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Total oscillation diminishing property for scalar conservation laws

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Summary.

We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone Φ:ℝ→ℝ, we have that |Φ(u)| TV (ℝ) is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently in some numerical computations based on TVD schemes. We also show that semi-discrete Total Variation Diminishing finite volume schemes are TOD and that the fully discrete Godunov scheme is TOD.

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Authors and Affiliations

  1. Département de Mathématiques et Applications, École Normale Supérieure, CNRS UMR 8553, 45 rue d’Ulm, 75230, Paris cedex 05, France

    Benoît Perthame

  2. Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstraße 10, 53115, Bonn

    Michael Westdickenberg

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  1. Benoît Perthame
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  2. Michael Westdickenberg
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Correspondence to Michael Westdickenberg.

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Mathematics Subject Classification (2000): 35L65, 35K55, 65M20

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Perthame, B., Westdickenberg, M. Total oscillation diminishing property for scalar conservation laws. Numer. Math. 100, 331–349 (2005). https://doi.org/10.1007/s00211-005-0602-9

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  • DOI: https://doi.org/10.1007/s00211-005-0602-9

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