As a variational phase-field model, the time-fractional Allen-Cahn (TFAC) equation enjoys the maximum bound principle (MBP) and a variational energy dissipation law. In this work, we develop and analyze linear, structure-preserving time-stepping schemes for TFAC, including first-order and $\min\{1+\alpha, 2-\alpha\}$-order L1 discretizations, together with fast implementations based on the sum-of-exponentials (SOE) technique. A central feature of the proposed linear schemes is their unconditional preservation of both the discrete MBP and the variational energy dissipation law on general temporal meshes, including graded meshes commonly used for these problems. Leveraging the MBP of the numerical solutions, we establish sharp error estimates by employing the time-fractional Gro\"nwall inequality. Finally, numerical experiments validate the theoretical results and demonstrate the effectiveness of the proposed schemes with an adaptive time-stepping strategy.

翻译：作为一类变分相场模型，时间分数阶Allen-Cahn（TFAC）方程满足最大值原理（MBP）及变分能量耗散律。本文针对TFAC方程，发展并分析了一类线性保结构时间步进格式，包括一阶及$\min\{1+\alpha, 2-\alpha\}$阶L1离散格式，并基于指数和（SOE）技术实现了快速计算。所提线性格式的核心特征在于，其在一般时间网格（包括此类问题中常用的分级网格）上无条件保持离散MBP与变分能量耗散律。借助数值解的最大值原理，我们通过应用时间分数阶Grönwall不等式建立了严格的误差估计。最后，数值实验验证了理论结果，并展示了所提格式结合自适应时间步进策略的有效性。