Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

• Published in: IMA journal of numerical analysis Vol. 39; no. 4; pp. 2096 - 2134
• Main Authors: Bréhier, Charles-Edouard, Cui, Jianbo, Hong, Jialin
• Format: Journal Article
• Language: English
• Published: Oxford University Press 16-10-2019
• Subjects: strong convergence rate; splitting scheme; exponential integrability; stochastic Allen–Cahn equation
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/dry052

Abstract
This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.