A highly efficient and accurate new SAV approach for the modified phase field crystal model

The modified phase field crystal (MPFC) model is a sixth-order evolutive nonlinear partial differential equation which can describe many crystal phenomena. In this paper, we propose a highly efficient and accurate numerical method to construct linear and unconditionally energy stable schemes for the MPFC model. In recent years, the scalar auxiliary variable (SAV) and SAV-based methods have attracted much attention in numerical solution for dissipative systems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. The considered numerical schemes are based on a new SAV-type approach which named new scalar auxiliary variable (nSAV) approach. We first give a first-order energy stable numerical scheme by introducing a new SAV R(t). Then, the high-order nSAV schemes based on the k-step backward differentiation formula (BDFk) are constructed. The considered nSAV schemes allow us to construct high-order schemes for both the phase variable phi and psi while only using a first-order approximation of the energy balance equation. To our knowledge, there is no careful research to give high-order accurate but energy stable schemes for the MPFC model. Meanwhile, the proposed approach only needs to solve linear equation with constant coefficients in one time step which is easy to use fast Fourier transform (FFT) to save more CPU time in calculation. Some numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.