Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations

Abstract

In this paper, we propose a new unified family of arbitrary high order accurate explicit one-step finite volume and discontinuous Galerkin schemes on unstructured triangular and tetrahedral meshes for the solution of the compressible Navier–Stokes equations. This new family of numerical methods has first been proposed in [16] for purely hyperbolic systems and has been called P N P M schemes, where N indicates the polynomial degree of the test functions and M is the degree of the polynomials used for flux and source computation. A particular feature of the general P N P M schemes is that they contain classical high order accurate finite volume schemes ( N = 0 ) as well as standard discontinuous Galerkin methods ( M = N ) just as special cases, which therefore allows for a direct efficiency comparison. In the application section of this paper we first show numerical convergence results on unstructured meshes obtained for the compressible Navier–Stokes equations with Sutherland’s viscosity law, comparing all third to sixth order accurate P N P M schemes with each other. In order to validate the method also in practice we show several classical steady and unsteady CFD applications, such as the laminar boundary layer flow over a flat plate at high Reynolds numbers, flow past a NACA0012 airfoil, the unsteady flows past a circular cylinder and a sphere, the unsteady flows of a compressible mixing layer in two space dimensions and finally we also show applications to supersonic flows with shock Mach numbers up to M s = 10 .