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On a General Definition of the Godunov Method for Nonconservative Hyperbolic Systems. Application to Linear Balance Laws

In: Numerical Mathematics and Advanced Applications

Author

Listed:
  • M.J. Castro

    (Universidad de Málaga, Dpt. Análisis Matem ático, Facultad de Ciencias)

  • J.M. Gallardo

    (Universidad de Málaga, Dpt. Análisis Matem ático, Facultad de Ciencias)

  • M.L. Muñoz

    (Universidad de Málaga, Dpt. Matemática Aplicada, E.T.S.I. Telecomunicación)

  • C. Parés

    (Universidad de Málaga, Dpt. Análisis Matem ático, Facultad de Ciencias)

Abstract

This work is concerned with the numerical approximation of Cauchy problems for one-dimensional nonconservative hyperbolic systems, for which it is assumed that each characteristic field is either genuinely nonlinear or linearly degenerate. The theory developed by Dal Maso, LeFloch and Murat [1] is used to define the concept of weak solutions of these systems, giving a sense to nonconservative products as Borel measures, based on the choice of a family of paths in the phases space. We establish some basic hypotheses concerning this family of paths which ensure the fulfilling of some good properties for weak solutions. A family of paths satisfying these hypotheses can be constructed at least for states that are close enough. In particular, we prove that the choice of such a family allows to write the Godunov method for a nonconservative system in a simple and general manner. The previous results are applied to a linear balance law, for which the Godunov method can be explicitly written and easily implemented.

Suggested Citation

  • M.J. Castro & J.M. Gallardo & M.L. Muñoz & C. Parés, 2006. "On a General Definition of the Godunov Method for Nonconservative Hyperbolic Systems. Application to Linear Balance Laws," Springer Books, in: Alfredo Bermúdez de Castro & Dolores Gómez & Peregrina Quintela & Pilar Salgado (ed.), Numerical Mathematics and Advanced Applications, pages 662-670, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-34288-5_64
    DOI: 10.1007/978-3-540-34288-5_64
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