In the recent breakthrough work \cite{xu2023lack}, a rigorous numerical analysis was conducted on the numerical solution of a scalar ODE containing a cubic polynomial derived from the Allen-Cahn equation. It was found that only the implicit Euler method converge to the correct steady state for any given initial value $u_0$ under the unique solvability and energy stability. But all the other commonly used second-order numerical schemes exhibit sensitivity to initial conditions and may converge to an incorrect equilibrium state as $t_n\to\infty$. This indicates that energy stability may not be decisive for the long-term qualitative correctness of numerical solutions. We found that using another fundamental property of the solution, namely monotonicity instead of energy stability, is sufficient to ensure that many common numerical schemes converge to the correct equilibrium state. This leads us to introduce the critical step size constant $h^*=h^*(u_0,\epsilon)$ that ensures the monotonicity and unique solvability of the numerical solutions, where the scaling parameter $\epsilon \in(0,1)$. We prove that the implicit Euler scheme $h^*=h^*(\epsilon)$, which is independent of $u_0$ and only depends on $\epsilon$. Hence regardless of the initial value taken, the simulation can be guaranteed to be correct when $h<h^*$. But for various other numerical methods, no mater how small the step size $h$ is in advance, there will always be initial values that cause simulation errors. In fact, for these numerical methods, we prove that $\inf_{u_0\in \mathbb{R}}h^*(u_0,\epsilon)=0$. Various numerical experiments are used to confirm the theoretical analysis.

翻译：在近期突破性工作\cite{xu2023lack}中，研究者对源自Allen-Cahn方程的三次多项式标量ODE数值解进行了严格数值分析。研究发现，在唯一可解性与能量稳定性条件下，仅隐式Euler方法能对任意给定初值$u_0$收敛至正确稳态。而所有其他常用二阶数值格式均表现出对初值的敏感性，当$t_n\to\infty$时可能收敛至错误平衡态。这表明能量稳定性对数值解长期定性正确性可能不具有决定性。我们发现利用解的另一个基本性质——单调性替代能量稳定性，足以保证多种常见数值格式收敛至正确平衡态。由此我们引入临界步长常数$h^*=h^*(u_0,\epsilon)$以保证数值解的单调性与唯一可解性，其中尺度参数$\epsilon \in(0,1)$。我们证明隐式Euler格式的$h^*=h^*(\epsilon)$与$u_0$无关而仅依赖于$\epsilon$。因此无论取何初值，当$h<h^*$时均可保证模拟正确性。但对于其他各类数值方法，无论预先取多小的步长$h$，总存在导致模拟误差的初值。事实上，对于这些数值方法，我们证明$\inf_{u_0\in \mathbb{R}}h^*(u_0,\epsilon)=0$。本文通过多种数值实验验证了理论分析。