On the discretisation in time of the stochastic Allen-Cahn equation
Provides rigorous convergence analysis for time discretization of a stochastic PDE with non-Lipschitz nonlinearity, which is important for numerical simulations in materials science.
The authors prove strong convergence with rate O(Δt^{1/2}) for an Euler split-step method and the backward Euler scheme applied to the stochastic Allen-Cahn equation with additive noise in dimensions d≤3.
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(Δt^{\frac12}) $. By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.