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Finite-Volume approximation of the invariant measure of a viscous stochastic scalar conservation law

Sébastien Boyaval 1, 2 Sofiane Martel 3 Julien Reygner 4 
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
3 SIMSMART - SIMulation pARTiculaire de Modèles Stochastiques
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
Abstract : We study the numerical approximation of the invariant measure of a viscous scalar conservation law, one-dimensional and periodic in the space variable, and stochastically forced with a white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz continuous and to have at most polynomial growth. The numerical scheme we employ discretises the SPDE according to a finite-volume method in space, and a split-step backward Euler method in time. As a first result, we prove the well-posedness as well as the existence and uniqueness of an invariant measure for both the semi-discrete and the split-step scheme. Our main result is then the convergence of the invariant measures of the discrete approximations, as the space and time steps go to zero, towards the invariant measure of the SPDE, with respect to the second-order Wasserstein distance. We investigate rates of convergence theoretically, in the case where the flux function is globally Lipschitz continuous with a small Lipschitz constant, and numerically for the Burgers equation.
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Submitted on : Wednesday, May 26, 2021 - 2:24:25 PM
Last modification on : Wednesday, June 8, 2022 - 12:50:04 PM


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Sébastien Boyaval, Sofiane Martel, Julien Reygner. Finite-Volume approximation of the invariant measure of a viscous stochastic scalar conservation law. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2021, ⟨10.1093/imanum/drab049⟩. ⟨hal-02291253v3⟩



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