It is known in \cite{beccari} 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 \textit{a priori} estimates in $\mathcal {C}(\mathcal {O})$-norm and $\dot{H}^{\beta}(\mathcal{O})$-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.

翻译：文献\cite{beccari}指出，标准显式欧拉型格式（如指数欧拉格式和线性隐式欧拉格式）虽在计算上高效，但使用均匀时间步长时可能导致随机Allen–Cahn方程的解发散。为克服这一发散问题，本文提出并分析了自适应时间步进方案，该方案在每次迭代中调整时间步长以控制数值解的不稳定性。在自适应时间步长函数适当有界的条件下，建立了数值解在$\mathcal{C}(\mathcal{O})$范数和$\dot{H}^{\beta}(\mathcal{O})$范数下的先验估计，这在收敛性分析中起关键作用。我们证明了自适应时间步进方案在时间方向上具有$\frac{\beta}{2}$阶强收敛性，在空间方向上具有$\frac{\beta}{d}$阶强收敛性，其中$d$（$d=1,2,3$）为空间维数，$\beta\in(0,2]$。数值实验表明，自适应时间步进方案实现简单，且计算成本低于具有均匀时间步长的方案。