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

M.J. Castro (), J.M. Gallardo (), M.L. Muñoz () and C. Parés ()
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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

A chapter in Numerical Mathematics and Advanced Applications, 2006, pp 662-670 from Springer

Abstract: 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.

Keywords: Weak Solution; Hyperbolic System; Borel Measure; Riemann Problem; Integral Curve (search for similar items in EconPapers)
Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-34288-5_64

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DOI: 10.1007/978-3-540-34288-5_64

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