关键词: 大时间步长方法, 修正的Cahn-Hilliard方程, 有界性, 收敛阶

Abstract: Over the past decades, the Cahn-Hilliard equation has attracted the attention of many scholars. This equation was originally used to describe the phase separation of two homogeneous mixtures that occurs when the temperature drops and the two mixtures automatically separate and occupy different regions. Along with the theory thorough research, it also has the widespread application in other aspects. The modified Cahn-Hilliard equation enriches the Cahn-Hilliard equation with more properties, and it is a fourth-order nonlinear parabolic equation. Coupled with the small parameter problem of the equation, it is difficult to obtain the exact solution of the equation. Therefore, numerical method can only be used to solve the numerical solution in a small time step. If the solution is carried out in a large time step, the numerical solution will be divergent. A large time step method is proposed in this paper. The proposed scheme is discretized by the finite element method in space and the semi-implicit scheme in time. Stability of the first-order semi-discrete scheme and error estimation of the full discrete scheme are proved. Finally, numerical examples are used to verify the accuracy and validity of the theoretical analysis.

Key words: large time-stepping method, Cahn-Hilliard equation, boundedness, convergence rate