In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time-fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist of (1) constructing first- and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; (2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time-fractional Allen-Cahn equation. In particular, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials.

在本文中，我们提出并分析了时间分数次Allen-Cahn方程的高阶有效格式。所提出的方案基于时间分数导数的L1离散化和最近开发的用于处理方程中非线性项的扩展标量辅助变量（SAV）方法。本文的主要贡献包括：（1）为不同网格类型构造一阶和高阶无条件稳定方案，并证明所构造的均匀网格方案的无条件稳定性；（2）进行数值实验，以验证方案的有效性，并研究由时间分数Allen-Cahn方程控制的粗化动力学。特别是，仔细检查了分数阶对粗化行为的影响。