Shock detection and capturing methods for high order Discontinuous-Galerkin Finite Element Methods

High-order Discontinuous Galerkin (DG) methods have shown a lot of promise in being able to provide high accuracy and efficiency as well as the flexibility to handle complex unstructured grids. However in order to solve compressible flow problems effectively, we need to be able to detect discontinuities/shocks in the flow and capture them effectively such that the solution accuracy is not affected elsewhere. Traditionally finite difference and finite volume schemes have used a variety of physical sensors. Such physical sensors are not best suited to distinguish between different flow phenomena in more complex viscous flows involving boundary layers, vortices etc. Also, owing to the availability of higher resolution in each element/cell in these high order finite element methods, it would be optimal to detect shocks/discontinuities through a mathematical procedure inside each element and capture the shock inside a single element . While there has been considerable progress in this direction, robust shock detection across the broad spectrum of compressible flow problems still remains an issue. While we have been able to get some of these methods to work quite well for certain cases, they need to be fine-tuned to each problem and there is lack of rigor in how one should do the same. Klöckner also points out certain problems with methods which infer the presence of discontinuities using just the decay (or not) of the highest (few) modal frequencies. The most important issue is that the modal co-efficient decay is not uniform for a small finite resolution and the situation is only worse in polynomial bases compared to Fourier spectra. All this amounts to unreliable performance in complex flows, with inadequate separation of scales between shocks and other flow features like vortices and boundary layers. These issues make it hard to build a generic solver for inviscid and viscous compressible flow computations.