Discontinuous Galerkin Method for the Numerical Solution of Inviscid and Viscous Compressible Flow

Abstract

In this work we are concerned with the numerical solution of a viscous compressible gas flow (compressible Navier-Stokes equations) with the aid of the discontinuous Galerkin finite element method (DGFEM). Our goal is to incorporate viscous terms into existing semi-implicit DGFEM scheme for the Euler equations, which is capable of solving flows with a wide range of Mach numbers [2, 4]. The nonsymmetric (NIPG), symmetric (SIPG) and incomplete interior penalty Galerkin method (IIPG) are generalized using the unified framework of [1] – derived for the Poisson equation – to the Navier-Stokes viscous terms. The resulting nonlinearities are linearized in a similar manner as nonlinear convective terms in the original scheme, thus enabling semi-implicit time stepping. The resulting scheme has very good stability properties and requires the solution of one sparse linear system per time level.