A discontinuous Galerkin approach for high-resolution simulations of three-dimensional flows

High order discontinuous Galerkin (DG) discretizations possess features making them attractive for high-resolution computations in three-dimensional flows that include strong discontinuities and embedded complex flow features. A key element, which could make the DG method more suitable for computations of these time-dependent flows in complex domains, is application of limiting procedures that ensure sharp and accurate capturing of discontinuities for unstructured mixed-type meshes. A unified limiting procedure for DG discretizations in unstructured three-dimensional meshes is developed. A total variation bounded (TVB) limiter is applied in the computational space for the characteristic variables. The performance of the unified limiting approach is shown for different element types employed in mixed-type meshes and for a number of standard inviscid flow test problems including strong shocks to demonstrate the potential of the method. Other alternatives, such as hierarchical three-dimensional limiting with the proposed limiting approach and TVD limiting, are developed and demonstrated. Furthermore, increased order of expansion and adaptive mesh refinement is introduced in the context of it h/ph/p-adaptivity in order to locally enhance resolution for three-dimensional flow simulations that include discontinuities and embedded complex flow features.