Variable order and stepsize in general linear methods

  • Saghir AhmadEmail author
  • J. C. Butcher
  • Winston L. Sweatman
Original Paper


This paper describes the implementation of a class of IRKS methods (Wright 2002). These GLM algorithms are practical with reliable error estimators (Butcher and Podhaisky, Appl. Numer. Math. 56, 345–357 2006). The current robust ODE solvers in variable stepsize as well as in variable-order mode are based upon heuristics. In this paper, we examine an optimisation approach, based on Euler-Lagrange theory (Butcher, IMA J. Numer. Anal. 6, 433–438 1986), (Butcher, Computing 44, 209–220 1990), to control the stepsize as well as the order and implement the GLMs in an efficient manner. A set of nonstiff to mildly stiff problems have been used to investigate this approach in fixed-order and variable-order modes.


General linear methods Variable order Variable stepsize Lagrange multiplier controller 

Mathematics Subject Classification (2010)



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Copyright information

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Authors and Affiliations

  • Saghir Ahmad
    • 1
    Email author
  • J. C. Butcher
    • 2
  • Winston L. Sweatman
    • 3
  1. 1.Auckland Institute of StudiesAucklandNew Zealand
  2. 2.The University of AucklandAucklandNew Zealand
  3. 3.Massey UniversityAucklandNew Zealand

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