A limiting approach for DG discretizations on mixed type meshes

A limiting procedure of Discontinuous Galerkin (DG) discretizations that is suitable for computations of high speed flows with strong shocks around complex geometries using a pp-adaptive procedure on mixed-type (quadrilateral/triangular) meshes is developed. A total variation bounded (TVB) approach is the building block of the limiting procedure. The main difference of this work with other recent limiting approaches for DG discretizations that were based on TVB limiting is that a unified approach for limiting on mixed-type meshes is obtained. This is achieved by applying the TVB limiter on the transformed canonical elements where appropriate estimates of the variation of the approximate solution derivatives are feasible. Applications for mixed-type meshes and pp-adaptive calculations with the DG method are performed for a number of standard test cases for shock capturing schemes to illustrate the potential of the proposed limiting approach. It is demonstrated in the computed results that good resolution of discontinuities and embedded smooth flow features is obtained with quadrilateral, triangular, and mixed-type meshes. Furthermore, the resolution of the present approach on arbitrary-type meshes is equally good with the resolution obtained with other limiting approaches of equivalent order obtained on rectangular meshes.