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

Abstract

We aim to give a numerical approximation of the invariant measure of a viscous scalar conservation law, onedimensional 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 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 spatial semi-discretisation and the fully discrete 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. A few numerical experiments are performed to illustrate these results.

Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

“…Let us review the existing literature concerning the numerical approximation of invariant distributions for parabolic semilinear SPDEs -see also the preprint [2] where stochastic viscous scalar conservation laws are considered, and the monograph [16] and references therein where some stochastic Schrödinger equations are studied. In [3], parabolic semilinear SPDEs, with Lipschitz nonlinearity, driven by space-time white noise, are considered; temporal discretization is performed using a linear implicit Euler scheme, and weak error estimates which are uniform in time are obtained using a Kolmogorov equation approach.…”