Stability-Window-Guided Energy-Monitored Solvers for Allen--Cahn and Cahn--Hilliard Phase-Field Dynamics

Phase-field simulations require time integrators that are simultaneously robust for stiff diffuse interfaces, faithful to energy and mass laws, and computationally efficient across parameter regimes. We develop a stability-window-guided phase-field simulation framework for Allen--Cahn and Cahn--Hilliard dynamics. The method combines Fourier pseudospectral discretization, semi-implicit baseline updates, convex-splitting energy-stable updates, and a computable window diagnostic that estimates when low-cost iterations are expected and when a robust nonlinear solver or time-step reduction should be used. The diagnostic is evaluated through physical observables: free-energy decay, mass conservation, interface-length proxies, droplet relaxation, spinodal decomposition, nonlinear iteration counts, and time-step/interface-width risk maps. For Allen--Cahn, we prove and test a convex-splitting update with unconditional discrete energy decrease. For Cahn--Hilliard, we prove and test a convex-splitting update with simultaneous energy decrease and discrete mass conservation. The numerical experiments show that the window diagnostic provides useful solver guidance while the energy-stable updates preserve the central phase-field dissipation structures in the tested regimes.