SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3305-A3327, January 2021.
In this paper, we propose and analyze a time stepping method for the time fractional Allen--Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for this type of equation. The unconditional stability is proved through establishing a discrete nonlocal free energy dissipation law, which is also true for the continuous problem. The main idea used in the analysis is to split the time fractional derivative into two parts: a local part and a history part, which are discretized by the well known L1, L1-CN, and $L1^{+}$-CN schemes. Then an extended auxiliary variable approach is used to deal with the nonlinear and history term. The main contributions of the paper are first, it is found that the time fractional Allen--Cahn equation is a dissipative system related to a nonlocal free energy. Second, we construct efficient time stepping schemes satisfying the same dissipation law at the discrete level. In particular, we prove that the proposed schemes are unconditionally stable for quite general meshes. Finally, the efficiency of the proposed method is verified by a series of numerical experiments.

中文翻译：

---

时间分数阶 Allen--Cahn 方程的高效耗能方案

SIAM 科学计算杂志，第 43 卷，第 5 期，第 A3305-A3327 页，2021 年 1 月。
在本文中，我们提出并分析了时间分数阶 Allen-Cahn 方程的时间步长方法。所提出方法的关键特性是它对一般网格的无条件稳定性，包括常用于此类方程的分级网格。通过建立离散的非局部自由能耗散定律证明了无条件稳定性，对于连续问题也是如此。分析中使用的主要思想是将时间分数阶导数分成两部分：局部部分和历史部分，它们由众所周知的 L1、L1-CN 和 $L1^{+}$-CN 方案离散化. 然后使用扩展辅助变量方法处理非线性和历史项。该论文的主要贡献是首先，发现时间分数阶Allen-Cahn方程是一个与非局域自由能相关的耗散系统。其次，我们构建了有效的时间步长方案，在离散水平上满足相同的耗散规律。特别是，我们证明了所提出的方案对于非常一般的网格是无条件稳定的。最后，通过一系列数值实验验证了所提方法的有效性。