Variance Reduced Particle Solution of the Fokker-Planck Equation with Application to Rarefied Gas and Plasma Dynamics

Abstract

We develop an importance-weight-based variance reduction method for direct Monte Carlo simulations of the Fokker-Planck equation with applications to rarefied gas and plasma dynamics. We show that there exists a class of Fokker-Planck equations that admit stable weight evolution processes along particle trajectories, including those with drift and diffusion coefficients independent of moments higher than the kinetic energy, such as the linearized Landau equation.  When drift or diffusion coefficients depend on higher-order moments, such as in the cubic Fokker-Planck equation, maintaining stability and accuracy of the weight evolution becomes challenging. In this work, stability and conservation for the cubic FP model are achieved using the maximum cross-entropy formulation introduced in recent work [M. Sadr and N. Hadjiconstantinou, J. Comput. Phys. 472 (2023): 111677]. Several test cases  show that significant speed-up is obtained using the proposed variance reduced method compared to the standard Monte Carlo solution in the low-signal limit.

Keywords: Variance reduction, Monte Carlo, Fokker-Planck equation, Maximum cross-entropy formulation