Efficient second-order unconditionally stable numerical schemes for the modified phase field crystal model with long-range interaction

Abstract

In this paper, we consider numerical approximations for the modified phase field crystal model with long-range interaction, which describes the micro-phase separation in diblock copolymers. The model is a nonlinear damped wave equation with a nonlocal term that includes both diffusive dynamics and elastic interaction. To develop easy-to-implement time-stepping schemes with unconditional energy stabilities, we employ the scalar auxiliary variable (SAV) approach to achieve two highly efficient and linear numerical schemes based on the second-order Crank–Nicolson and backward differentiation formula. These schemes lead to decoupled linear equations with constant coefficients at each time step and their unconditional energy stabilities are proved rigorously. The stabilization technique is adopted to further improve the stability of the numerical schemes. Various 2D and 3D numerical experiments are performed to demonstrate the accuracy, stability, and efficiency.