An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property

TL;DR

This paper develops an entropy stable discontinuous Galerkin spectral element method for non-linear conservation laws on non-conforming meshes, ensuring stability and high-order accuracy through a modified mortar approach and SBP operators.

Contribution

It introduces a novel entropy stable DG method with a modified mortar approach that guarantees stability on non-conforming meshes for non-linear systems.

Findings

The method achieves high-order accuracy on non-conforming meshes.

The modified mortar approach ensures entropy stability for non-linear hyperbolic systems.

Numerical tests confirm the stability and accuracy of the proposed method.

Abstract

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre-Gauss-Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between nonconforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h/p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.