On efficient semi-implicit auxiliary variable methods for the six-order Swift–Hohenberg model

Abstract

The Swift–Hohenberg model is a very important phase field crystal model which can describe 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. In this paper, we consider several linear, second-order and unconditionally energy stable schemes based on the traditional and modified scalar auxiliary variable (SAV) approaches. The two novel modified SAV approaches which called step-by-step solving scalar auxiliary variable (3S-SAV) and exponential scalar auxiliary variable (E-SAV) approaches have been proved to have several advantages than the traditional SAV method. We proved the unconditional energy stability for all the semi-discrete schemes carefully and rigorously. In calculation, we show that the novel semi-implicit schemes save many computational CPU time compared with the traditional SAV scheme. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.