Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation

The phase-field crystal equation is a sixth-order nonlinear parabolic equation and can be applied to simulate various phenomena such as epitaxial growth, material hardness, and phase transition. We propose a series of efficient modified stabilized invariant energy quadratization approaches with unconditional energy stability for the phase-field crystal model. Firstly, we propose a more suitable positive preserving function strictly in square root and consider a modified invariant energy quadratization (MIEQ) approach. Secondly, a series of efficient and suitable functionalsH(phi) in square root are considered and the modified stabilized invariant energy quadratization (MSIEQ) approaches are developed. We prove the unconditional energy stability and optimal error estimates for the semi-discrete schemes carefully and rigorously. A comparative study of classical IEQ, MIEQ, and MSIEQ approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.