【Lecture】Improved Shock capturing with high-order discontinuous Galerkin discretizations using adaptive mesh refinement and sub-cell discontinuity resolution

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. Key elements, which could make the DG method more suitable for computations of such time-dependent flows in complex domains, is use of adaptive mesh refinement (AMR) and application of limiting procedures that ensure sharp and accurate capturing of discontinuities for unstructured mixed-type meshes. A unified limiting approach suitable for 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, a nonlinear filter is employed in the finite element context of discontinuous DG discretizations to achieve low dissipative, well-balanced, high-order discontinuity capturing. 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 higher-order hierarchical limiting. The proposed approach is suitable for p–adaptivity in order to locally enhance resolution of three-dimensional flow simulations that include discontinuities and complex flow features.