A fully discrete approximation of the one-dimensional stochastic heat equation

TL;DR

This paper introduces a fully discrete numerical scheme for the one-dimensional stochastic heat equation driven by multiplicative white noise, combining finite differences in space with a stochastic exponential method in time, and proves convergence properties.

Contribution

It presents a novel explicit exponential time integrator that avoids CFL restrictions and provides error bounds and convergence results for the stochastic heat equation.

Findings

Error bounds in $L^q(\,Omega)$ for the scheme

Almost sure convergence of the numerical solution

Convergence in probability under non-globally Lipschitz conditions

Abstract

A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space-time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in $L^q(\Omega)$, for all $q\geq2$, improving some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of non-globally Lipschitz coefficients, we provide sufficient conditions under which the numerical solution converges in probability to the exact solution. Numerical experiments are presented to illustrate the theoretical results.