A Third Order Accurate in Time, BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation

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 − A 0 ∆ t 2 ∆ N ( φ n +1 − φ 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 6 N bound of the numerical solution is obtained. And also, the optimal rate convergence analysis and error estimate are provided, in the L ∞ ∆ t (0 , T ; L 2 N ) ∩ L 2∆ t (0 , T ; H 2 h ) norm, with the help of the L 6 N 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.