New paper submitted: A Sub-Element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

In this paper,a new strategy for a sub-element based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low to high order discretisations on this set of data, including a first order finite volume (FV) scheme up to the full order DG scheme. The different DG discretisations are then blended according to sub-element troubled cell indicators, resulting in a final discretisation that adaptively blends from low to high order within a single DG element. The goal is to retain as much high order accuracy as possible, even in simulations with very strong shocks, as e.g. presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement (AMR) and parallel computing. The numerical tests demonstrate the sub-element adaptive behaviour of the new shock capturing approach and its high accuracy.