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


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


CR classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations