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Multilevel Monte Carlo simulation of Coulomb collisions. (English) Zbl 1351.82085
Summary: 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 \(\varepsilon\), the computational cost of the method is \(\mathcal{O}(\varepsilon^{- 2})\) or \(\mathcal{O}(\varepsilon^{- 2}(\ln \varepsilon)^2)\), depending on the underlying discretization, Milstein or Euler-Maruyama respectively. This is to be contrasted with a cost of \(\mathcal{O}(\varepsilon^{- 3})\) 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 \(\varepsilon = 10^{- 5}\). We discuss the importance of the method for problems in which collisions constitute the computational rate limiting step, and its limitations.

MSC:
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
82D10 Statistical mechanical studies of plasmas
Software:
GenFoo; ISDEP
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