Structure-preserving particle methods for the Landau collision operator using the metriplectic framework

TL;DR

This paper introduces a new family of particle discretisation methods for the Landau collision operator that preserve physical invariants and thermodynamic laws by leveraging the metriplectic structure of the underlying system.

Contribution

It develops structure-preserving particle methods using spline or finite element representations combined with discrete gradient techniques, ensuring physical and thermodynamic consistency.

Findings

Methods automatically preserve mass, momentum, and energy.

Guarantee monotonic entropy dissipation.

Ensure thermodynamic laws are respected in discretisation.

Abstract

We present a novel family of particle discretisation methods for the nonlinear Landau collision operator. We exploit the metriplectic structure underlying the Vlasov-Maxwell-Landau system in order to obtain disretisation schemes that automatically preserve mass, momentum, and energy, warrant monotonic dissipation of entropy, and are thus guaranteed to respect the laws of thermodynamics. In contrast to recent works that used radial basis functions and similar methods for regularisation, here we use an auxiliary spline or finite element representation of the distribution function to this end. Discrete gradient methods are employed to guarantee the aforementioned properties in the time discrete domain as well.