Transport based particle methods for the Fokker-Planck-Landau equation

TL;DR

This paper introduces a novel particle method inspired by transport modeling to solve the Landau equation, preserving key physical properties and demonstrating effectiveness through numerical experiments.

Contribution

The paper presents a new particle-based numerical method for the Landau equation that maintains physical invariants and improves upon traditional methods.

Findings

Method preserves mass, momentum, and energy.

Method effectively captures entropy decay.

Numerical experiments show competitive accuracy.

Abstract

We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.