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 has been 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 and three dimensional applications 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/p-adaptivity in order to locally enhance resolution for three-dimensional flow simulations that include discontinuities and embedded complex flow features.

Furthermore in the spirit of p-adaptivity the nonlinear filter introduced by Yee et al. (JCP Vol. 150, 1999) and extensively used in the development of low dissipative well-balanced high order accurate finite-difference schemes is adapted to the finite element context of discontinuous Galerkin (DG) discretizations. The filter operator is constructed in the canonical computational domain for the standard cubical element where it can be applied in a direction per direction basis, and it is then extended for all element types in unstructured meshes using collapsed coordinate transformations. The performance of the proposed nonliner filter for DG discretizations is demonstrated and evaluated for different orders of expansions for one-dimensional and multidimensional problems with exact solutions. It is shown that for higher order discretizations discontinuity resolution within the cell is achieved and the design order of accuracy is preserved. The filter is applied for a number of standard inviscid flow test problems including strong shocks interactions to demonstrate that the proposed dissipative mechanism for DG discretizations yields superior results compared to the results obtained with the TVB limiter and high-order hierarchical limiting. The proposed approach is suitable for p–type adaptivity in order to locally enhance resolution of three-dimensional flow simulations that include discontinuities and complex flow features.