An adaptive shock-capturing HDG method for compressible flows

Abstract

We introduce a hybridizable discontinuous Galerkin (HDG) method for the numerical solution of the compressible Euler equations with shock waves. By locally condensing the approximate conserved variables the HDG method results in a final system involving only the degrees of freedom of the approximate traces of the conserved variables. The HDG method inherits the geometric flexibility and high-order accuracy of discontinuous Galerkin methods, and offers a significant reduction in the computational cost. In order to treat compressible fluid flows with discontinuities, the HDG method is equipped with an artificial viscosity term based on an extension of existing artificial viscosity methods. Moreover, the artificial viscosity can be used as an indicator for adaptive grid refinement to improve shock profiles. Numerical results for subsonic, transonic, supersonic, and hypersonic flows are presented to demonstrate the performance of the proposed approach.