A Second Order BDF Numerical Scheme with Variable Steps for the Cahn--Hilliard Equation

Abstract

We present and analyze a second order in time variable step BDF2 numerical scheme for the Cahn--Hilliard equation. The construction relies on a second order backward difference, convex-splitting technique and viscous regularizing at the discrete level. We show that the scheme is unconditionally stable and uniquely solvable. In addition, under mild restriction on the ratio of adjacent time-steps, an optimal second order in time convergence rate is established. The proof involves a novel generalized discrete Gronwall-type inequality. As far as we know, this is the first rigorous proof of second order convergence for a variable step BDF2 scheme, even in the linear case, without severe restriction on the ratio of adjacent time-steps. Results of our numerical experiments corroborate our theoretical analysis.