Variance reduced particle solution of the Fokker-Planck equation with application to rarefied gas and plasma

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 Sadr and Hadjiconstantinou (2023) [30]. 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.& COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

We propose an importance-weight-based variance reduction method for Monte Carlo simulations of the Fokker-Planck equation, applied to rarefied gas and plasma dynamics. We demonstrate the stability and accuracy of weight evolution for a class of Fokker-Planck equations, including the linearized Landau equation. However, maintaining stability and accuracy becomes challenging when drift or diffusion coefficients depend on higher-order moments, such as in the cubic Fokker-Planck equation. The proposed variance reduced method shows significant speed-up compared to the standard Monte Carlo solution in the low-signal limit.