A discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries

Abstract

We present a discontinuous Galerkin method (DGM) for solutions of the Euler equations on Cartesian grids with embedded geometries. Cartesian grid methods can provide considerable computational savings for computationally intensive schemes like the DGM. Cartesian mesh generation is also simplified compared to the body fitted meshes. However, cutting an embedded geometry out of the grid creates cut cells. These are difficult to deal with for two reasons: the restrictive CFL number and irregular shapes. Both of these issues are more involved for the DG than for finite volume methods, which most Cartesian grid techniques have been developed for. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We provide a number of computational examples for smooth flows to demonstrate the accuracy of our approach.