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Convergence of the Upwind Interface Source Method for Hyperbolic Conservation Laws

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Hyperbolic Problems: Theory, Numerics, Applications

Abstract

This paper deals with typical questions arising in the analysis of numerical approximations for scalar conservation laws with a source term. We focus our attention on semi-discrete finite volume schemes, in the general case of a nonuniform spatial mesh. To define appropriate discretizations of the source term, we introduce the formalism peculiar to the Upwind Interface Source method and we establish conditions on the numerical functions so that the discrete solver preserves the steady state solutions. Then we formulate a rigorous definition of consistency, adapted to the class of well-balanced schemes, for which we are able to prove a Lax-Wendroff type convergence theorem. Some examples of numerical methods are discussed, in order to validate the arguments we propose.

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Author information

Authors and Affiliations

  1. Département de Mathématiques et Applications, École Normale Supérieure, 45, rue d’Ulm, 75230, Paris Cedex 05, France

    Benoît Perthame & Chiara Simeoni

Authors
  1. Benoît Perthame
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  2. Chiara Simeoni
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Editor information

Editors and Affiliations

  1. Department of Applied and Computational Mathematics 217-50, California Institute of Technology, Pasadena, CA, 91125, USA

    Thomas Y. Hou

  2. Center for Scientific Computation and Mathematical Modeling (CSCAMM) Department of Mathematics, University of Maryland, College Park, MD, 20742-3289, USA

    Eitan Tadmor

  3. Institute for Physical Science & Technology (IPST), University of Maryland, College Park, MD, 20742-3289, USA

    Eitan Tadmor

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© 2003 Springer-Verlag Berlin Heidelberg

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Perthame, B., Simeoni, C. (2003). Convergence of the Upwind Interface Source Method for Hyperbolic Conservation Laws. In: Hou, T.Y., Tadmor, E. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55711-8_5

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  • DOI: https://doi.org/10.1007/978-3-642-55711-8_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62929-7

  • Online ISBN: 978-3-642-55711-8

  • eBook Packages: Springer Book Archive

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