Mesh-robust stability and convergence of variable-step deferred correction methods based on the BDF2 formula
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909099347673088 |
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| author | Yue, Jiahe Liao, Hong-lin Liu, Nan |
| author_facet | Yue, Jiahe Liao, Hong-lin Liu, Nan |
| contents | We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06129 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mesh-robust stability and convergence of variable-step deferred correction methods based on the BDF2 formula Yue, Jiahe Liao, Hong-lin Liu, Nan Numerical Analysis 65M06, 65M12 We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory. |
| title | Mesh-robust stability and convergence of variable-step deferred correction methods based on the BDF2 formula |
| topic | Numerical Analysis 65M06, 65M12 |
| url | https://arxiv.org/abs/2402.06129 |