Aspects of discontinuous Galerkin methods for hyperbolic conservation laws

  • Authors:

  • Joseph E. Flaherty;Lilia Krivodonova;Jean-Francois Remacle;Mark S. Shephard

  • Affiliations:

  • Scientific Computation Research Center, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, NY;Scientific Computation Research Center, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, NY;Scientific Computation Research Center, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, NY;Scientific Computation Research Center, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, NY

  • Venue:
  • Finite Elements in Analysis and Design - Robert J. Melosh medal competition
  • Year:
  • 2002

Quantified Score

Hi-index 0.02

Visualization

Abstract

We review several properties of the discontinuous Galerkin method for solving hyperbolic systems of conservation laws including basis construction, flux evaluation, solution limiting, adaptivity, and a posteriori error estimation. Regarding error estimation, we show that the leading term of the spatial discretization error using the discontinuous Galerkin method with degree p piecewise polynomials is proportional to a linear combination of orthogonal polynomials on each element of degrees p and p+1. These are Radau polynomials in one dimension. The discretization errors have a stronger superconvergence of order O(h2p+1), where h is a mesh-spacing parameter, at the outflow boundary of each element. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors in regions where solutions are smooth.We present the results of applying the discontinuous Galerkin method to unsteady, two-dimensional, compressible, inviscid flow problems. These include adaptive computations of Mach reflection and mixing-instability problems.