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

Mihály Kovács, Stig Larsson, Fredrik Lindgren

Published 2015-10-13Version 1

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^{\gamma}) $ for any $\gamma<\frac12$. By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.