A stabilized fully-discrete scheme for phase field crystal equation

A finite difference scheme is developed for solving the phase field crystal equation with the Dirichlet boundary condition. The second-order stabilized semi-implicit scheme is applied to discretize the time variable. The mass conservation, unique solvability of the semi-discrete scheme are proved. Then, the spatial discretization is attained by the fourth-order compact difference scheme. The unconditional energy stability and convergence analysis of the fully-discrete scheme are presented. In the implementation, a fast sine transform technique is utilized to reduce the computational cost. Several numerical examples are presented to verify the effectiveness of the proposed scheme.