Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo’s derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time steps are always incorporated into our approaches so that a adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Grönwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.

中文翻译：

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分数阶Allen-Cahn方程的简单的最大原理保时分步方法

对于具有Caputo导数的时间分数次Allen-Cahn方程，建议了两种快速的L1时间步长方法，包括后向Euler和稳定的半隐式方案。时间网格在初始时间附近被精炼，以解决解决方案固有的初始奇异性，并且始终将不相等的时间步长合并到我们的方法中，以便可以在长时间仿真中使用自适应时间步长策略。结果表明，所提出的使用快速L1公式的方案保留了离散最大原理。通过应用离散分数Grönwall不等式和全局一致性分析，可以建立反映解决方案时间规律性的尖锐误差估计。数值实验表明了我们方法的有效性并证实了我们的分析。