A Structure Preserving Schemes For Nonlocal Allen-Cahn Equations

The Allen-Cahn equation,as a model of binary alloy reverse phase domain coarsening,has a wide range of applications in many fields.This paper mainly studies the initial boundary value problem of the non-local Allen-Cahn equation proposed by Rubinstein and Stenbeg to describe the phase separation in binary mixtures.This paper mainly uses the Discrete Variational Derivative Method(DVDM).First,in the continuous situation,a local energy functionis defined and the variational derivative is obtained,and then the original equation is rewritten with the variational derivative to obtain the dissipative properties of the original equation;then in the discrete case,based on the variational structure of the original equation,combined with the linearization technique,a discrete local energy function G_d is established and the expression of the discrete variational derivative is calculated,thereby constructing the structure-preserving linear scheme of the non-local Allen-Cahn equation.This scheme not only inherits the conservation of mass of the original equation,but also inherits the nature of the original equation with decreasing energy.At the same time,this paper also proves the stability of the scheme and the existence and uniqueness of the solution,and gives the error estimate of the scheme.The final numerical example illustrates the validity of the scheme.The structure-preserving scheme constructed in this paper is linear.Compared with the original nonlinear structure-preserving scheme,the two scheme have the same accuracy,that is,both the time direction and the space direction have second-order accuracy.However,the linear format constructed in this paper has obvious advantages both in terms of calculation amount and calculation time.The paper is divided into four parts:Chapter 1,we introduces several common methods of solving partial differential equation,including finite difference method,finite element method,spectrum method and finite volume method,and also explains the research background,research status,innovation and main content of Allen Cahn equation;Chapter 2,introduces the basic symbols,the basic process of constructing numerical format,the common formulas and other preparatory knowledge;Chapter 3,on the basis of the discrete variable derivative method(DVDM),the linear structure preserving scheme of Allen Cahn is constructed by using the linearization technique,and it is proved that the scheme not only inherits the properties of the original equation,but also is stable,has a unique solution,and gives the error estimate.Numerical examples are given to show the reliability of the scheme;In the last part,we summarize the above research and discuss the next research direction.