In this paper, we give a stabilized second order scheme for the time fractional Allen-Cahn equation. The scheme uses the fractional backward difference formula (FBDF) for the time fractional derivative and the Legendre spectral method for the space approximation. The nonlinear terms are treated implicitly with a second order stabilized term. Based on the fractional Grönwall inequality, we strictly prove that the proposed scheme converges to second order accuracy in time and spectral accuracy in space. To save computation time and storage, a fast evaluation is developed. Finally, we give some numerical examples to show the configurations of phase field evolution and verify the effectiveness of the proposed methods.

在本文中，我们给出了时间分数Allen-Cahn方程的稳定二阶格式。该方案将分数后向差分公式（FBDF）用于时间分数导数，将Legendre谱方法用于空间逼近。非线性项用二阶稳定项隐式处理。基于分数Grönwall不等式，我们严格证明了该方案收敛于时间的二阶精度和空间的频谱精度。为了节省计算时间和存储空间，开发了一种快速评估方法。最后，我们给出一些数值例子来说明相场演化的配置并验证所提出方法的有效性。