Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model
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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 α∈(0,1), under a weak step-ratio constraint 0.4753≤τ_k/τ_k-1<r^*(α), where τ_k is the k-th time-step size and r^*(α)≥4.660 for α∈(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 α→1^-, 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.
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