On the discretisation in time of the stochastic Allen-Cahn equation

TL;DR

This paper analyzes the time discretization of the stochastic Allen-Cahn equation with Gaussian noise, proving strong convergence rates for Euler-type methods in spatial dimensions up to three.

Contribution

It establishes the strong convergence rate of $O( frac{1}{2})$ for Euler-type split-step and backward Euler schemes applied to the stochastic Allen-Cahn equation.

Findings

Euler split-step method converges strongly with rate $O( frac{1}{2})$

Backward Euler scheme also converges strongly with the same rate

Results hold for spatial dimensions up to three

Abstract

We consider the stochastic Allen--Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$, and study the semidiscretisation in time of the equation by an Euler type split-step method. We show that the method converges strongly with a rate $O(\Delta t^{\frac12}) $. By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.