The Swift-Hohenberg model is a very important phase field crystal model which can be described many crystal phenomena. This model with quadratic-cubic nonlinearity based on the H^− 1-gradient flow approach is a sixth-order system which satisfies mass conservation and energy dissipation law. The negative energy of this model will bring huge difficulties to energy stability for many existing approaches. In this paper, we consider two linear, second-order and unconditionally energy stable schemes by linear invariant energy quadratization (LIEQ) and modified scalar auxiliary variable (MSAV) approaches. These two schemes will be effective for all negative E₁. Furthermore, we proved that all the semi-discrete schemes are unconditionally energy stable with respect to a modified energy. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

Swift-Hohenberg模型是一个非常重要的相场晶体模型，可以描述许多晶体现象。这个基于H ^-1梯度流方法的二次立方非线性模型是一个满足质量守恒和能量耗散定律的六阶系统。对于许多现有方法，该模型的负能量将给能量稳定性带来巨大困难。在本文中，我们通过线性不变能量平方（LIEQ）和改进的标量辅助变量（MSAV）方法考虑了两种线性，二阶和无条件的能量稳定方案。这两个方案将对所有负E ₁有效。。此外，我们证明了所有半离散方案相对于修改后的能量都是无条件的能量稳定。最后，我们提出各种2D数值模拟以证明稳定性和准确性。