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Journal of Scientific Computing

, Volume 48, Issue 1–3, pp 274–295 | Cite as

On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws

  • María Luz Muñoz-Ruiz
  • Carlos Parés
Article

Abstract

This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balance laws. We consider these systems as a particular case of hyperbolic systems in nonconservative form, for which we use the theory introduced by Dal Maso, LeFloch and Murat (J. Math. Pures Appl. 74:483, 1995) in order to define the concept of weak solutions. This theory is based on the prescription of a family of paths in the phases space. We also consider path-conservative schemes, that were introduced in Parés (SIAM J. Numer. Anal. 44:300, 2006). The first goal is to prove a Lax-Wendroff type convergence theorem. In Castro et al. (J. Comput. Phys. 227:8107, 2008) it was shown that, for general nonconservative systems a rather strong convergence assumption is needed to prove such a result. Here, we prove that the same hypotheses used in the classical Lax-Wendroff theorem are enough to ensure the convergence in the particular case of systems of balance laws, as the numerical results shown in Castro et al. (J. Comput. Phys. 227:8107, 2008) seemed to suggest. Next, we study the relationship between the well-balanced properties of path-conservative schemes applied to systems of balance laws and the family of paths.

Keywords

Hyperbolic systems of balance laws Hyperbolic nonconservative systems Path-conservative schemes Convergence Well-balanced schemes 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. Matemática AplicadaUniversidad de MálagaMálagaSpain
  2. 2.Dept. Análisis MatemáticoUniversidad de MálagaMálagaSpain

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