Numerical Methods for Gasdynamic Systems on Unstructured Meshes

Abstract

This article considers stabilized finite element and finite volume discretization techniques for systems of conservation laws. Using newly developed techniques in entropy symmetrization theory, simplified forms of the Galerkin least-squares (GLS) and the discontinuous Galerkin (DG) finite element method are developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE system. Detailed consideration is given to symmetrization of the Euler, Navier-Stokes, and magneto-hydrodynamic (MHD) equations. Numerous calculations are presented to evaluate the spatial accuracy and feature resolution capability of the simplified DG and GLS discretizations. Next, upwind finite volume methods are reviewed. Specifically considered are generalizations of Godunov’s method to high order accuracy and unstructured meshes. An important component of high order accurate Godunov methods is the spatial reconstruction operator. A number of reconstruction operators are reviewed based on Green-Gauss formulas as well as least-squares approximation. Several theoretical results using maximum principle analysis are presented for the upwind finite volume method. To assess the performance of the upwind finite volume technique, various numerical calculations in computational fluid dynamics are provided.