High order discontinuous Galerkin discretizations with discontinuity resolution within the cell

The nonlinear filter of Yee et al. and used for low dissipative well-balanced high order accurate finite-difference schemes is adapted to the finite element context of discontinuous Galerkin (DG) discretizations. The performance of the proposed nonlinear filter for DG discretizations is demonstrated for different orders of expansions for one- and multi-dimensional 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 inviscid and viscous 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 [2]. The proposed approach is suitable for p-adaptivity in order to locally enhance resolution of three-dimensional flow simulations. [1] H.C. Yee, N.D. Sandham, M.J. Djomehri, J. Comput. Phys., 150 (1999) 199-238. [2] K. Panourgias, J.A. Ekaterinaris, Comput. Methods Appl. Mech. Engrg., 299 (2016) 254-282.