Strong convergence rates of semi-discrete splitting approximations for stochastic Allen--Cahn equation

Abstract

This article analyzes 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 $d\le 3$. 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 $\OO(δt^{\frac 14})$. When $d\le 3$ 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 $\OO(δt)$.

arXiv comment33 pages

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Cited by in corpus (5)

• Strong Approximation of Stochastic Allen-Cahn Equation with White Noise
• Strong Convergence Rate of Splitting Schemes for Stochastic Nonlinear Schrödinger Equations
• Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
• Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations
• Error estimates of semi-discrete and fully discrete finite element methods for the Cahn-Hilliard-Cook equation