We build an asymptotically compatible energy of the variable-step L2-$1_{\sigma}$ scheme for the time-fractional Allen-Cahn model with the Caputo's fractional derivative of order $\alpha\in(0,1)$, under a weak step-ratio constraint $\tau_k/\tau_{k-1}\geq r_{\star}(\alpha)$ for $k\ge2$, where $\tau_k$ is the $k$-th time-step size and $r_{\star}(\alpha)\in(0.3865,0.4037)$ for $\alpha\in(0,1)$. It provides a positive answer to the open problem in [J. Comput. Phys., 414:109473], and, to the best of our knowledge, it is the first second-order nonuniform time-stepping scheme to preserve both the maximum bound principle and the energy dissipation law of time-fractional Allen-Cahn model. The compatible discrete energy is constructed via a novel discrete gradient structure of the second-order L2-$1_{\sigma}$ formula by a local-nonlocal splitting technique. It splits the discrete fractional derivative into two parts: one is a local term analogue to the trapezoid rule of the first derivative and the other is a nonlocal summation analogue to the L1 formula of Caputo derivative. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our scheme and the asymptotic properties of the associated modified energy.

翻译：针对Caputo分数阶导数阶数$\alpha\in(0,1)$的时间分数阶Allen-Cahn模型，在弱步长比约束$\tau_k/\tau_{k-1}\geq r_{\star}(\alpha)$（$k\ge2$）下，我们构建了变步长L2-$1_{\sigma}$格式的渐近相容能量，其中$\tau_k$为第$k$个时间步长，且对于$\alpha\in(0,1)$有$r_{\star}(\alpha)\in(0.3865,0.4037)$。这为[J. Comput. Phys., 414:109473]中的开放问题提供了肯定回答，且据我们所知，这是首个同时保持时间分数阶Allen-Cahn模型最大值原理和能量耗散律的二阶非均匀时间步进格式。该相容离散能量通过二阶L2-$1_{\sigma}$公式的局部-非局部分裂技术构建出新型离散梯度结构，将离散分数阶导数分解为两部分：一部分是类似一阶导数梯形法则的局部项，另一部分是类似Caputo导数L1公式的非局部求和项。数值算例采用自适应时间步进策略，验证了该格式的有效性以及关联修正能量的渐近性质。