The Study Of Large Time Stepping Numerical Methods For The Modified Cahn-Hilliard Equation

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 CahnHilliard equation is a fourth-order nonlinear parabolic equation.Coupled with the small parameter problem of this equation,it is difficult to obtain the exact solution of this equation.Therefore,numerical method can only be used to solve the numerical solution in a small time step.The modified Cahn-Hilliard equation enriches the Cahn-Hilliard equation with more properties.The main content of this paper is as follows.Firstly,the background and time step method of the modified Cahn-Hilliard are introduced.Secondly,the paper puts forward the method of increasing the time step for characteristic of the nonlinear term and small parameters of the modified Cahn-Hilliard equation.The main idea is to use conforming finite element method in space,the semiimplicit scheme in time,the nonlinear term is treated with explicit method.The study found that this method did have a role to increase the time step,and the evidence was given in the theory of theory.Next,in order to avoid the difficulty of dealing with the fourth-order problem,the mixed element method is used to reduce the order of the equation.Mixing finite element method is used in space,and the semi-implicit scheme in time,and the nonlinear term is used implicit method.It is proved that this scheme is energy stable and gives a error analysis.Finally,the validity of the two methods is verified by numerical examples.