Abstract

This article presents a high-order Discontinuous Galerkin method for compressible flow problems, in which is very frequent the formation of shocks. The stabilization is introduced by a new basis functions. This base has the flexibility to vary locally (within each element) between continuous polynomial functions space and a space of piecewise polynomial functions. Thus, the proposed method provides a bridge between the standard methods of high-order Discontinuous Galerkin and classical Finite Volume methods, maintaining the locality and compactness of the scheme. The variation of basis functions is automatically set according to the regularity of the solution and the stabilization is introduced by the jump operator, standard in Discontinuous Galerkin methods. Unlike the classical methods of slope limiting, the strategy here presented is very local, robust, and applies to any order of approximation. Moreover, the proposed method does not require adaptive mesh refinement techniques and it can be used with any temporal integration scheme. Several applications of the Euler equations are shown, demonstrating the validity and effectiveness of the method, especially for high orders of approximation.