The convergence of two linearized finite difference schemes for the modified phase field crystal equation

The modified phase field crystal model is a sixth order nonlinear generalized damped wave equation. Two linearized difference schemes are presented based on the method of order reduction. One scheme is second order both in time and space, and the other scheme is convergent with the second temporal order and fourth spatial order. A theoretical analysis is carried out by the energy argument and mathematical induction. The uniqueness of the numerical solution and unconditional convergence in discrete L-infinity-norm are proved rigorously. Numerical results demonstrate that the presented schemes for the modified phase field crystal equation can achieve the theoretical convergence order.