Efficient and energy stable numerical schemes for the two-mode phase field crystal equation

Abstract

In this paper, we propose four time marching schemes for the two-mode phase field crystal equation with the periodic boundary condition. The first- and second-order schemes are based on the stabilized semi-implicit approach with a combination of the backward Euler method, Crank–Nicolson method, and second-order backward differentiation formula, respectively. Based on the convex splitting method and third-order backward differentiation formula, a third-order scheme is developed. Different stabilized terms are added so that the energy stability of the proposed schemes can be guaranteed. Theoretically, the unique solvability and energy stability of the proposed schemes are rigorously proved. The error estimate of the first-order scheme is also given. Various numerical examples and simulations in the 2D and 3D cases are presented to demonstrate the accuracy, stability, and efficiency of the proposed schemes.