A second-order, uniquely solvable, energy stable BDF numerical scheme for the phase field crystal model

In this paper, we propose a second-order time accurate convex splitting scheme for the phase field crystal model. The temporal discretization is based on the second-order backward differentiation formula (BDF) and a convex splitting of the energy functional. The mass conservation, unconditionally unique solvability, unconditionally energy stability and convergence of the numerical scheme are proved rigorously. Mixed finite element method is employed to obtain the fully discrete scheme due to a sixth-order spatial derivative. Numerical experiments are presented to demonstrate the accuracy, mass conservation, energy stability and effectiveness of the proposed scheme.