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

Abstract

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 the design of several auxiliary ODEs, combined with the finite element method for spatial discretization, the projection method for the Navier-Stokes equations, and the IEQ type method for the nonlinear potentials. At each time step, by using the nonlocal splitting technique, only a few decoupled elliptic constant-coefficient equations need to be solved. We further prove that the developed scheme is unconditionally energy stable, and the detailed implementation process is given as well. To verify the effectiveness of the developed scheme, various numerical experiments are carried out, including the crystal growth under the action of shear flow and the sedimentation process of a large number of particles.