The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn Model
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910624287555584 |
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| author | Chen, YiMing Luo, Dianlun Pei, Wenlong Xing, Yulong |
| author_facet | Chen, YiMing Luo, Dianlun Pei, Wenlong Xing, Yulong |
| contents | We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditional non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial discretization. For the non-linear term, we combine the DLN scheme with two efficient temporal algorithms: partially implicit modified algorithm and scalar auxiliary variable algorithm. For both approaches, we prove the unconditional, long-term stability of the model energy under any arbitrary time step sequence. Moreover, we provide rigorous error analysis for the partially implicit modified algorithm with variable time-stepping. Efficient time adaptive algorithms based on these schemes are also proposed. Several one- and two-dimensional numerical tests are presented to verify the properties of the proposed time adaptive DLN methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn Model Chen, YiMing Luo, Dianlun Pei, Wenlong Xing, Yulong Numerical Analysis 65M12, 65M15, 35K35, 35K55 We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditional non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial discretization. For the non-linear term, we combine the DLN scheme with two efficient temporal algorithms: partially implicit modified algorithm and scalar auxiliary variable algorithm. For both approaches, we prove the unconditional, long-term stability of the model energy under any arbitrary time step sequence. Moreover, we provide rigorous error analysis for the partially implicit modified algorithm with variable time-stepping. Efficient time adaptive algorithms based on these schemes are also proposed. Several one- and two-dimensional numerical tests are presented to verify the properties of the proposed time adaptive DLN methods. |
| title | The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn Model |
| topic | Numerical Analysis 65M12, 65M15, 35K35, 35K55 |
| url | https://arxiv.org/abs/2409.19481 |