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Analysis of adaptive BDF2 scheme for diffusion equations

Authors: Hong-lin Liao and Zhimin Zhang
Journal: Math. Comp. 90 (2021), 1207-1226
MSC (2020): Primary 65M06, 65M12
Published electronically: December 28, 2020
MathSciNet review: 4232222
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Abstract: The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k\coloneq \tau _k/\tau _{k-1}\le (3+\sqrt {17})/2\approx 3.561$, the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the $L^2$ norm. The second-order temporal convergence can be recovered if almost all of time-step ratios $r_k\le 1+\sqrt {2}$ or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the $H^1$ seminorm) and the $L^2$ norm monotonicity at the discrete levels. An example is included to support our analysis.

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Additional Information

Hong-lin Liao
Affiliation: Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, People’s Republic of China
ORCID: 0000-0003-0777-6832

Zhimin Zhang
Affiliation: Beijing Computational Science Research Center, Beijing 100193, People’s Republic China; and Department of Mathematics, Wayne State University, Detroit, Michigan 48202
MR Author ID: 303173

Keywords: Linear diffusion equations, adaptive BDF2 scheme, orthogonal convolution kernels, positive semi-definiteness, stability and convergence
Received by editor(s): December 19, 2019
Received by editor(s) in revised form: June 21, 2020
Published electronically: December 28, 2020
Additional Notes: The first author was supported in part by the NSFC grant 12071216 and the grant 1008-56SYAH18037 from NUAA Scientific Research Starting Fund of Introduced Talent.
The second author is the corresponding author.
Article copyright: © Copyright 2020 American Mathematical Society