Stability And Convergence Of Several Finite Difference Schemes For The Cahn-Hilliard Equation

In this thesis, a class of stable finite difference schemes for solving the initial-boundary value problem of the Cahn-Hilliard equation are proposed. The unique existence, conservation laws, stability and convergence of the numerical solutions are analyzed in detail. First of all, the mass conservation and energy dissipation of the numerical solution in the discrete sense are proved, based which the a priori estimates of the numerical solutions in the H1 norm are obtained, which implies that the schemes are completely stable in the H1 norm, then the unique existence of the numerical solution is proved by using a fixed point theorem and the energy method, then the local truncation errors are given and based them the optimal error estimates in the maximum norm are established by using the energy method as well as the a priori estimates of the numerical solutions, without any constrains on the grid ratios, the error estimates in the maximum are O(h2+τ2) with time step τ and mesh step h.Because the proposed schemes are nonlinear, the iterations at each time step are unavoidable in the practical computation. Thus, an efficient iterative algorithm for solving the proposed nonlinear schemes is proposed and discussed in detail.Numerical examples show that the proposed scheme is stable and effective.