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

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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.

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Authors and Affiliations

  1. Dept. Matemática Aplicada, Universidad de Málaga, 29071, Málaga, Spain

    María Luz Muñoz-Ruiz

  2. Dept. Análisis Matemático, Universidad de Málaga, 29071, Málaga, Spain

    Carlos Parés

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  1. María Luz Muñoz-Ruiz
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  2. Carlos Parés
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Correspondence to María Luz Muñoz-Ruiz.

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Muñoz-Ruiz, M.L., Parés, C. On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws. J Sci Comput 48, 274–295 (2011). https://doi.org/10.1007/s10915-010-9425-7

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