Asymptotically compatible energy and dissipation law of the nonuniform L2-$1_σ$ scheme for time fractional Allen-Cahn model
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| author | Liao, Hong-lin Zhu, Xiaohan Sun, Hong |
| author_facet | Liao, Hong-lin Zhu, Xiaohan Sun, Hong |
| contents | We build an asymptotically compatible energy of the variable-step L2-$1_σ$ scheme for the time-fractional Allen-Cahn model with the Caputo's fractional derivative of order $α\in(0,1)$, under a weak step-ratio constraint $τ_k/τ_{k-1}\geq r_{\star}(α)$ for $k\ge2$, where $τ_k$ is the $k$-th time-step size and $r_{\star}(α)\in(0.3865,0.4037)$ for $α\in(0,1)$. It provides a positive answer to the open problem in [J. Comput. Phys., 414:109473], and, to the best of our knowledge, it is the first second-order nonuniform time-stepping scheme to preserve both the maximum bound principle and the energy dissipation law of time-fractional Allen-Cahn model. The compatible discrete energy is constructed via a novel discrete gradient structure of the second-order L2-$1_σ$ formula by a local-nonlocal splitting technique. It splits the discrete fractional derivative into two parts: one is a local term analogue to the trapezoid rule of the first derivative and the other is a nonlocal summation analogue to the L1 formula of Caputo derivative. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our scheme and the asymptotic properties of the associated modified energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_13216 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotically compatible energy and dissipation law of the nonuniform L2-$1_σ$ scheme for time fractional Allen-Cahn model Liao, Hong-lin Zhu, Xiaohan Sun, Hong Numerical Analysis 65M12, 65M06, 35Q99, 74A50 We build an asymptotically compatible energy of the variable-step L2-$1_σ$ scheme for the time-fractional Allen-Cahn model with the Caputo's fractional derivative of order $α\in(0,1)$, under a weak step-ratio constraint $τ_k/τ_{k-1}\geq r_{\star}(α)$ for $k\ge2$, where $τ_k$ is the $k$-th time-step size and $r_{\star}(α)\in(0.3865,0.4037)$ for $α\in(0,1)$. It provides a positive answer to the open problem in [J. Comput. Phys., 414:109473], and, to the best of our knowledge, it is the first second-order nonuniform time-stepping scheme to preserve both the maximum bound principle and the energy dissipation law of time-fractional Allen-Cahn model. The compatible discrete energy is constructed via a novel discrete gradient structure of the second-order L2-$1_σ$ formula by a local-nonlocal splitting technique. It splits the discrete fractional derivative into two parts: one is a local term analogue to the trapezoid rule of the first derivative and the other is a nonlocal summation analogue to the L1 formula of Caputo derivative. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our scheme and the asymptotic properties of the associated modified energy. |
| title | Asymptotically compatible energy and dissipation law of the nonuniform L2-$1_σ$ scheme for time fractional Allen-Cahn model |
| topic | Numerical Analysis 65M12, 65M06, 35Q99, 74A50 |
| url | https://arxiv.org/abs/2311.13216 |