New adaptive modal and DTV filtering routines for the DG method on triangular grids applied to the Euler equations

Abstract

In this paper, we continue our investigations started in Meister et al. (Numer. Methods Partial Differ. Equ. 2011) on the development of adaptive filtering techniques for DG methods on unstructured triangular grids solving hyperbolic conservation laws. While in our former work the focus was put on scalar conservation laws in order to study the principle mechanisms of the proposed techniques, new results are now presented concerning the application of adaptive modal and DTV filtering to the Euler equations of gas dynamics. To demonstrate the applicability of this approach for practically relevant problems, we perform experiments on standard test cases such as shock-vortex interactions, a double mach reflection as well as flow through a wind tunnel containing a forward facing step. The constructed modal filter is based on a specific spectral viscosity formulation on triangular grids. In this context, this paper gives a new result on the energy dissipation of the involved damping step which is based on an interesting estimate on weighted L 2-norms on the reference triangle. With respect to adaptive modal filtering, we investigate the behaviour of two different shock indicators which are new candidates for elementwise modal filtering in the DG context. Furthermore, we propose a novel adaptive strategy for the original DTV iteration as well as the modified DTV filter on subtriangle graphs constructed in Meister et al. (2011). For both filtering techniques, convergence studies away from the discontinuities of the exact entropy solution are additionally carried out for a scalar nonlinear test case yielding an improvement on the former results in Meister et al. (2011).