Unstructured three-dimensional High Order Grids for Discontinuous Galerkin Schemes

Abstract

Discontinuous Galerkin (DG) methods are a prominent candidate for high order accurate schemes for advection dominated problems in threedimensional complex geometries, since they sustain high order spatial accuracy even on general unstructured grids. To maintain the high order accuracy at curved wall boundaries, a high order representation of the elements near the wall surface is required, i.e. high order grids. Regarding three-dimensional curved geometries, the construction of such unstructured curved grids is a subtle task. The implementation of the DG scheme will be discussed, focusing on curved element mappings. The mappings of curved element sides as well as the curved element volume are described. We present our approach of the high order grid generation process in detail. The main idea is to rely on established unstructured grid generators for a basic volume grid consisting of straight sided elements and provide additionally high order information for the curved boundaries. We analyze the properties of the geometry approximation with a simple test case using the Maxwell equations. Finally, we present an unstructured curved mesh of a fairly complex aircraft, showing the applicability of the approach.