Mortar-based entropy-stable discontinuous Galerkin methods on non-conforming quadrilateral and hexahedral meshes

TL;DR

This paper develops a mortar-based entropy-stable discontinuous Galerkin method for nonlinear conservation laws on non-conforming quadrilateral and hexahedral meshes, improving computational efficiency and maintaining high-order accuracy.

Contribution

It extends entropy-stable Gauss collocation DG schemes to non-conforming meshes using mortar-based interface treatment, reducing flux evaluation costs.

Findings

Method is stable and accurate in 2D and 3D experiments.

Reduces flux evaluation complexity on non-conforming interfaces.

Maintains high-order accuracy with mortar-based correction.

Abstract

High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining entropy conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed by collocating at Lobatto quadrature points. Recent work has extended the construction of entropy-stable DG schemes to collocation at more accurate Gauss quadrature points. In this work, we extend entropy-stable Gauss collocation schemes to non-conforming meshes. Entropy-stable DG schemes require computing entropy conservative numerical fluxes between volume and surface quadrature nodes. On conforming tensor product meshes where volume and surface nodes are aligned, flux evaluations are required only between "lines" of nodes. However, on non-conforming meshes, volume and surface nodes are no longer aligned, resulting in a larger number of flux evaluations. We reduce this expense by introducing an entropy-stable mortar-based treatment of non-conforming interfaces via a face-local correction term, and provide necessary conditions for high-order accuracy. Numerical experiments in both two and three dimensions confirm the stability and accuracy of this approach.