Maximum bound principle preserving and energy decreasing exponential time differencing schemes for the matrix-valued Allen-Cahn equation
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| Format: | Preprint |
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2023
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| _version_ | 1866916488249606144 |
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| author | Liu, Yaru Quan, Chaoyu Wang, Dong |
| author_facet | Liu, Yaru Quan, Chaoyu Wang, Dong |
| contents | This work delves into the exponential time differencing (ETD) schemes for the matrix-valued Allen-Cahn equation. In fact, the maximum bound principle (MBP) for the first- and second-order ETD schemes is presented in a prior publication [SIAM Review, 63(2), 2021], assuming a symmetric initial matrix field. Noteworthy is our novel contribution, demonstrating that the first- and second-order ETD schemes for the matrix-valued Allen-Cahn equation -- both being linear schemes -- unconditionally preserve the MBP, even in instances of nonsymmetric initial conditions. Additionally, we prove that these two ETD schemes preserve the energy dissipation law unconditionally for the matrix-valued Allen-Cahn equation. Some numerical examples are presented to verify our theoretical results and to simulate the evolution of corresponding matrix fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15613 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Maximum bound principle preserving and energy decreasing exponential time differencing schemes for the matrix-valued Allen-Cahn equation Liu, Yaru Quan, Chaoyu Wang, Dong Numerical Analysis This work delves into the exponential time differencing (ETD) schemes for the matrix-valued Allen-Cahn equation. In fact, the maximum bound principle (MBP) for the first- and second-order ETD schemes is presented in a prior publication [SIAM Review, 63(2), 2021], assuming a symmetric initial matrix field. Noteworthy is our novel contribution, demonstrating that the first- and second-order ETD schemes for the matrix-valued Allen-Cahn equation -- both being linear schemes -- unconditionally preserve the MBP, even in instances of nonsymmetric initial conditions. Additionally, we prove that these two ETD schemes preserve the energy dissipation law unconditionally for the matrix-valued Allen-Cahn equation. Some numerical examples are presented to verify our theoretical results and to simulate the evolution of corresponding matrix fields. |
| title | Maximum bound principle preserving and energy decreasing exponential time differencing schemes for the matrix-valued Allen-Cahn equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2312.15613 |