Explicit discontinuous Galerkin methods for unsteady problems

In this work we consider a special implementation of a discontinuous Galerkin (DG) method for general unstructured hexahedral element meshes called the discontinuous Galerkin Spectral Element Method (DGSEM). We are solving the compressible Navier–Stokes equations for unsteady turbulent flow simulations. We use explicit time stepping because of the high parallel scalability and also because the physical time scale of the simulation is in the range of the explicit time step restriction. In the explicit DGSEM framework, the efficiency of element-wise operations is highly improved compared to standard DG implementations. This improvement is due to collocated interpolation and integration points and tensor product nodal basis functions inside the hexahedron. In the first part of this paper, we describe the DGSEM scheme and derive the element-wise operators. We will conclude this part with accuracy and convergence analysis. The locality of the explicit DGSEM scheme is highly attractive for parallel computing, thus the second part is dedicated to a parallel performance analysis of the code. In the last part, we show the applicability of the scheme with a direct numerical simulation of a weak turbulent flow past a sphere at Reynolds number 1000.