A Split-Form, Stable CG/DG-SEM for Wave Propagation Modeled by Linear Hyperbolic Systems

TL;DR

This paper introduces a hybrid spectral element method combining continuous and discontinuous Galerkin approaches, ensuring stability, conservation, and spectral accuracy for wave propagation in complex geometries.

Contribution

It develops a split-form hybrid CG/DG spectral element method that guarantees stability and conservation on unstructured curved meshes for hyperbolic systems.

Findings

Achieves spectral accuracy in wave scattering examples.

Ensures stability with split-form and two-point fluxes.

Maintains conservation and constant state preservation.

Abstract

We present a hybrid continuous and discontinuous Galerkin spectral element approximation that leverages the advantages of each approach. The continuous Galerkin approximation is used on interior element faces where the equation properties are continuous. A discontinuous Galerkin approximation is used at physical boundaries and if there is a jump in properties at a face. The approximation uses a split form of the equations and two-point fluxes to ensure stability for unstructured quadrilateral/hexahedral meshes with curved elements. The approximation is also conservative and constant state preserving on such meshes. Spectral accuracy is obtained for all examples, which include wave scattering at a discontinuous medium boundary.