修正的Cahn-Hilliard方程的大时间步长方法

Large Time Stepping Method for the Modified Cahn-Hilliard Equation

过去的几十年中,Cahn-Hilliard方程引起了很多学者的关注.该方程最早被用来描述在温度降低时两种均匀的混合物所发生的相分离现象.随着理论的深入研究,该方程在其他方面也有广泛的应用.修正的Cahn-Hilliard方程是一个四阶非线性抛物方程,再加上该方程的小参数问题,使得该方程在求精确解时,具有一定的难度,只能利用数值方法在较小的时间步长上求解数值解,若在较大的时间步长上进行求解会造成数值解的发散.本文提出了求解修正的Cahn-Hilliard方程的大时间步长方法.所提格式在空间上采用有限元方法离散,在时间上采用一阶半隐格式进行离散,证明了一阶半离散格式的稳定性及全离散格式的误差估计.最终通过数值算例来验证理论分析的准确性及有效性.

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.