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

Source 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 $−A_0\Delta t^2\Delta_N (\phi^{n+1}−\phi^n),$ 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 $L^6_N$ 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.