A mass-energy-conserving discontinuous Galerkin scheme for the isotropic multispecies Rosenbluth--Fokker--Planck equation

TL;DR

This paper introduces a novel discontinuous Galerkin scheme for the isotropic multispecies Rosenbluth--Fokker--Planck equation that conserves mass and energy, ensuring structure preservation and accurate equilibrium reproduction.

Contribution

It presents a structure-preserving discretization that maintains mass-energy conservation and symmetry properties for the Rosenbluth--Fokker--Planck equation using a discontinuous Galerkin approach.

Findings

The scheme conserves mass and energy with round-off errors.

It accurately reproduces analytic equilibria within truncation errors.

The method effectively incorporates nonlinear upwind fluxes without violating conservation laws.

Abstract

Structure-preserving discretization of the Rosenbluth-Fokker-Planck equation is still an open question especially for unlike-particle collision. In this paper, a mass-energy-conserving isotropic Rosenbluth-Fokker-Planck scheme is introduced. The structure related to the energy conservation is skew-symmetry in mathematical sense, and the action-reaction law in physical sense. A thermal relaxation term is obtained by using integration-by-parts on a volume integral of the energy moment equation, so the discontinuous Galerkin method is selected to preserve the skew-symmetry. The discontinuous Galerkin method enables ones to introduce the nonlinear upwind flux without violating the conservation laws. Some experiments show that the conservative scheme maintains the mass-energy-conservation only with round-off errors, and analytic equilibria are reproduced only with truncation errors of its formal accuracy.