Highly efficient and accurate numerical schemes for the anisotropic phase field crystal models by using the improved scalar auxiliary variable (iSAV) approach

The two-dimensional anisotropic phase field crystal (APFC) model is a sixth-order nonlinear parabolic equation that can be used to simulate various phenomena such as epitaxial growth, material hardness, and phase transition. The scalar auxiliary variable method (SAV) is a common method to solve various nonlinear dissipative systems, and the improved SAV (iSAV) method is not only completely linear, but also strictly guarantees the original dissipation law. In this paper, we construct several efficient, accurate linear and original energy-stable numerical schemes of the APFC model based on the iSAV method. Firstly, a first-order iSAV scheme is considered to keep the original energy stability for the APFC model. Secondly, we propose a new stabilized iSAV scheme and give its rigorous energy stability analysis to keep its original dissipation law. Finally, several interesting numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed methods.