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.

翻译：能量耗散定律和最大值界原理是Allen-Cahn方程的重要特征。为了在离散层面保持这些性质，目标系统的线性部分通常采用隐式离散，从而产生大型线性或非线性方程组。快速傅里叶变换（FFT）算法常用于求解此类线性或非线性系统，每时间步计算复杂度为$\mathcal{O}(M^d \log M)$，其中$M$为每个方向空间网格点数，$d$为问题维度。本文结合Saul'yev方法和稳定化技术，提出并分析了Allen-Cahn方程的新型一阶和二阶数值格式。与传统方法相比，所提方法可逐分量求解，每时间步仅需$\mathcal{O}(M^d)$计算成本。此外，该格式在离散层面保持最大值界原理和原始能量耗散定律。我们同时给出了格式相容性和收敛性的严格分析。数值实验验证了理论分析结果并展示了所提方法的高效性。