This paper develops three linear and energy-stable schemes for a modified phase field crystal model with a strong nonlinear vacancy potential (VMPFC model). This sixth-order phase-field model enables realistic crystal growth simulation. Starting from a Crank-Nicolson scheme based on the stabilized-SAV (S-SAV) method, we optimize it via the generalized positive auxiliary variable (GPAV) and modified exponential scalar auxiliary variable (ESAV) methods, thereby reducing computational complexity or eliminating the requirement for the nonlinear free energy potential to be bounded from below. The newly developed Energy-Variation Moving Average (EV-MA) adaptive time-stepping strategy resolves numerical instabilities and mitigates the high parameter sensitivity of the conventional adaptive time algorithm during rapid energy decay in the strongly nonlinear system. Unlike conventional instantaneous energy-derivative monitors, the EV-MA technique incorporates a moving average of the energy variation. Additionally, the rate of change between adjacent time steps is constrained by a maximum change factor. This design effectively dampens spurious oscillations and enhances the robustness of time step selection. Extensive numerical experiments are conducted to validate the accuracy and energy stability of the proposed schemes. The EV-MA strategy is also demonstrated to perform robustly across a wide range of parameters.

翻译：本文针对具有强非线性空位势的修正相场晶体模型（VMPFC模型）提出了三种线性且能量稳定的数值格式。该六阶相场模型能够实现真实的晶体生长模拟。从基于稳定标量辅助变量（S-SAV）法的Crank-Nicolson格式出发，我们通过广义正辅助变量（GPAV）法和修正指数标量辅助变量（ESAV）法对其进行优化，从而降低了计算复杂度或消除了非线性自由能势需有下界的要求。新开发的能量变分移动平均（EV-MA）自适应时间步进策略解决了数值不稳定性问题，并缓解了传统自适应时间算法在强非线性系统能量快速衰减期间的高参数敏感性。与传统的瞬时能量导数监测器不同，EV-MA技术引入了能量变分的移动平均。此外，相邻时间步之间的变化率受到最大变化因子的约束。这种设计有效地抑制了寄生振荡，增强了时间步长选择的鲁棒性。我们进行了大量的数值实验以验证所提格式的精度和能量稳定性。EV-MA策略也被证明能在广泛的参数范围内稳健运行。