Mathematical Modelling of Natural Phenomena

Research Article

Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method

A. Klöcknera1 c1, T. Warburtona2 and J. S. Hesthavena3

a1 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012

a2 Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005

a3 Division of Applied Mathematics, Brown University, Providence, RI 02912


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.

(Online publication May 16 2011)

Key Words:

  • shock detection;
  • Euler’s equations;
  • discontinuous Galerkin;
  • explicit time integration;
  • shock capturing;
  • artificial viscosity

Mathematics Subject Classification:

  • 65N30;
  • 65N35;
  • 65N40;
  • 35F61


c1 Corresponding author. E-mail: