A Provably Stable Discontinuous Galerkin Spectral Element Approximation for Moving Hexahedral Meshes

TL;DR

This paper introduces a new stable discontinuous Galerkin spectral element method for solving conservation laws on moving domains, ensuring stability, conservation, and free-stream preservation through a skew-symmetric ALE formulation.

Contribution

The paper presents a novel provably stable DGSEM approach for moving hexahedral meshes using a skew-symmetric ALE formulation, with theoretical proofs and implementation guidance.

Findings

Proves stability, conservation, and free-stream preservation of the method.

Provides numerical validation supporting theoretical claims.

Details integration of the new method into existing DGSEM codes.

Abstract

We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian formulation to map the governing equations to a fixed reference domain. The approximation is made stable by a discretization of a skew-symmetric formulation of the problem. We prove that the discrete approximation is stable, conservative and, for constant coefficient problems, maintains the free-stream preservation property. We also provide details on how to add the new skew-symmetric ALE approximation to an existing discontinuous Galerkin spectral element code. Lastly, we provide numerical support of the theoretical results.