This study addresses numerical stability challenges in cubic polynomial ODEs derived from the Allen-Cahn equation. In the recent breakthrough work [29], it was found that only the implicit Euler method converges 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 \(t_n\rightarrow \infty \) . We reveal that energy stability alone cannot guarantee long-term solution accuracy. Through a monotonicity-based analysis framework, we establish conditions for correct equilibrium convergence across common numerical schemes. Our key innovation introduces a critical step size \(h^* = h^*(u_0, \epsilon )\) that ensures solution monotonicity and unique solvability, where \(\epsilon \in (0,1]\) is a scaling parameter. We prove that (i) Universal positivity: \(h^* > 0\) for any initial value \(u_0 \in \mathbb {R}\) ; (ii) Implicit Euler: \(h^*\) depends solely on \(\epsilon \) , ensuring universal convergence when \(h < h^*\) and other methods: \(\inf _{u_0\in \mathbb {R}}h^*(u_0,\epsilon )=0\) , making error-free simulation impossible for certain initial values regardless of step size; (iii) In the six numerical methods examined in this note, the numerical solution exhibits monotonicity without crossing equilibrium, it can satisfy energy stability. However, the converse is not necessarily true. Numerical experiments confirm our theoretical framework. The results establish monotonicity as a fundamental principle for designing reliable nonlinear ODE solvers, with particular significance for phase-field simulations.
