A Discontinuous Galerkin method for Shock Capturing using a mixed high-order and sub-grid low-order approximation space

TL;DR

This paper introduces a novel discontinuous Galerkin discretization for conservation laws that combines high-order polynomials with sub-grid low-order modes, using a sensor-based penalty to adaptively suppress high-order modes for improved shock capturing.

Contribution

It presents a new hybrid approximation space and a sensor-based penalty approach within a discontinuous Galerkin framework for better shock capturing in conservation laws.

Findings

Effective shock capturing demonstrated in numerical tests

Adaptive suppression of high-order modes improves stability

Hybrid approximation space enhances accuracy and robustness

Abstract

This article considers a new discretization scheme for conservation laws. The discretization setting is based on a discontinuous Galerkin scheme in combination with an approximation space that contains high-order polynomial modes as well as piece-wise constant modes on a sub-grid. The high-order modes can continuously be suppressed with a penalty function that is based on a sensor which is intertwined with the approximation space. Numerical tests finally illustrate the performance of this scheme.