A Sub-Element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

TL;DR

This paper introduces a novel sub-element adaptive shock capturing method for discontinuous Galerkin schemes that blends discretizations of varying order within elements to improve shock resolution and accuracy.

Contribution

It presents a hierarchical blending strategy for DG methods using sub-element troubled cell indicators, enhancing shock capturing while maintaining high order accuracy.

Findings

Effective shock capturing demonstrated in numerical tests

High accuracy preserved even with strong shocks

Compatible with adaptive mesh refinement and parallel computing

Abstract

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 discretizations on this set of data, including a first order finite volume scheme up to the full order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization 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 and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.