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 |$\mathcal{O}(\delta t)$|⁠.

中文翻译：

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随机Allen-Cahn方程的半离散分裂逼近的强收敛速度

本文分析了随机Allen-Cahn方程的显式时间分裂数值格式，该方程由加性噪声驱动，边界空间域的边界为| $ d \ leqslant 3 $ |⁠。分裂策略与辅助问题的指数欧拉方案结合。当| $ d = 1 $ | 而驱动噪声是时空白噪声，我们首先显示对该拆分方案的一些先验估计。然后使用漂移非线性的单调性证明，在对初始数据的非常温和的假设下，该方案可实现最佳的强收敛速度| $ \ mathcal {O}（\ delta t ^ {\ frac 14}）$ |⁠。当| $ d \ leqslant 3 $ |并且驱动噪声在空间中具有一定规律性，我们研究精确解和数值解的指数可积性。最后，在维| $ d = 1 $ |⁠中，这些属性用于证明拆分方案具有很强的收敛速度| $ \ mathcal {O}（\ delta t）$ |⁠。