An entropy stable high-order discontinuous Galerkin method on cut meshes

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 [1], we present a high-order accurate, entropy stable scheme for hyperbolic conservation laws on cut meshes. Our formulation requires positive/non-negative weight quadrature rules on cut elements, which we construct via explicit parameterizations, subtriangulations, and Carathéodory pruning. We numerically verify the accuracy and stability of our method using the shallow water and compressible Euler equations. While we do not address entropy stability for treatments of the small cell problem in this work, we do explore the effect of pairing state redistribution with an entropy stable method to relax the CFL condition and show it can (but does not always) result in a loss of entropy stability.