A study on conserving invariants of the Vlasov equation in semi-Lagrangian computer simulations

TL;DR

This paper analyzes the conservation properties and long-term behavior of a semi-Lagrangian discontinuous Galerkin method for the Vlasov--Poisson equation, highlighting its advantages over traditional schemes in preserving physical invariants.

Contribution

It provides a theoretical analysis and numerical evidence that the discontinuous Galerkin method better conserves invariants and maintains entropy properties compared to traditional spline-based methods.

Findings

Entropy is nondecreasing in the DG scheme.

Traditional spline methods exhibit unphysical entropy oscillations.

DG method shows improved long-term conservation of invariants.

Abstract

The semi-Lagrangian discontinuous Galerkin method, coupled with a splitting approach in time, has recently been introduced for the Vlasov--Poisson equation. Since these methods are conservative, local in space, and able to limit numerical diffusion, they are considered a promising alternative to more traditional semi-Lagrangian schemes. In this paper we study the conservation of important physical invariants and the long time behavior of the semi-Lagrangian discontinuous Galerkin method. To that end we conduct a theoretical analysis and perform a number of numerical simulations. In particular, we find that the entropy is nondecreasing for the discontinuous Galerkin scheme, while unphysical oscillations in the entropy are observed for the traditional method based on cubic spline interpolation.