Numerically, classical and conservative Allen--Cahn equations have disparate requirements because their invariant properties are different -- namely, phase relaxation via a nonconservative evolution process and phase redistribution in a mean-preserving fashion. An energy-projected zero-mode correction operator on Fourier modes is developed for periodic Allen--Cahn phase field dynamics. In each step, it advances the field using a semi-implicit Fourier predictor, calculates the nonlinear residual defect of the field at its predicted state, obtains a regularization of the correction modes of the field, selects only an energy-admissible correction amplitude of the field, performs a smooth projection of the maximum bound of the field, and reestablishes the mean-preserving property of the field via a scalar zero-mode correction. The resulting propagator examines whether the long-time evolution of Allen--Cahn equations can preserve residual accuracy, dissipative property, admissible point-wise range, and mean-preserving property without the use of fully nonlinear update at each step. The numerical findings are two-dimensional checkpointing errors of the classical and conservative Allen--Cahn equations with double well free energy, mesh size $h=1/256$, and time step sizes $\dt=0.2$, $0.1$, and $0.05$ at time $t=5$, $20$, $100$, and $500$. The retained step size $\dt=0.1$ produces the least mean error and least worst-checkpointing error of both equations. Relative to $\dt=0.2$, it decreases the mean error of the classical equation by $43.54\%$ and that of the conservative equation by $46.01\%$. Relative to $\dt=0.05$, it decreases the mean error of the classical equation by $31.61\%$ and that of the conservative equation by $32.72\%$. The findings indicate that reliability in the long term arises from the combined effects of residual correction, energy descent, permissible range control, and zero mode recovery and not from nominal step size alone.

Allen--Cahn equation,conservative phase field,Fourier spectral method,energy stability,maximum bound principle,mass conservation,long-time propagation,materials morphology