It is known from Beccari et al. (2019) that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen–Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration to control numerical solutions from instability. The a priori estimates in $\mathscr{C}(\mathscr{O})$-norm and $\dot{H}^{\beta }(\mathscr{O})$-norm of numerical solutions are established provided the adaptive timestep function is suitably bounded, which plays a key role in the convergence analysis. We show that the adaptive time-stepping schemes converge strongly with order $\frac{\beta }{2}$ in time and $\frac{\beta }{d}$ in space with $d$ ($d=1,2,3$) being the dimension and $\beta \in (0,2]$. Numerical experiments show that the adaptive time-stepping schemes are simple to implement and at a lower computational cost than a scheme with the uniform timestep.

中文翻译：

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随机 Allen–Cahn 方程的自适应时间步进方案的强收敛性

从 Beccari 等人得知。 (2019) 具有统一时间步长的标准显式欧拉型方案（例如指数欧拉和线性隐式欧拉方案）虽然计算效率高，但对于随机 Allen–Cahn 方程可能会发散。为了克服这种发散，本文提出并分析了自适应时间步长方案，该方案在每次迭代时调整时间步长以控制数值解的不稳定。数值解的 $\mathscr{C}(\mathscr{O})$-范数和 $\dot{H}^{\beta }(\mathscr{O})$-范数的先验估计成立，前提是自适应时间步函数有适当的有界，这在收敛分析中起着关键作用。我们表明，自适应时间步进方案在时间上以 $\frac{\beta }{2}$ 的阶数强烈收敛，在空间上以 $\frac{\beta }{d}$ 的阶数强烈收敛，$d$ ($d=1, 2,3$) 为维度，$\beta \in (0,2]$。数值实验表明，自适应时间步长方案实现起来很简单，并且比统一时间步长的方案计算成本更低。