Unconditional Dissipative Difference Scheme For The Cahn-hilliard Equation

In this work, the initial-boundary value problem of two-dimensional Cahn-Hilliard equation is considered. A class of fully discrete dissipative finite difference schemes is proposed. Moreover, semi-implicit prediction-correction schemes are presented. The numerical simulations are performed to demonstrate the effective-ness of the proposed schemes. In the latter part of the paper, we discuss the existence and uniquess of the numerical solution of Cahn-Hilliard equation. We apply the five-points difference scheme to get the numerical simulation. The nine-points scheme numerical simulation results is also shown. Cahn-Hilliard equation with a variable mobility is discussed in the last part. Also, the numerical results are performed. From the tables, we get the conclusions about orders of time stepping.Chapter 1 introduces the background and the current situations of Cahn-Hilliard equation.Chapter 2 we use finite-difference approximation. Propose five-point difference formula for Cahn-Hilliard equation. Also, periodic boundary conditions on the squareΩ= [0,L]×[0,L] are considered. We describe some notations and the approximation solution of (1.1), then some lemmas are also established in this chapter. Especially, We propose dissipative five-point difference schemes.Chapter 3 proof the existence and the uniqueness of the numerical solution of (2.8). Lemmas and theorem are also established in this chapter.Chapter 4 is about the convergence of the numerical solution.Chapter 5 is about the results of the numerical solution. In this chapter. We propose the nine-point difference scheme in spatial for Cahn-Hilliard equation. Introduce the methods to solve Cahn-Hilliard equation with variable mobility.