In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete $H^{1}$ error estimate and energy stability for the classic constant mobility case and the $L^{\infty}$ error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy.

翻译：本文提出并分析了一种求解具有一般迁移率的Allen-Cahn方程的线性二阶数值方法。该方法通过结合一阶和二阶向后差分公式（采用非均匀时间步长进行时间逼近）与中心有限差分（用于空间离散），精心构建了全离散格式。利用核重组技术，在时间步长及相邻时间步长比满足特定温和约束条件下，证明了该格式的离散最大界原理。进一步，针对经典常数迁移率情形严格推导了离散$H^{1}$误差估计与能量稳定性，针对一般迁移率情形推导了$L^{\infty}$误差估计。通过多种数值实验验证了理论结果，并展示了采用时间自适应策略时该方法的性能。