A horizontally explicit, vertically implicit (HEVI) discontinuous Galerkin scheme for the 2-dimensional Euler and Navier-Stokes equations using terrain-following coordinates

Abstract

A solver for the Euler equations with optional diffusion based on the Discontinuous Galerkin (DG) method is presented. To be usable for simulation problems in the atmosphere from the global scale down to the meso-scale, the horizontally explicit, vertically implicit (HEVI) approach is applied to the DG discretization, to avoid tiny time-steps by thin grid cells. To consider orography in the correct approximation order, terrain-following coordinates are used, and to nevertheless keep the local conservation properties of DG, the equations are formulated in the strong conservation form. IMEX-Runge-Kutta time integration enables at least third order approximation in time. In a similar manner, diffusion both as a physical process (for turbulence parameterization) and as a physically motivated stabilization mechanism is treated by the Bassi, Rebay (1997) approach in terrain-following coordinates and in a conserving manner, too. Several test cases relevant for the atmosphere demonstrate the validity of the approach. Although most of the presented theory is applicable both in two and three dimensions (2D and 3D), these test cases are purely 2D. They show, in particular, that the proposed scheme seems to tolerate very steep terrain.