Random batch particle methods for the homogeneous Landau equation

TL;DR

This paper introduces stochastic random batch particle methods for efficiently solving the homogeneous Landau equation, reducing computational cost while preserving key physical invariants and entropy decay.

Contribution

It develops a novel stochastic particle method using random batching that maintains conservation laws and entropy decay for the Landau equation.

Findings

Computational cost is reduced to O(N) per time step.

The method preserves mass, momentum, energy, and entropy decay.

Numerical examples validate the effectiveness of the approach.

Abstract

We consider in this paper random batch particle methods for efficiently solving the homogeneous Landau equation in plasma physics. The methods are stochastic variations of the particle methods proposed by Carrillo et al. [J. Comput. Phys.: X 7: 100066, 2020] using the random batch strategy. The collisions only take place inside the small but randomly selected batches so that the computational cost is reduced to $O(N)$ per time step. Meanwhile, our methods can preserve the conservation of mass, momentum, energy and the decay of entropy. Several numerical examples are performed to validate our methods.