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Stability ordinates of Adams predictor-corrector methods

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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 \(p = 3, 4, \ 7, 8, \ 11, 12, \ldots \).

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References

  1. Atkinson, K.: An Introduction to Numerical Analysis. Wiley, New York (1989)

    MATH  Google Scholar 

  2. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  3. Ghrist, M: High-order Finite Difference Methods for Wave Equations. Ph.D. thesis, Department of Applied Mathematics, University of Colorado-Boulder, Boulder, CO (2000)

  4. Ghrist, M., Fornberg, B., Driscoll, T.A.: Staggered time integrators for wave equations. SIAM J. Num. Anal. 38, 718–741 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ghrist, M., Fornberg, B.: Two results concerning the stability of staggered multistep methods. SIAM J. Num. Anal. 50, 1849–1860 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Springer, Berlin (1991)

    Book  Google Scholar 

  7. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, 2nd edn. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  8. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1962)

    MATH  Google Scholar 

  9. Iserles, A.: Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  10. Jeltsch, R.: A necessary condition for A-stability of multistep multiderivative methods. Math. Comp. 30, 739–746 (1976)

    MATH  MathSciNet  Google Scholar 

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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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Authors and Affiliations

  1. Department of Mathematical Sciences, United States Air Force Academy, 2354 Fairchild, Suite 6D2, USAF Academy, CO, 80840, USA

    Michelle L. Ghrist

  2. Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, CO, 80309, USA

    Bengt Fornberg

  3. Department of Mathematics and Statistics, Air Force Institute of Technology, 2950 Hobson Way, WPAFB, OH, 45433, USA

    Jonah A. Reeger

Authors
  1. Michelle L. Ghrist
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  2. Bengt Fornberg
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  3. Jonah A. Reeger
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Corresponding author

Correspondence to Michelle L. Ghrist.

Additional information

Communicated by Mechthild Thalhammer.

Support for M. Ghrist and J. Reeger provided by the United States Air Force. Support for B. Fornberg provided by NSF DMS-0914647.

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Ghrist, M.L., Fornberg, B. & Reeger, J.A. Stability ordinates of Adams predictor-corrector methods. Bit Numer Math 55, 733–750 (2015). https://doi.org/10.1007/s10543-014-0528-7

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  • DOI: https://doi.org/10.1007/s10543-014-0528-7

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