Novel linear decoupled and unconditionally energy stable numerical methods for the modified phase field crystal model

Abstract

In this paper, we propose a novel numerical approach to construct unconditionally energy stable schemes for the modified phase field crystal (MPFC) model. The new technique is based on the invariant energy quadratization (IEQ) method. The numerical schemes based on IEQ approach lead to time-dependent dense matrices, thus the fast Fourier transform (FFT) is difficult to be applied to solve the systems directly. By introducing a new auxiliary variable to replace the original one, we derive a novel equivalent MPFC system. A step-by-step solving approach, termed 3S-IEQ method is considered to solve this equivalent system. Some totally decoupled, linear and unconditional energy stable semi-implicit schemes based on 3S-IEQ method are very easy to construct. More importantly, the phase function ϕ and auxiliary variable η can be calculated step-by-step. Meanwhile, the proposed approach only needs to solve linear equation with constant coefficients which is easy to use FFT directly in calculation. Some numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.