In this paper, a non-uniform time-stepping convex-splitting numerical algorithm for solving the widely used time-fractional Cahn–Hilliard equation is introduced. The proposed numerical scheme employs the $L{1}^{+}$ formula for discretizing the time-fractional derivative and a second-order convex-splitting technique to deal with the non-linear term semi-implicitly. Then the pseudospectral method is utilized for spatial discretization. As a result, the fully discrete scheme has several advantages: second-order accurate in time, spectrally accurate in space, uniquely solvable, mass preserving, and unconditionally energy stable. Rigorous proofs are given, along with several numerical results to verify the theoretical results, and to show the accuracy and effectiveness of the proposed scheme. Also, some interesting phase separation dynamics of the time-fractional Cahn–Hilliard equation has been investigated.

本文介绍了一种求解时间分数维Cahn-Hilliard方程的非均匀时步凸分解数值算法。拟议的数值方案采用$\mathrm{大号}{\mathrm{1个}}^{+}$离散时间分数导数的公式和二阶凸分解技术半隐式地处理非线性项。然后将伪谱方法用于空间离散化。结果，完全离散的方案具有几个优点：时间精确的二阶，空间光谱精确的，唯一可解的，质量守恒的和无条件的能量稳定的。给出了严格的证明，并提供了一些数值结果来验证理论结果，并证明所提方案的准确性和有效性。此外，还研究了时间分数Cahn-Hilliard方程的一些有趣的相分离动力学。