The Pseudo-spectral Method Of Approximatively Solving Cahn-Hilliard Equation

This paper gives out the numerical solution to Cahn-Hilliard equation with invariable mobility and to Cahn-Hilliard equation with variable mobility, hence validates the character of solution to Cahn-Hilliard equation with the spectral method of the numerical solution.This paper is divided into six chapters. The first chapter describes the important back-ground of Cahn-Hilliard and recent great achievement in the research of Cahn-Hilliard equa-tion. The second chapter describes the definitions and lemmas which are used in the process of solving the equation. From the third chapter to the fifth chapter we introduce the spec-tral method and the pseudo-spectral method to solve following Cahn-Hilliard equation with invariable mobility in detail. where u=u(x, t) is the density of one of the two inter diffusion materials, gamma>0 is the mobility.Φ(u)= H'(u), where H(u) is a typical double well potential: And here we only consider the case ofγ1=0,thenΦ(u)=-u+γ2u3In the third chapter we set up the variational form of the problem (1)(2)(3), and proof that wheny> 0,γ2> 0, a function u= u(x, t)∈C4(QT) is the solution of (1)(2)(3) if only if u∈HE2(0,π) and satisfying:In the fourth chapter we introduce the Pseudo-spectral method to solve the variational form of the problem (1)(2)(3). We set up a function space which choosing trigonometric polynomial as its basic functions, we define SN= span{cos jx}j=02N-1. Then, we use this finite dimensional space SN to close in upon the infinite dimensional space HE2(0,π). We define the discrete inner product in SN as follow: Firstly, we apply the Pseudo-spectral method to variational equation (4)(5) to obtain a semi-discrete approximation of the problem:Find out an Its coefficients {aj(t)}j=02N-1 satisfy the following equation for all j= 0,1...,2N-1: On a basis of semi-discrete scheme, we using Back Euler-Galerkin scheme to set up a com-plete discrete scheme of Cahn-Hilliard equation. And then materialize numerical solving to equation with initial function uo(x)=cosx or u0(x)=cosx2.Through the numerical result we found that:whenγ2> 0, the semi-discrete problem has a unique global solution; whenγ2< 0, the solution of semi-discrete problem can probably blow up. This phenomenon validate the character of solution.In the fifth chapter, we use Spectral method to solve this Cahn-Hilliard. Different with pseudo-spectral method, the semi-discrete approximation of the spectral method is:find out Its coefficients{aj(t)}j=02N-1 satisfy the following equation where is the orthonormal Basis of SN under the inner product L2(·,·). uOM=PNuO is the projection of u0(x) on the subspace SN,which means for allu∈L2(0,1) we have (PNu, v)= (u, v), (?)v∈SN. Similarly, we set up the complete discrete scheme of solving the variational form of Cahn-Hilliard equation. And then, perform numeric computations by means of Matlab, compared this numerical solution with the numerical solution under pseudo-spectral methodThe last chapter, the use the pseudo-spectral method which is introduced in the fifth chapter to solve the following C-H equation with varying mobility. where D= (?), A(u)=-u+γ1u2+γ2u3,γ2>0 (13) where, u(x,t)is the density of one of the two inter diffusion materials, m(x, t) is mobility, defination QT= (0,1) x (0, T), and we suppose satisfied: where m0, M0 and M1 are constant.We set up its variational form and discrete scheme, also we find the numerical solution to the equation whenγ1=1,γ2= 1, m(x, t)= 1+xsin100πt, uo(x)= x4(1-x)4.