In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=\tau_k/\tau_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.

中文翻译：

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关于 Allen--Cahn 方程的变步长能量稳定、最大守恒、二阶 BDF 方案

在这项工作中，我们研究了 Allen-Cahn 方程的具有非均匀网格的两步向后微分公式 (BDF2)。我们表明非均匀 BDF2 方案在时间步长比率限制下是能量稳定的 $r_k:=\tau_k/\tau_{k-1}<(3+\sqrt{17})/2\approx3.561.$ 此外，通过开发一种新的核重组和互补技术，我们首次展示了 BDF2 方案在时间步长比限制 $r_k<1+\sqrt{2}\approx 2.414$ 下的离散最大值原理和一个实用的时间步长约束。还给出了最大范数中的二阶收敛速度。提供了数值实验来支持理论发现。