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BIT Numerical Mathematics

, Volume 55, Issue 3, pp 733–750 | Cite as

Stability ordinates of Adams predictor-corrector methods

  • Michelle L. GhristEmail author
  • Bengt Fornberg
  • Jonah A. Reeger
Article

Abstract

How far the stability domain of a numerical method for approximating solutions to differential equations extends along the imaginary axis indicates how useful the method is for approximating solutions to wave equations; this maximum extent is termed the imaginary stability boundary, also known as the stability ordinate. It has previously been shown that exactly half of Adams-Bashforth (AB), Adams-Moulton (AM), and staggered Adams-Bashforth methods have nonzero stability ordinates. In this paper, we consider two categories of Adams predictor-corrector methods and prove that they follow a similar pattern. In particular, if \(p\) is the order of the method, AB\(p\)-AM\(p\) methods have nonzero stability ordinate only for \(p = 1, 2, \ 5, 6,\ 9, 10, \ldots \), and AB(\(p-\)1)-AM\(p\) methods have nonzero stability ordinates only for \(p = 3, 4, \ 7, 8, \ 11, 12, \ldots \).

Keywords

Adams methods Predictor-corrector Imaginary stability boundary Linear multistep methods Finite difference methods Stability region Stability ordinate 

Mathematics Subject Classification

65L06 65L12 65L20 65M06 65M12 

Notes

Acknowledgments

The authors are extremely grateful to Ernst Hairer for suggesting major simplifications in a previous form of this manuscript, in particular with regard to using the backward difference forms of AB and AM methods. We are also grateful for helpful comments from the referees that allowed us to improve our manuscript.

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Copyright information

© Springer Science+Business Media Dordrecht (outside the USA) 2014

Authors and Affiliations

  • Michelle L. Ghrist
    • 1
    Email author
  • Bengt Fornberg
    • 2
  • Jonah A. Reeger
    • 3
  1. 1.Department of Mathematical SciencesUnited States Air Force AcademyUSAF AcademyUSA
  2. 2.Department of Applied MathematicsUniversity of ColoradoBoulderUSA
  3. 3.Department of Mathematics and StatisticsAir Force Institute of TechnologyWPAFBUSA

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