In this paper, a second-order numerical scheme for the time-fractional phase field models is proposed. In this scheme, the fractional backward difference formula is used to approximate the time-fractional derivative and the extended scalar auxiliary variable method is used to deal with the nonlinear terms. The energy dissipation property for the numerical scheme is proved. Our discussion includes the time-fractional Allen–Cahn equation, the time-fractional Cahn–Hilliard equation, and the time-fractional molecular beam epitaxy model. In the numerical implementation, a fast method based on a globally uniform approximation of the trapezoidal rule for the integral on the real line is adopted to decrease the memory requirement and computational cost. Finally, some numerical examples are given to confirm the effectiveness of the proposed methods.

本文提出了时间分数相场模型的二阶数值格式。在该方案中，使用分数后向差分公式来近似时间分数导数，并使用扩展的标量辅助变量方法来处理非线性项。证明了数值方案的耗能特性。我们的讨论包括时间分数Allen–Cahn方程，时间分数Cahn–Hilliard方程和时间分数分子束外延模型。在数值实现中，采用了一种基于梯形规则整体均匀逼近实线积分的快速方法，以减少存储需求和计算成本。最后，通过数值算例验证了所提方法的有效性。