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Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise

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Abstract

We study here the discretization by monotone finite volume schemes of multi-dimensional nonlinear scalar conservation laws forced by a multiplicative noise with a time and space dependent flux-function and a given initial data in \(L^{2}(\mathbb {R}^d)\). After establishing the well-posedness theory for solutions of such kind of stochastic problems, we prove under a stability condition on the time step the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation.

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Notes

  1. We consider that \(u^{\Delta t}(s)=u_0\) if \(s<0\).

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Acknowledgments

This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A\(^*\)MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR).

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Authors and Affiliations

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

    C. Bauzet, J. Charrier & T. Gallouët

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  1. C. Bauzet
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  2. J. Charrier
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  3. T. Gallouët
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Correspondence to J. Charrier.

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Bauzet, C., Charrier, J. & Gallouët, T. Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise. Stoch PDE: Anal Comp 4, 150–223 (2016). https://doi.org/10.1007/s40072-015-0052-z

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