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Convergence of Approximations to Stochastic Scalar Conservation Laws

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Abstract

We develop a general framework for the analysis of approximations to stochastic scalar conservation laws. Our aim is to prove, under minimal consistency properties and bounds, that such approximations are converging to the solution to a stochastic scalar conservation law. The weak probabilistic convergence mode is convergence in law, the most natural in this context. We use also a kinetic formulation and martingale methods. Our result is applied to the convergence of the finite volume method in the companion paper (Dotti and Vovelle in Convergence of the finite volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation, 2016).

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Author information

Authors and Affiliations

  1. Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453, Marseille, France

    Sylvain Dotti

  2. Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Julien Vovelle

Authors
  1. Sylvain Dotti
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  2. Julien Vovelle
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Correspondence to Julien Vovelle.

Additional information

Communicated by A. Bressan

Julien Vovelle was supported by ANR Projects STOSYMAP and STAB.

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Dotti, S., Vovelle, J. Convergence of Approximations to Stochastic Scalar Conservation Laws. Arch Rational Mech Anal 230, 539–591 (2018). https://doi.org/10.1007/s00205-018-1252-2

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