In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with 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:在本文中,我们提出并分析一种线性二级数字方法,用一般流动性解决艾伦-卡恩方程式。拟议的完全分解办法,是在将一级和二级后向偏差公式与非统一时间步骤的时间近似和空间离散的中央限值差相结合的基础上精心设计的。通过使用内核再组合技术,在时间步骤和相邻时间步骤大小比率的某些轻度限制下使用离散的最大约束原则证明了拟议办法。此外,我们严格地为典型的常态流动案例得出离散的$H ⁇ 1}的误差估计数和能源稳定性,为一般流动案例提出“L ⁇ infty}$”的误差估计数。还进行了各种数字实验,以验证理论结果,并用时间适应战略展示拟议方法的性能。