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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Analysis of adaptive BDF2 scheme for diffusion equations
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by Hong-lin Liao and Zhimin Zhang HTML | PDF
Math. Comp. 90 (2021), 1207-1226 Request permission


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
  • Email:,
  • 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
  • Email:,
  • 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.
  • © Copyright 2020 American Mathematical Society
  • Journal: Math. Comp. 90 (2021), 1207-1226
  • MSC (2020): Primary 65M06, 65M12
  • DOI:
  • MathSciNet review: 4232222