  The energy dissipation law and maximum bound principle are significant
characteristics of the Allen-Chan equation. To preserve discrete counterpart of
these properties, the linear part of the target system is usually discretized
implicitly, resulting in a large linear or nonlinear system of equations. The
Fast Fourier Transform (FFT) algorithm is commonly used to solve the resulting
linear or nonlinear systems with computational costs of $\mathcal{O}(M^d log
M)$ at each time step, where $M$ is the number of spatial grid points in each
direction, and $d$ is the dimension of the problem. Combining the Saul'yev
methods and the stabilized technique, we propose and analyze novel first- and
second-order numerical schemes for the Allen-Cahn equation in this paper. In
contrast to the traditional methods, the proposed methods can be solved by
components, requiring only $\mathcal{O}(M^d)$ computational costs per time
step. Additionally, they preserve the maximum bound principle and original
energy dissipation law at the discrete level. We also propose rigorous analysis
of their consistency and convergence. Numerical experiments are conducted to
confirm the theoretical analysis and demonstrate the efficiency of the proposed
methods.

Numerical Analysis;Numerical Analysis