An adaptive time-stepping strategy for the modified phase field crystal model with a strong nonlinear vacancy potential

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.