Smooth and compactly supported viscous sub-cell shock capturing for Discontinuous Galerkin methods

TL;DR

This paper introduces a new smooth, compactly supported artificial viscosity technique for Discontinuous Galerkin methods, enhancing shock capturing with sharper profiles and better small-scale feature resolution while ensuring stability.

Contribution

The work develops a novel $C_0^inity$ artificial viscosity approach using functions from robust reprojection and mollifiers, improving shock resolution and stability in DG methods.

Findings

Sharper shock profiles achieved

Enhanced resolution of small-scale features

Maintains stability with improved viscosity functions

Abstract

In this work, a novel artificial viscosity method is proposed using smooth and compactly supported viscosities. These are derived by revisiting the widely used piecewise constant artificial viscosity method of Persson and Peraire as well as the piecewise linear refinement of Kl\"ockner et al. with respect to the fundamental design criteria of conservation and entropy stability. Further investigating the method of modal filtering in the process, it is demonstrated that this strategy has inherent shortcomings, which are related to problems of Legendre viscosities to handle shocks near element boundaries. This problem is overcome by introducing certain functions from the fields of robust reprojection and mollififers as viscosity distributions. To the best of our knowledge, this is proposed for the first time in this work. The resulting $C_0^\infty$ artificial viscosity method is demonstrated to provide sharper profiles, steeper gradients and a higher resolution of small-scale features while still maintaining stability of the method.