Global-in-time energy stability: a powerful analysis tool for the gradient flow problem without maximum principle or Lipschitz assumption
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914832641425408 |
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| author | Sun, J. Wang, H. Zhang, H. Qian, X. Song, S. |
| author_facet | Sun, J. Wang, H. Zhang, H. Qian, X. Song, S. |
| contents | Before proving (unconditional) energy stability for gradient flows, most existing studies either require a strong Lipschitz condition regarding the non-linearity or certain $L^{\infty}$ bounds on the numerical solutions (the maximum principle). However, proving energy stability without such premises is a very challenging task. In this paper, we aim to develop a novel analytical tool, namely global-in-time energy stability, to demonstrate energy dissipation without assuming any strong Lipschitz condition or $L^{\infty}$ boundedness. The fourth-order-in-space Swift-Hohenberg equation is used to elucidate the theoretical results in detail. We also propose a temporal second-order accurate scheme for efficiently solving such a strongly stiff equation. Furthermore, we present the corresponding optimal $L^2$ error estimate and provide several numerical simulations to demonstrate the dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07941 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global-in-time energy stability: a powerful analysis tool for the gradient flow problem without maximum principle or Lipschitz assumption Sun, J. Wang, H. Zhang, H. Qian, X. Song, S. Numerical Analysis Before proving (unconditional) energy stability for gradient flows, most existing studies either require a strong Lipschitz condition regarding the non-linearity or certain $L^{\infty}$ bounds on the numerical solutions (the maximum principle). However, proving energy stability without such premises is a very challenging task. In this paper, we aim to develop a novel analytical tool, namely global-in-time energy stability, to demonstrate energy dissipation without assuming any strong Lipschitz condition or $L^{\infty}$ boundedness. The fourth-order-in-space Swift-Hohenberg equation is used to elucidate the theoretical results in detail. We also propose a temporal second-order accurate scheme for efficiently solving such a strongly stiff equation. Furthermore, we present the corresponding optimal $L^2$ error estimate and provide several numerical simulations to demonstrate the dynamics. |
| title | Global-in-time energy stability: a powerful analysis tool for the gradient flow problem without maximum principle or Lipschitz assumption |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2406.07941 |