Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model

A new discrete energy dissipation law of the variable-step fractional BDF2 (second-order backward differentiation formula) scheme is established for time-fractional Cahn-Hilliard model with the Caputo's fractional derivative of order $\alpha\in(0,1)$, under a weak step-ratio constraint $0.4753\le \tau_k/\tau_{k-1}<r^*(\alpha)$, where $\tau_k$ is the $k$-th time-step size and $r^*(\alpha)\ge4.660$ for $\alpha\in(0,1)$.We propose a novel discrete gradient structure by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo's derivative. More interestingly, in the sense of the limit $\alpha\rightarrow1^-$, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn-Hilliard equation, respectively. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.