Viscous stabilization of discontinuous Galerkin solutions of hyperbolic conservation laws

Authors:
Jianguo Xin;Joseph E. Flaherty
Affiliations:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180, USA;Scientific Computation Research Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
Venue:
Applied Numerical Mathematics
Year:
2006

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Abstract

We use an artificial viscosity term to stabilize discontinuous Galerkin solutions of hyperbolic conservation laws in the presence of discontinuities. Viscous coefficients are selected to minimize spurious oscillations when a kinematic wave equation is subjected to piecewise constant initial data. The same strategy is used with a local linearization in more complex situations. Several one and two-dimensional flow problems illustrate performance. A shock detection scheme [L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, J.E. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math. 48 (2004) 323-338] further sharpens results near discontinuities.