Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method

¹ Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
² Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005
³ Division of Applied Mathematics, Brown University, Providence, RI 02912

^⋆ Corresponding author. E-mail: kloeckner@cims.nyu.edu

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.