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

TL;DR

The paper addresses the challenge of simulating collisional Landau dynamics in a way that respects fundamental thermodynamic structure. It introduces a structure-preserving, particle-based discretisation grounded in the metriplectic framework, combining a conservative Poisson bracket with a dissipative metric bracket and a projected spline/finite-element representation of the distribution function. The authors prove that the semi-discrete scheme conserves mass, momentum, and energy while dissipating entropy monotonically, and that a discrete gradient time integrator preserves these properties at the fully discrete level. The approach yields Maxwellian equilibria via an energy-Casimir principle and offers a clear path to coupling with full Vlasov–Maxwell dynamics for robust, thermodynamically consistent plasma simulations.

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.