Mesh-robust stability and convergence of variable-step deferred correction methods based on the BDF2 formula
<|reference_start|>Mesh-robust stability and convergence of variable-step deferred correction methods based on the BDF2 formula: 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.<|reference_end|>
@article{yue2024mesh-robust,
    title={Mesh-robust stability and convergence of variable-step deferred
  correction methods based on the BDF2 formula},
    author={Jiahe Yue, Hong-lin Liao, Nan Liu},
    journal={arXiv preprint arXiv:2402.06129},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.06129},
    primaryClass={math.NA cs.NA}
}