A high-precision numerical method based on spectral deferred correction for solving the time-fractional Allen-Cahn equation

Abstract

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.