Spectral Method For Spacial Periodic One-dimensional Cahn-Hilliard Equation

In this paper,we study the spectral methods for a class of Cahn-Hilliard equations in one dimensional space with periodic boundary condition where T:= R/Z is an one dimensional torus,ε is spatial scale parameter,W is smoothing potential.In this paper,we use the double well potential:Let QT≡ Ix(0,T),where I =[0,1],D =(?)/(?)x,Equations(0.1)and(0.2)is equivalent toLet L2(I)represent the square integrable space with norm ||u|| =∫01 u2dx on I,H2(I)represent the space H2 on I.Let Cp∞(I)= {v ∈C∞(I);Dkv(0)＝Dkv(1),E∈Z+}.HP2(I)is the closure of Cp∞(I)in the H2(I)norm.The function u is called the weak solution of the problem(0.3)-(0.5),if u(·,t)∈Hp2(I)satisfies initial conditions(0.4)and the following equation:In this paper,we apply the spectral method to approximate equation(0.6).For any integer N>0,let SN span{1,cos2kπx,sin2kπx，k 1,2,…,N}.The spacial method for equation(0.6)is to find uN∈SN,such that:Where UNj,approaches uN at time of t = tj.We proved the boundedness of the solution uN in L∞ norm and proved convergence property.Where C is a constant.Furthermore,we discrete time variable to construct full discretization.Let 0<t0<t1<…<tJ= T,where tj = jk,k represents the time stepsize.The full discrete scheme of equation(0.7)is:find UNj(x),j = 0,1,…,J,such that(?)v∈SN,We proved the boundedness of fully discrete form solution in L∞ norm.and proved convergence property.Where C is a constant.