This work is concerned with the numerical solution of an inviscid compressible flow. Our goal is to develop a sufficiently accurate and robust method allowing the solution of problems with a wide range of Mach numbers. The main ingredients are the discontinuous Galerkin finite element method (DGFEM), semi-implicit time stepping, special treatment of transparent boundary conditions and a suitable limiting procedure near discontinuities. Numerical experiments show the robustness of the method.