Novel energy stable schemes for Swift-Hohenberg model with quadratic-cubic nonlinearity based on the H−1-gradient flow approach

Abstract

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.