Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model
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| Format: | Preprint |
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2022
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| author | Liao, Hong-lin Liu, Nan Zhao, Xuan |
| author_facet | Liao, Hong-lin Liu, Nan Zhao, Xuan |
| contents | 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 $α\in(0,1)$, under a weak step-ratio constraint $0.4753\le τ_k/τ_{k-1}<r^*(α)$, where $τ_k$ is the $k$-th time-step size and $r^*(α)\ge4.660$ for $α\in(0,1)$.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 $α\rightarrow1^-$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_12514 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model Liao, Hong-lin Liu, Nan Zhao, Xuan Numerical Analysis 35Q99, 65M06, 65M12, 74A50 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 $α\in(0,1)$, under a weak step-ratio constraint $0.4753\le τ_k/τ_{k-1}<r^*(α)$, where $τ_k$ is the $k$-th time-step size and $r^*(α)\ge4.660$ for $α\in(0,1)$.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 $α\rightarrow1^-$, 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. |
| title | Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model |
| topic | Numerical Analysis 35Q99, 65M06, 65M12, 74A50 |
| url | https://arxiv.org/abs/2210.12514 |