Energy-conserving discontinuous Galerkin methods for the Vlasov-Amp\`{e}re system

TL;DR

This paper introduces novel energy-conserving numerical schemes for the Vlasov-Ampère system, ensuring total energy preservation in simulations to prevent unphysical plasma heating or cooling over long time periods.

Contribution

It presents the first Eulerian solvers that fully conserve discrete total energy using explicit or implicit temporal discretizations, operator splitting, and discontinuous Galerkin methods.

Findings

Successfully preserve total energy in numerical simulations

Accurately simulate Landau damping and plasma instabilities

Demonstrate stability over long simulation times

Abstract

In this paper, we propose energy-conserving numerical schemes for the Vlasov-Amp\`{e}re (VA) systems. The VA system is a model used to describe the evolution of probability density function of charged particles under self consistent electric field in plasmas. It conserves many physical quantities, including the total energy which is comprised of the kinetic and electric energy. Unlike the total particle number conservation, the total energy conservation is challenging to achieve. For simulations in longer time ranges, negligence of this fact could cause unphysical results, such as plasma self heating or cooling. In this paper, we develop the first Eulerian solvers that can preserve fully discrete total energy conservation. The main components of our solvers include explicit or implicit energy-conserving temporal discretizations, an energy-conserving operator splitting for the VA equation and discontinuous Galerkin finite element methods for the spatial discretizations. We validate our schemes by rigorous derivations and benchmark numerical examples such as Landau damping, two-stream instability and bump-on-tail instability.