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

Author:

Boyaval Sébastien1,Martel Sofiane2,Reygner Julien3

Affiliation:

1. LHSV, Ecole des Ponts, CEREMA, EDF R et D, 78401 Chatou, France, and Inria Paris, Matherials, 75012 Paris, France

2. Inria Rennes–Bretagne Atlantique, SIMSMART, 35042, Rennes, France

3. CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée,France

Abstract

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 discretizes the stochastic partial differential equation (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 semidiscrete 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.

Funder

French National Research Agency

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Mathematics,General Mathematics

Reference32 articles.

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5. Approximation of the invariant measure with a Euler scheme for stochastic PDEs driven by space-time white noise;Bréhier;Potential Anal.,2014

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