Stability of variable-step BDF2 and BDF3 methods
Zhaoyi Li, Hong-lin Liao
TL;DR
This paper establishes new stability results for variable-step BDF2 and BDF3 methods, relaxing previous mesh restrictions and enhancing understanding of their stability properties on arbitrary and nearly arbitrary grids.
Contribution
It proves stability of BDF2 on any grid and BDF3 under relaxed step ratio conditions, advancing the theoretical understanding of variable-step BDF methods.
Findings
BDF2 is stable on arbitrary time grids.
BDF3 is stable if most step ratios are below 2.553.
The results relax previous mesh restrictions.
Abstract
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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Taxonomy
TopicsNumerical methods for differential equations · Advanced Numerical Methods in Computational Mathematics · Matrix Theory and Algorithms