The Semi-implicit Alternating Direction Discontinuous Galerkin Method For The Cahn-Allen Equation And Cahn-Hilliard Equation

This paper mainly studies the Cahn-Allen model and Cahn-Hilliard model inthe phase field models.Usually,the numerical methods that are used to study thesetwo equations,are mainly applicable to the situation that the solution'smoothness isgood.This paper mainly adopts new semi-implicit alternating direction discontinuousGalerkin(DG) method to study the equations with poor smoothness.The method is based on the variational principle, considering the local weak form,looking for some properties of discontinuous points of the solution. The main researchcontents are:the use of DG method to obtain the one-dimensional and two-dimensionalsemi-implicit difference scheme of Cahn-Allen equation and Cahn-Hilliard equation;todiscuss the numerical stability and convergence order of these schemes.Specifically:inallied to the Cahn-Allen equation and Cahn-Hilliard equation,we offer a semi-implicitdifference method about the time direction,develop some new skills to overcome thedifficulty which is caused by the space direction'derivative of the Cahn-Hilliard equa-tion,and put forward alternating direction method;finally,to be aimed at different kindsof DG methods,to compare the errors and precisions of these equations,and to do thenumerical simulation of the phase field.