A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation (Communications in Computational Physics, 2018)
In this paper we present a second order accurate (in time) energy stable numericalscheme for the Cahn-Hilliard (CH) equation, with a mixed finite element approximationin space. Instead of the standard second order Crank-Nicolson methodology,we apply the implicit backward differentiation formula (BDF) concept to derivesecond order temporal accuracy, but modified so that the concave diffusion term istreated explicitly. This explicit treatment for the concave part of the chemical potentialensures the unique solvability of the scheme without sacrificing energy stability. Anadditional term $A$τ∆($u^{k+1}$−$u^k$) is added, which represents a second order Douglas-Dupont-typeregularization, and a careful calculation shows that energy stability isguaranteed, provided the mild condition $A$≥$\frac{1}{16}$ is enforced. In turn, a uniform in time $H^1$ bound of the numerical solution becomes available. As a result, we are able toestablish an $ℓ^∞$(0,$T$;$L^2$) convergence analysis for the proposed fully discrete scheme,with full $\mathcal{O}$ (τ2+$h^2$) accuracy. This convergence turns out to be unconditional; no scalinglaw is needed between the time step size $τ$ and the spatial grid size $h$. A fewnumerical experiments are presented to conclude the article.
Keywords: Cahn-Hilliard equation; energy stable BDF; Douglas-Dupont regularization; mixed finite element; energy stability.