A structure-preserving local discontinuous Galerkin method for the Fokker-Planck-Landau equation

TL;DR

This paper develops a structure-preserving local discontinuous Galerkin method for the Fokker-Planck-Landau equations, ensuring conservation and stability, and demonstrates its effectiveness through benchmark tests.

Contribution

It extends the structure-preserving strategy to the LDG framework, combining conservation with stabilization via upwind flux in a novel way.

Findings

The scheme is conservative and stabilized.

Numerical experiments validate the method.

The method effectively solves benchmark problems.

Abstract

In this work, we introduce a structure-preserving local discontinuous Galerkin (LDG) method \cite{cockburn1998local} for solving the non-local non-linear Fokker-Planck-Landau (FPL) equations. We rephrase the structure-preserving strategy of Shiroto and Sentoku\cite{shiroto2019structure} in the language of numerical analysis, and extend it to the LDG framework. We propose a method that is not only conservative, but also stabilized through upwind flux. The apparent contradiction between conservation laws and numerical stabilization is elegantly resolved by leveraging the properties of the jump terms inherent to the LDG framework. In the numerical experiments, our scheme is tested with benchmark examples.