Stability of variable-step BDF2 and BDF3 methods

@article{Li2022StabilityOV,
  title={Stability of variable-step BDF2 and BDF3 methods},
  author={Zhaoyi Li and Hong-lin Liao},
  journal={ArXiv},
  year={2022},
  volume={abs/2201.00527}
}
We prove that the two-step backward differentiation formula (BDF2) method is stable on arbitrary time grids; while the variable-step BDF3 scheme is stable if almost all adjacent step ratios are less than 2.553. These results relax the severe mesh restrictions in the literature and provide a new understanding of variable-step BDF methods. Our main tools include the discrete orthogonal convolution kernels and an elliptic-type matrix norm. 

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