随机Allen-Cahn方程半离散分裂逼近的Strong convergence速率
Strong convergence rates of semi-discrete splitting approximations for stochastic Allen--Cahn equation
摘要：
本文在空间有界区域中，在$d\le 3$维光滑边界上，分析了加性噪声驱动的随机Allen-Cahn方程的显式时间分裂数值格式.分裂策略与辅助问题的指数欧拉格式相结合。当$d=1$和驱动噪声是空时白噪声时，我们首先给出了这种分裂方案的一些先验估计。利用漂移非线性的单调性，我们证明了在对初始数据非常温和的假设下，该方案达到了最优的强收敛速度$\OO(δt^{\frac 14})$。当$d\le 3$和驱动噪声在空间上具有某种规律性时，我们研究了精确解和数值解的指数可积性。最后，在$d=1$维上，利用这些性质证明了分裂格式具有强收敛速度$\OO(δt)$。
英文摘要：
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 $\OO(δt)$.
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