A structure-preserving collisional particle method for the Landau kinetic equation

TL;DR

This paper introduces a novel structure-preserving stochastic particle method for the Landau equation that maintains physical invariants and offers high accuracy and efficiency, applicable to both homogeneous and non-homogeneous cases.

Contribution

It presents a new particle method that exactly preserves the Landau equation's structure and physical properties without additional temporal discretization, improving stability and accuracy.

Findings

Achieves $O(N)$ complexity per time step.

Preserves mass, momentum, energy, and entropy dissipation.

Demonstrates strong long-time stability and accuracy.

Abstract

In this paper, we propose and implement a structure-preserving stochastic particle method for the Landau equation. The method is based on a particle system for the Landau equation, where pairwise grazing collisions are modeled as diffusion processes. By exploiting the unique structure of the particle system and a spherical Brownian motion sampling, the method avoids additional temporal discretization of the particle system, ensuring that the discrete-time particle distributions exactly match their continuous-time counterparts. The method achieves $O(N)$ complexity per time step and preserves fundamental physical properties, including the conservation of mass, momentum and energy, as well as entropy dissipation. It demonstrates strong long-time accuracy and stability in numerical experiments. Furthermore, we also apply the method to the spatially non-homogeneous equations through a case study of the Vlasov--Poisson--Landau equation.