Abstract
In this paper we propose and analyze a backward differentiation formula (BDF) type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy. The Fourier pseudo-spectral method is used to discretize space. The surface diffusion and the nonlinear chemical potential terms are treated implicitly, while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont regularization term, in the form of -A0 Delta t(2) Delta(N)(phi(n+1) - phi(n)), is added in the numerical scheme. In particular, the energy stability is carefully derived in a modified version, so that a uniform bound for the original energy functional is available, and a theoretical justification of the coefficient A becomes available. As a result of this energy stability analysis, a uniform-in-time L-N(6) bound of the numerical solution is obtained. And also, the optimal rate convergence analysis and error estimate are provided, in the L-Delta t(infinity)(0, T; L-N(2)) boolean AND L-Delta t(2)(0, T; H-h(2)) norm, with the help of the L-N(6) bound for the numerical solution. A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.