A fully discrete approximation of the one-dimensional stochastic heat equation
Journal article, 2020

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(\varOmega) $, for all $q\geqslant 2$, 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 nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

finite difference scheme

Lq(ω)-convergence

multiplicative noise

stochastic heat equation

stochastic exponential integrator

Author

Rikard Anton

Umeå University

David Cohen

Umeå University

Lluís Quer-Sardanyons

Universitat Autonoma de Barcelona (UAB)

IMA Journal of Numerical Analysis

0272-4979 (ISSN) 1464-3642 (eISSN)

Vol. 40 1 247-284

Subject Categories

Mathematics

Computational Mathematics

DOI

10.1093/imanum/dry060

More information

Latest update

2/18/2021