Advertisement

Trees and B-series

  • J. C. ButcherEmail author
Original Paper
  • 21 Downloads

Abstract

The connection between trees and differential equations was pointed out in the classic paper by Cayley (Phil. Mag. 13, 172–176 1857). Trees were also used in the work of Merson (1957), on the order of Runge–Kutta methods. The paper by Hairer and Wanner (Computing 13, 1–15 1974), where the term B-series was introduced, followed papers by the present author (Butcher J. Austral. Math. Soc. 3, 185–201 1963, Math. Comput. 26, 79–106 1972). The present paper will survey the use of trees in the formulation of B-series and illustrate the results by constructing and analysing some examples of general linear methods.

Keywords

Trees B-series 

Mathematics Subject Classification (2010)

65L05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

I am grateful to an anonymous referee for a helpful review of the paper.

References

  1. 1.
    Butcher, J.C.: Coefficients for the study of Runge–Kutta integration processes. J. Austral. Math. Soc. 3, 185–201 (1963)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Butcher, J.C.: An algebraic theory of integration methods. Math. Comput. 26, 79–106 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cayley, A.: On the theory of the analytical forms called trees. Phil. Mag. 13, 172–176 (1857)CrossRefGoogle Scholar
  4. 4.
    Ditkowski, D., Gottlieb, S.: Error inhibiting block one-step schemes for ordinary differential equations. J. Sci. Comput. 73, 691–711 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hairer, E., Wanner, G.: On the Butcher group and general multi-value methods. Computing 13, 1–15 (1974)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Łukasiewicz, J., Tarski, J.: Comp. rend. Soc. Sci. Lett. Warsaw 23(Class III), 31–32 (1930)Google Scholar
  7. 7.
    Merson, R.H.: An operational method for the study of integration processes. In: Proc. Symp. Data Processing, Weapons Research Establishment. Salisbury (1957)Google Scholar
  8. 8.
    Sweedler, M.: Hopf Algebras, Benjamin. New York (1969)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

Personalised recommendations