Entropy Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Euler equations

Abstract: The entropy conservative/stable algorithm of Friedrich~\etal (2018) for hyperbolic conservation laws on nonconforming p-refined/coarsened Cartesian grids, is extended to curvilinear grids for the compressible Euler equations. The primary focus is on constructing appropriate coupling procedures across the curvilinear nonconforming interfaces. A simple and flexible approach is proposed that uses interpolation operators from one element to the other. On the element faces, the analytic metrics are used to construct coupling terms, while metric terms in the volume are approximated to satisfy a discretization of the geometric conservation laws. The resulting scheme is entropy conservative/stable, elementwise conservative, and freestream preserving. The accuracy and stability properties of the resulting numerical algorithm are shown to be comparable to those of the original conforming scheme (~p+1 convergence) in the context of the isentropic Euler vortex and the inviscid Taylor-Green vortex problems on manufactured high order grids.