A novel second-order time accurate fully discrete finite element scheme with decoupling structure for the hydrodynamically-coupled phase field crystal model

In this article, we construct a fully discrete finite element numerical scheme with linearity, decoupling, unconditional energy stability, and second-order time accuracy for the Navier-Stokes coupled phase-field crystal model. The key idea is based on design of several auxiliary ODEs, combined method spatial discretization, projection equations, IEQ type nonlinear potentials. At each step, by using nonlocal splitting technique, only few decoupled elliptic constant-coefficient equations need to be solved. We further prove that developed unconditionally stable, detailed implementation process given as well. To verify effectiveness scheme, various experiments are carried out, including growth under action shear flow sedimentation large number particles.