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 Klöckner 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 mollifiers 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.
Highlights
In the last decades, great efforts have been made to develop accurate and stable numerical methods for time dependent partial differential equations (PDEs), especially for hyperbolic conservation laws
We extend the artificial viscosity method to the Euler equations of gas dynamics
In order to derive them, widely used artificial viscosity methods, such as the ones of Persson and Peraire as well as Klöckner et al, were analytically revisited with respect to the essential design criteria of conservation and entropy stability. It was proved for the viscosity extension that conservation carries over if the viscosity is continuous and compactly supported, while entropy stability already holds for positive viscosities
Summary
Great efforts have been made to develop accurate and stable numerical methods for time dependent partial differential equations (PDEs), especially for hyperbolic conservation laws. A novel artificial viscosity method is proposed for sub-cell shock capturing in Discontinuous Galerkin and related methods. This new artificial viscosity method, referred to as the C0∞ artificial viscosity method, essentially relies on the idea to replace commonly used viscosities in the artificial viscosity method [47,58] by certain weight functions from the field of robust reprojection [25] and mollifiers [66]. Summarizing the characteristic features of the proposed new artificial viscosity methods and discussing possible future applications
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