A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation

In this paper we present a second order accurate (in time) energy stable numerical scheme for the Cahn-Hilliard (CH) equation, with a mixed finite element approximation in space. Instead of the standard second order Crank-Nicolson methodology, we apply the implicit backward differentiation formula (BDF) concept to derive second order temporal accuracy, but modified so that the concave diffusion term is treated explicitly. This explicit treatment for the concave part of the chemical potential ensures the unique solvability of the scheme without sacrificing energy stability. An additional term A tau Delta(u(k+1) - u(k)) is added, which represents a second order Douglas-Dupont-type regularization, and a careful calculation shows that energy stability is guaranteed, provided the mild condition A >= 1/16 is enforced. In turn, a uniform in time H-1 bound of the numerical solution becomes available. As a result, we are able to establish an l(infinity)(0, T; L-2) convergence analysis for the proposed fully discrete scheme, with full O(tau(2) + h(2)) accuracy. This convergence turns out to be unconditional; no scaling law is needed between the time step size tau and the spatial grid size h. A few numerical experiments are presented to conclude the article.