The Seudo-spectral Method For Cahn-Hilliard Equation With Periodic Boundary Condition

In this paper we consider the initial-boundary value problem of the following forth order parabolic Cahn-Hillard equation with constant rate of transitionwhere u = u(x,t) is the density of one of the two inter diffusion materials, 7 > 0 is the rate of transitio and it is assumed to be constant. (u) = H'(u), H(u) is the standard double-well potentialand in this paper we only consider the case 1 = 0, which gives (u) = -u + u3.The main purpose of this paper is to set up the semi-discrete approximation to the equation (1),(2),(3) with the seudo-spectral method and analyze its convergence property when 2 > 0.Cahn-Hillard equation is a class of important nonlinear diffusion equation with fourth order, which was derived by Cahn and Hillard when they studiedthe phenomenon of inter-diffusion between two materials (such as alloy) in energetic. Later on, such mathematical model is also used by people in describing the phenomenon of the competition and blackball of biology grow groups, the process of riverbed transplant and diffusion of a tiny drop on the surface of a solid. Since 80' in this century, people have studied the Cahn-Hilliard equation systematically. Recently for the important background of Cahn-Hilliard equation in chemistry, materials, etc, there are great achievement in the research of Cahn-Hillard equation. In this paper we will study Cahn-Hillard equation by the way of numerical method. The seudo-spectral method is a Ritz-Galerkin method, which sets up the semi-discrete scheme for the problem by choosing gonometrical polynomials as the basic functions of the trial function space.We first set up a equivalent variational equation for the initial-boundary value problem of the Cahn-Hilliard equation (1),(2),(3). We prove that if u : [0,T] - C4[0, TT] is a solution of equation (1),(2),(3), u must be a solution of the following equationIf u : [0, T] - HP is a solution of equation (4), and if for any t, u(, t) C4[0, ] and satisfy condition (2),(3), it is a solution of equation (1),(2),(3). We use the seudo-spectral method to deal with the space variable x and get the following semi-discrete problem: find a uN : [0, T] - SN such thatwhere UN0 = INu0 is the interpolation of u0(x) onto the space SN with discrete norm. Then we get the uniqueness and existence property in the whole interval[0,T] for the sime-discrete problem (5),(6). We prove that for 7 > 0, 72 > 0 there is a unique solution in the whole interval for equations (5),(6). Then we obtain the convergence property of the solution of the semi-discrete system (5),(6). We obtain the following estimatewhere e = PNu - UN, u is the solution of (1),(2),(3) and C is a constant depends on 7, 72 and U0(x). We also prove...