An embedded discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations

Abstract

We present an Embedded Discontinuous Galerkin (EDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the approximate conserved variables defining the discontinuous Galerkin solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the solution. The EDG method can be seen as a particular form of a Hybridizable Discontinuous Galerkin (HDG) method in which the hybrid fluxes are required to belong to a smaller space than in standard HDG methods. In our EDG method, the hybrid unknown is taken to be continuous at the vertices, thus resulting in an even smaller number of coupled degrees of freedom than in the HDG method. In fact, the resulting stiffness matrix has the same structure as that of the statically condensed continuous Galerkin method. In exchange for the reduced number of degrees of freedom, the EDG method looses the optimal converge property of the flux which characterizes other HDG methods. Thus, for convection-diffusion problems, the EDG solution converges optimally for the primal unknown but suboptimally for the flux.