Pointwise Monotonicity of the Allen--Cahn Flow and Dynamical Limitations of Energy-Stable Schemes

Abstract

Energy stability is fundamental in the numerical approximation of phase-field models, but it does not by itself guarantee faithful reproduction of local dynamics. Recent ODE-level studies have shown that energy dissipation alone does not ensure dynamical fidelity for spatially homogeneous phase-field models (Xu and Xu, 2023; Li and Wang, 2026). However, these ODE arguments do not extend directly to spatially inhomogeneous Allen--Cahn solutions: diffusion enters the full residual $\mathcal R_\varepsilon(u):=Δu+\varepsilon^{-2}(u-u^3)$ and may change its pointwise sign even when $0<u\leq1$. Motivated by this distinction, we introduce sign-dependent admissible classes determined by the full Allen--Cahn residual and prove that the exact PDE flow preserves the corresponding pointwise monotone-growth or monotone-decay direction. The fully implicit Euler method inherits this structure in its standard unique-solvability regime. We then examine several widely used energy-stable schemes. First-order convex splitting and sufficiently strong first-order stabilization can preserve the correct direction, but on a reduced effective time scale, causing artificial delay or damping. By contrast, second-order convex splitting, stabilized CN/AB, IEQ, and SAV schemes admit monotone data for which large time steps generate wrong-signed pointwise increments despite dissipation of the original or a modified energy. For all four schemes, these strict reversals persist under sufficiently small smooth spatially nonhomogeneous admissible perturbations, so the counterexamples remain valid with genuinely active diffusion. Numerical experiments confirm this classification. Thus, pointwise monotonicity is a local dynamical criterion complementary to energy stability and maximum-bound preservation.