An Entropy Stable Discontinuous Galerkin Finite-Element Moment Method for the Boltzmann Equation

TL;DR

This paper introduces an entropy-stable discontinuous Galerkin finite-element method for the Boltzmann equation, utilizing a moment system approach with divergence-based closure to ensure structural properties and entropy dissipation.

Contribution

It develops a novel entropy-stable numerical scheme for the Boltzmann equation using a divergence-based closure and a Galerkin approximation in a renormalized form.

Findings

The method preserves entropy dissipation.

Numerical results demonstrate stability and accuracy.

The scheme effectively approximates the Boltzmann equation in 1D.

Abstract

This paper presents a numerical approximation technique for the Boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous Galerkin finite-element approximation in position dependence. The closure relation for the moment systems derives from minimization of a suitable {\phi}-divergence. This divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The resulting combined discontinuous Galerkin moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. We present a new class of numerical flux functions, based on the underlying renormalized Boltzmann equation, that ensure entropy dissipation of the approximation scheme. Numerical results are presented for a one-dimensional test case.