Asymptotic stability of many numerical schemes for phase-field modeling

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.