In this paper, a linear second order numerical scheme is developed and investigated for the Allen-Cahn equation with a general positive mobility. In particular, our fully discrete scheme is mainly constructed based on the Crank-Nicolson formula for temporal discretization and the central finite difference method for spatial approximation, and two extra stabilizing terms are also introduced for the purpose of improving numerical stability. The proposed scheme is shown to unconditionally preserve the maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stability simultaneously. With the help of uniform boundedness of the numerical solutions due to MBP, we then successfully derive $H^{1}$-norm and $L^{\infty}$-norm error estimates for the Allen-Cahn equation with a constant and a variable mobility, respectively. Moreover, the energy stability of the proposed scheme is also obtained in the sense that the discrete free energy is uniformly bounded by the one at the initial time plus a {\color{black}constant}. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the performance of the proposed scheme with a time adaptive strategy.

翻译：本文针对具有一般正迁移率的Allen-Cahn方程，提出并研究了一种线性二阶数值格式。具体而言，本文的全离散格式主要基于时间离散的Crank-Nicolson公式和空间逼近的中心有限差分法构建，并引入了两个额外稳定项以提高数值稳定性。在稳定化参数的温和限制下，该格式被证明能无条件保持最大界原理（MBP），这对于同时实现良好精度与稳定性具有实际重要性。借助MBP导致的数值解一致有界性，我们成功推导了常数迁移率和变迁移率Allen-Cahn方程的$H^{1}$范数与$L^{\infty}$范数误差估计。此外，该格式的能量稳定性也得到了证明，即离散自由能一致有界于初始时刻的自由能加上一个{\color{black}常数}。最后，通过数值实验验证了理论结果，并展示了该格式结合时间自适应策略时的性能表现。