for problems in which collisions constitute the computational rate limiting step, and its limitations.