BIT Numerical Mathematics

, Volume 27, Issue 2, pp 181–189 | Cite as

Linear and non-linear stability for general linear methods

  • J. C. Butcher
Part II Numerical Mathematics

Abstract

We explore the interrelation between a number of linear and non-linear stability properties. The weakest of these,A-stability, is shown by counterexample not to imply any of the various versions ofAN-stability introduced in the paper and two of these properties, weak and strongAN-stability, are also shown not to be equivalent. Finally, another linear stability property defined here, EuclideanAN-stability, is shown to be equivalent to algebraic stability.

AMS classification

65L20 

CR classification

5.17 

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References

  1. 1.
    K. Burrage and J. C. Butcher,Stability criteria for implicit Runge-Kutta methods, Siam J. Numer. Anal. 16 (1979), 46–57.CrossRefGoogle Scholar
  2. 2.
    K. Burrage and J. C. Butcher,Non-linear stability of a general class of differential equation methods, BIT 20 (1980, 185–203.Google Scholar
  3. 3.
    G. Dahlquist,G-stability is equivalent to A-stability, BIT 18 (1978), 384–401.Google Scholar
  4. 4.
    J. D. Lambert,Computational Methods in Ordinary Differential Equations, John Wiley & Sons, London (1973).Google Scholar

Copyright information

© BIT Foundations 1987

Authors and Affiliations

  • J. C. Butcher
    • 1
  1. 1.Department of Computer ScienceThe University of AucklandAucklandNew Zealand

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