Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.