Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908683957436416 |
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| author | Chen, Wenbin Fu, Zhaohui Wang, Shun Wang, Xiaoming |
| author_facet | Chen, Wenbin Fu, Zhaohui Wang, Shun Wang, Xiaoming |
| contents | We analyze a variable-step extension of a family of arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes recently developed by the authors. We prove that the schemes are unconditionally stable in the sense that a modified energy-representing a slight perturbation of the original energy-decreases monotonically over time, provided the nonlinearity is Lipschitz continuous in some appropriate sense. Moreover, we establish optimal-order convergence under mild conditions on the time-step size and local time-step ratio. Numerical experiments on the thin film epitaxial growth model without slope selection, employing a novel variable-step second-order scheme, validate the theoretical findings as well as its potential in developing highly efficient time-adaptive solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity Chen, Wenbin Fu, Zhaohui Wang, Shun Wang, Xiaoming Numerical Analysis 65M12, 65M70, 65Z05 We analyze a variable-step extension of a family of arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes recently developed by the authors. We prove that the schemes are unconditionally stable in the sense that a modified energy-representing a slight perturbation of the original energy-decreases monotonically over time, provided the nonlinearity is Lipschitz continuous in some appropriate sense. Moreover, we establish optimal-order convergence under mild conditions on the time-step size and local time-step ratio. Numerical experiments on the thin film epitaxial growth model without slope selection, employing a novel variable-step second-order scheme, validate the theoretical findings as well as its potential in developing highly efficient time-adaptive solution. |
| title | Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity |
| topic | Numerical Analysis 65M12, 65M70, 65Z05 |
| url | https://arxiv.org/abs/2512.01601 |