On a robust discontinuous Galerkin technique for the solution of compressible flow

In the paper, we describe a numerical technique allowing the solution of compressible inviscid flow with a wide range of Mach numbers. The method is based on the application of the discontinuous Galerkin finite element method for the space discretization of the Euler equations written in the conservative form, combined with a semi-implicit time discretization. Special attention is paid to the treatment of boundary conditions and to the stabilization of the method in the vicinity of discontinuities avoiding the Gibbs phenomenon. As a result we obtain a technique allowing the numerical solution of flows with practically all Mach numbers without any modification of the Euler equations. This means that the proposed method can be used for the solution of high speed flows as well as low Mach number flows. Presented numerical tests prove the accuracy of the method and its robustness with respect to the Mach number.