In this paper, we propose a novel numerical approach to construct unconditionally energy stable schemes for the modified phase field crystal (MPFC) model. The new technique is based on the invariant energy quadratization (IEQ) method. The numerical schemes based on IEQ approach lead to time-dependent dense matrices, thus the fast Fourier transform (FFT) is difficult to be applied to solve the systems directly. By introducing a new auxiliary variable to replace the original one, we derive a novel equivalent MPFC system. A step-by-step solving approach, termed 3S-IEQ method is considered to solve this equivalent system. Some totally decoupled, linear and unconditional energy stable semi-implicit schemes based on 3S-IEQ method are very easy to construct. More importantly, the phase function ϕ and auxiliary variable η can be calculated step-by-step. Meanwhile, the proposed approach only needs to solve linear equation with constant coefficients which is easy to use FFT directly in calculation. Some numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

在本文中，我们提出了一种新颖的数值方法，用于为修正相场晶体（MPFC）模型构建无条件能量稳定方案。新技术基于不变能量平方（IEQ）方法。基于IEQ方法的数值方案导致了时间相关的稠密矩阵，因此难以将快速傅立叶变换（FFT）直接用于求解系统。通过引入一个新的辅助变量来代替原来的辅助变量，我们得出了一个新颖的等效MPFC系统。考虑使用称为3S-IEQ方法的逐步解决方法来解决该等效系统。基于3S-IEQ方法的一些完全解耦，线性和无条件的能量稳定半隐式方案非常容易构造。更重要的是，相位函数ϕ辅助变量η可以逐步计算。同时，所提出的方法只需要求解具有恒定系数的线性方程，这很容易直接在计算中使用FFT。一些数值模拟被证明来验证我们提出的方案的准确性和效率。