A linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908601495322624 |
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| author | Hou, Dianming Qiao, Zhonghua Tang, Tao |
| author_facet | Hou, Dianming Qiao, Zhonghua Tang, Tao |
| contents | As a variational phase-field model, the time-fractional Allen-Cahn (TFAC) equation enjoys the maximum bound principle (MBP) and a variational energy dissipation law. In this work, we develop and analyze linear, structure-preserving time-stepping schemes for TFAC, including first-order and $\min\{1+α, 2-α\}$-order L1 discretizations, together with fast implementations based on the sum-of-exponentials (SOE) technique. A central feature of the proposed linear schemes is their unconditional preservation of both the discrete MBP and the variational energy dissipation law on general temporal meshes, including graded meshes commonly used for these problems. Leveraging the MBP of the numerical solutions, we establish sharp error estimates by employing the time-fractional Gronwall inequality. Finally, numerical experiments validate the theoretical results and demonstrate the effectiveness of the proposed schemes with an adaptive time-stepping strategy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16496 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation Hou, Dianming Qiao, Zhonghua Tang, Tao Numerical Analysis 65M12, 65M15, 35K35 As a variational phase-field model, the time-fractional Allen-Cahn (TFAC) equation enjoys the maximum bound principle (MBP) and a variational energy dissipation law. In this work, we develop and analyze linear, structure-preserving time-stepping schemes for TFAC, including first-order and $\min\{1+α, 2-α\}$-order L1 discretizations, together with fast implementations based on the sum-of-exponentials (SOE) technique. A central feature of the proposed linear schemes is their unconditional preservation of both the discrete MBP and the variational energy dissipation law on general temporal meshes, including graded meshes commonly used for these problems. Leveraging the MBP of the numerical solutions, we establish sharp error estimates by employing the time-fractional Gronwall inequality. Finally, numerical experiments validate the theoretical results and demonstrate the effectiveness of the proposed schemes with an adaptive time-stepping strategy. |
| title | A linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation |
| topic | Numerical Analysis 65M12, 65M15, 35K35 |
| url | https://arxiv.org/abs/2510.16496 |