Maximum bound preservation of exponential integrators for Allen-Cahn equations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908317587079168 |
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| author | Quan, Chaoyu Zheng, Pingzhong Zhou, Zhi |
| author_facet | Quan, Chaoyu Zheng, Pingzhong Zhou, Zhi |
| contents | We develop and analyze a class of arbitrarily high-order, maximum bound preserving time-stepping schemes for solving Allen-Cahn equations. These schemes are constructed within the iterative framework of exponential integrators, combined with carefully chosen numerical quadrature rules, including the Gauss-Legendre quadrature rule and the left Gauss-Radau quadrature rule. Notably, the proposed schemes are rigorously proven to unconditionally preserve the maximum bound without requiring any additional postprocessing techniques, while simultaneously achieving arbitrarily high-order temporal accuracy. A thorough error analysis in the $L^2$ norm is provided. Numerical experiments validate the theoretical results, demonstrate the effectiveness of the proposed methods, and highlight that an inappropriate choice of quadrature rules may violate the maximum bound principle, leading to incorrect dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09874 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximum bound preservation of exponential integrators for Allen-Cahn equations Quan, Chaoyu Zheng, Pingzhong Zhou, Zhi Numerical Analysis 65M12, 65M15, 65R20 We develop and analyze a class of arbitrarily high-order, maximum bound preserving time-stepping schemes for solving Allen-Cahn equations. These schemes are constructed within the iterative framework of exponential integrators, combined with carefully chosen numerical quadrature rules, including the Gauss-Legendre quadrature rule and the left Gauss-Radau quadrature rule. Notably, the proposed schemes are rigorously proven to unconditionally preserve the maximum bound without requiring any additional postprocessing techniques, while simultaneously achieving arbitrarily high-order temporal accuracy. A thorough error analysis in the $L^2$ norm is provided. Numerical experiments validate the theoretical results, demonstrate the effectiveness of the proposed methods, and highlight that an inappropriate choice of quadrature rules may violate the maximum bound principle, leading to incorrect dynamics. |
| title | Maximum bound preservation of exponential integrators for Allen-Cahn equations |
| topic | Numerical Analysis 65M12, 65M15, 65R20 |
| url | https://arxiv.org/abs/2504.09874 |