A Roe-like numerical method for weakly hyperbolic systems of equations in conservation and non-conservation form

Timothy A. Smith, David J. Petty, Carlos Pantano

Research output: Contribution to journalArticlepeer-review

Abstract

We describe a scheme for the numerical approximation of uniformly or locally weakly hyperbolic systems of equations in conservation and non-conservation forms. The strategy focuses on generalizations of the Roe scheme for non-conservation systems that can be considered as upstream differencing discretizations. The present work explicitly treats an advection matrix that is not diagonalizable by extending previous ideas that introduce a perturbation into the weakly hyperbolic problem to convert it to a nearby strongly hyperbolic problem. The perturbative expansion allows one to identify and remove the terms that would otherwise result in an apparent singularity and isolate the regular part of the numerical method. The performance of the method is showcased for uniformly and locally weakly hyperbolic systems in one and two dimensions. Generalization to higher dimensional systems can be performed by following the usual practice of dimension-by-dimension splitting.

Original languageEnglish (US)
Pages (from-to)117-138
Number of pages22
JournalJournal of Computational Physics
Volume316
DOIs
StatePublished - Jul 1 2016

Keywords

  • Non-conservation form
  • Repeated eigenvalues
  • Weakly hyperbolic systems

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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