Numerical approximation of the invariant distribution for a class of stochastic damped wave equations

TL;DR

This paper develops and analyzes numerical methods to approximate the invariant distribution of stochastic damped wave equations, providing error estimates and establishing the convergence of these schemes.

Contribution

It introduces the first numerical schemes with proven convergence for invariant distributions of stochastic damped wave equations.

Findings

Numerical schemes admit unique invariant distributions.

Error estimates with convergence orders are established.

First known results for numerical approximation of invariant distributions in this context.

Abstract

We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.