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In this work, a novel artificial viscosity method is proposed using smoothand compactly supported viscosities. These are derived by revisiting the widelyused piecewise constant artificial viscosity method of Persson and Peraire aswell as the piecewise linear refinement of Kl\"ockner et al. with respect tothe fundamental design criteria of conservation and entropy stability. Furtherinvestigating the method of modal filtering in the process, it is demonstratedthat this strategy has inherent shortcomings, which are related to problems ofLegendre viscosities to handle shocks near element boundaries. This problem isovercome by introducing certain functions from the fields of robustreprojection and mollififers as viscosity distributions. To the best of ourknowledge, this is proposed for the first time in this work. The resulting$C_0^\infty$ artificial viscosity method is demonstrated to provide sharperprofiles, steeper gradients and a higher resolution of small-scale featureswhile still maintaining stability of the method.

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