An Entropy Stable High-Order Discontinuous Galerkin Method on Cut Meshes

TL;DR

This paper develops a high-order entropy stable discontinuous Galerkin method tailored for cut meshes, enabling robust and accurate simulations of hyperbolic conservation laws on complex geometries.

Contribution

It extends entropy stable SBP schemes to cut meshes using a skew-hybridized formulation and constructs positive quadrature rules for arbitrary cut elements.

Findings

Numerical verification of accuracy and stability on shallow water and Euler equations.

Demonstration of the method's robustness on arbitrarily shaped cut elements.

Promising results for state redistribution with entropy stable schemes.

Abstract

High-order entropy stable summation-by-parts (SBP) schemes are a class of robust and accurate numerical methods for hyperbolic conservation laws that are numerically stable at arbitrary order without the need for artificial stabilization. While SBP schemes are well-established on simplicial and tensor-product elements, they have not been extended to cut meshes. Cut meshes provide a convenient and efficient means of mesh generation for domains with embedded boundaries but can be difficult to use due to their arbitrarily shaped cut elements. Using the skew-hybridized SBP formulation of Chan ["Skew-symmetric entropy stable...", JSC, 2019], we present a high-order accurate, entropy stable scheme for hyperbolic conservation laws on cut meshes. The formulation requires positive/non-negative weight quadrature rules on cut elements, which we construct via explicit parameterizations, subtriangulations, and Caratheodory pruning. We numerically verify the accuracy and stability of our method using the shallow water and compressible Euler equations and note promising results for the use of state redistribution with entropy stable methods.