Symmetric Interior Penalty DG Methods for the Compressible Navier--Stokes Equations I: Method Formulation

Abstract

In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier--Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second--order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton--GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher--order polynomials are presented.