First part : We are going to see artificial boundary conditions for the numerical simu- lation of compressible flows. The construction of the proposed boundary conditions is based on characteristic analysis and are applied for boundaries with arbitrary shape and orientation. Numerical solutions of complex flow features, small-amplitude acoustic disturbances and viscous flow problems will be presented. Second part: We will speak for a DGFEM for the numerical solution of flow problems with discontinuities. This method is based on the limitation in every cell of the difference between the extrema values and the mean value of the solution. Numerical results computed by the application of the proposed method for classical one- and two- dimensional problems are going to be presented.