Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method

TL;DR

This paper introduces a new cell-local shock detector for discontinuous Galerkin methods, enabling effective artificial viscosity application for capturing shocks in nonlinear conservation laws with validated performance on benchmarks.

Contribution

It presents a novel shock detector and a systematic approach to integrate artificial viscosity into DG schemes, improving shock capturing capabilities.

Findings

Effective shock detection across benchmark problems

Reliable scaling and smoothing for local viscosity application

Enhanced shock capturing in nonlinear conservation law solutions

Abstract

We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechanism. Using an artificial viscosity in the latter role, we obtain a DG scheme for the numerical solution of nonlinear systems of conservation laws. Building on work by Persson and Peraire, we thoroughly justify the detector's design and analyze its performance on a number of benchmark problems. We further explain the scaling and smoothing steps necessary to turn the output of the detector into a local, artificial viscosity. We close by providing an extensive array of numerical tests of the detector in use.