Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

Abstract

Strong convergence rates for fuly discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities, such as the stochastic Allen–Cahn equation with space-time white noise, are shown. The obtained strong rates of convergence are essentially sharp.

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“…They are also an important class of nonlinear stochastic partial differential equations from a mathematical perspective. Recently, researchers in the field of numerical analysis have shown considerable interest in this topic; see [1,2,4,5,8,12,19,20,22,23,24,32] and the references therein. We briefly summarize some closely related works of numerical analysis of three-dimensional stochastic Allen-Cahn equations as follows.…”

“…• For the model problem (1) with a non-smooth initial value v ∈ L ∞ , we prove that it admits a unique mild solution and derive some regularity estimates. The main tools are as follows: the stochastic integration and Itô's formula in UMD spaces and the stochastic maximal L p -regularity estimate developed in [7,30,31]; the theory of stochastic partial differential equations with Lyapunov condition in [17,18]; the maximal L p -regularity estimate of the deterministic parabolic equations (see [33]).…”

“…For the stochastic Allen-Cahn equation with additive noise, Becker and Jentzen [3] obtained strong convergence rates for a nonlinearitytruncated Euler-type approximation. Becker et al [2] established sharp strong convergence rates for an explicit space-time discrete numerical approximation. Bréhier et al [4] analyzed an explicit temporal splitting numerical scheme.…”