On the discontinuous Galerkin method for the numerical solution of compressible high-speed flow

Abstract

The paper deals with an application of the discontinuous Galerkin finite element (DG-FE) method to the numerical solution of a system of hyperbolic equations. We extend our results from [8,9], where two versions of the DG-FE method were applied to the scalar convection-diffusion equation. In order to avoid spurious oscillations near discontinuities we develop a new limiting which is based on the control of interelement jumps and switches from piecewise linear to piecewise constant approximations. Isoparametric finite elements are used near a curved boundary of nonpolygonal computational domain in order to achieve a physically admissible and sufficiently accurate numerical solution. Numerical examples of transonic flow through the GAMM channel and around the NACA0012 airfoil are presented. Finally, we mention some theoretical results obtained for a modified DG-FE method applied to a nonlinear convection-diffusion problem.