论文标题
具有一般移动性的Allen-CAHN方程的线性二阶最大结合原理BDF方案
A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility
论文作者
论文摘要
在本文中，我们提出并分析了一种线性二阶数值方法，用于求解具有一般迁移率的艾伦-CAHN方程。根据一阶和二阶向后分化公式与时间近似和二阶向后分化公式的组合仔细构建了所提出的全差异方案，以进行时间近似和空间离散的中心有限差。通过在某些轻度约束下对相邻时间步长大小的时间步骤和比率，使用内核重组技术证明了拟议方案的离散最大结合原理。此外，我们严格地得出了经典恒定移动性案例的离散$ h^{1} $错误估计和能量稳定性，以及$ l^{\ infty} $通用移动性案例的错误估计。还提出了各种数值实验，以验证理论结果并通过时间自适应策略证明所提出的方法的性能。
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.
