The Allen-Cahn equation is a fundamental model for phase transitions, offering critical insights into the dynamics of interface evolution in various physical systems. This paper investigates the stability and robustness of frequently utilized time-discretization numerical schemes for solving the Allen-Cahn equation, with focuses on the Backward Euler, Crank-Nicolson (CN), convex splitting of modified CN, and Diagonally Implicit Runge-Kutta (DIRK) methods. Our stability analysis reveals that the Convex Splitting of the Modified CN scheme exhibits unconditional stability, allowing greater flexibility in time step selection, while the other schemes are conditionally stable. Additionally, our robustness analysis highlights that the Backward Euler method converges to correct physical solutions regardless of initial conditions. In contrast, the other methods studied in this work show sensitivity to initial conditions and may converge to incorrect physical solutions if the initial conditions are not carefully chosen. This study introduces a comprehensive approach to assessing stability and robustness in numerical methods for solving the Allen-Cahn equation, providing a new perspective for evaluating numerical techniques for general nonlinear differential equations.

翻译：艾伦-卡恩方程是研究相变现象的基础模型，为理解各类物理系统中界面演化的动力学行为提供了关键见解。本文针对求解艾伦-卡恩方程常用的时间离散化数值格式，重点研究了后向欧拉法、克兰克-尼科尔森（CN）法、修正CN凸分裂法以及对角隐式龙格-库塔（DIRK）法的稳定性与鲁棒性。稳定性分析表明，修正CN凸分裂格式具有无条件稳定性，在时间步长选取上具有更大灵活性，而其他格式均为条件稳定。鲁棒性分析进一步揭示，后向欧拉法对初始条件具有普适性，总能收敛至正确的物理解；相比之下，本文研究的其他方法对初始条件较为敏感，若初始条件选取不当可能收敛至错误的物理解。本研究提出了一种评估艾伦-卡恩方程数值解法稳定性与鲁棒性的综合分析方法，为评估一般非线性微分方程的数值技术提供了新视角。