AI 中文总结
本文针对Allen-Cahn方程，提出基于完整残差的逐点单调性判据，证明精确流与全隐式欧拉法保持该性质，并揭示多种能量稳定格式在大步长下产生错误符号增量，补充了能量稳定性与最大界保持的动力学局限性。
AI 中文摘要
能量稳定性是相场模型数值逼近中的基本要求，但它本身并不能保证对局部动力学的忠实再现。最近的常微分方程层面的研究表明，仅凭能量耗散并不能确保空间均匀相场模型的动力学保真度（Xu and Xu, 2023; Li and Wang, 2026）。然而，这些常微分方程论证并不能直接推广到空间非均匀的 Allen--Cahn 解：扩散项进入完整残差 $\mathcal R_\varepsilon(u):=\Delta u+\varepsilon^{-2}(u-u^3)$，即使当 $0<u\leq1$ 时，也可能改变其逐点符号。受这一区别的启发，我们引入了由完整 Allen--Cahn 残差决定的符号依赖的容许类，并证明了精确偏微分方程流保持相应的逐点单调增长或单调衰减方向。全隐式欧拉方法在其标准的唯一可解性区域内继承了这一结构。随后，我们考察了几种广泛使用的能量稳定格式。一阶凸分裂和足够强的一阶稳定化可以保持正确方向，但会在缩小的有效时间尺度上，导致人为的延迟或阻尼。相比之下，二阶凸分裂、稳定化 CN/AB、IEQ 和 SAV 格式允许存在单调数据，对于这些数据，大步长会产生错误符号的逐点增量，尽管原始能量或修正能量在耗散。对于所有四种格式，这些严格反转在足够小的光滑空间非均匀容许扰动下仍然存在，因此反例在扩散真正活跃的情况下依然有效。数值实验证实了这一分类。因此，逐点单调性是能量稳定性和最大界保持之外的补充性局部动力学判据。
英文摘要
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.
