This paper presents a high-precision numerical method based on spectral deferred correction (SDC) for solving the time-fractional Allen-Cahn equation. In the temporal direction, we establish a stabilized variable-step L1 semi-implicit scheme which satisfies the discrete variational energy dissipation law and the maximum principle. Through theoretical analysis, we prove that this numerical scheme is convergent and unconditionally stable. In the spatial direction, we apply the Fourier-Galerkin spectral method for discretization and conduct an error analysis of the fully discretized scheme. Since the stabilized variable-step L1 semi-implicit scheme is only of first-order accuracy in the time direction, to improve the accuracy, we combine explicit and implicit schemes (linear terms are handled implicitly, while nonlinear terms are handled explicitly) to establish a stabilized semi-implicit spectral deferred correction scheme. Finally, we verify the validity and feasibility of the numerical scheme through numerical examples.

中文翻译：

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一种基于谱延迟校正的高精度数值求解时间分数 Allen-Cahn 方程


该文提出了一种基于频谱延迟校正 （SDC） 的高精度数值方法，用于求解时间分数 Allen-Cahn 方程。在时间方向上，我们建立了一个稳定的变步长 L1 半隐式方案，它满足离散变分能量耗散定律和最大值原理。通过理论分析，我们证明了该数值方案是收敛的，并且是无条件稳定的。在空间方向上，我们应用 Fourier-Galerkin 谱法进行离散化，并对完全离散化方案进行误差分析。由于稳定的变步长 L1 半隐式方案在时间方向上仅具有一阶精度，为了提高精度，我们将显式和隐式方案（线性项是隐式处理的，而非线性项是显式处理的）相结合，建立了稳定的半隐式频谱延迟校正方案。最后，通过数值算例验证了该数值方案的有效性和可行性。