A Hybridized Discontinuous Petrov-Galerkin scheme for compressible flows

Abstract

The Hybridized Discontinuous Petrov-Galerkin scheme (HDPG) for compressible flows is presented. The HDPG method stems from a combination of the Hybridized Discontinuous Galerkin (HDG) method and the theory of the optimal test functions, suitably modified to enforce the conservativity at the element level. The new scheme maintains the same number of globally coupled degrees of freedom as the HDG method while increasing the stability in the presence of discontinuities or under-resolved features. The new scheme has been successfully tested in several problems involving shocks such as Burgers equation and the Navier-Stokes equations and delivers solutions with reduced oscillation at the shock. When combined with artificial viscosity, the oscillation can be completely eliminated using one order of magnitude less viscosity than that required by other Finite Element methods. Also, convergence studies in the sequence of meshes proposed by Peterson [49] show that, unlike other DG methods, the HDPG method is capable of breaking the suboptimal k+1/2 rate of convergence for the convective problem and thus achieve optimal k+1 convergence.