Skip to main content
SpringerLink
Log in

Nordsieck Methods with an Off-Step Point

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Hybrid methods, incorporating one or more off-step points, are difficult to implement in a variable stepsize situation using the standard representation of input and output data in each step. However, instead of representing this data in terms of solution values and derivative values at a sequence of step points, it is possible to reformulate the method so that it operates on a Nordsieck vector. This has the consequence of reducing stepsize adjustments to nothing more than rescaling the components of the Nordsieck vector. This paper shows how to derive methods in both formulations and considers some implementation details. It is also possible to derive a new type of hybrid method using the Norsieck representation as the starting point and this is also discussed in the paper. The new method is found to have comparable accuracy for corresponding work expended as for standard methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P.R. Beaudet, Multi-off-grid methods in multi-step integration of ordinary differential equations, in: Proc. of Conf. on Numer. Solution of ODEs, University of Texas, 1972, pp. 128-148.

  2. J.C. Butcher, A modified multistep method for the numerical integration of ordinary differential equations, J. Assoc. Comput. Mach. 12(1) (1965) 124–135.

    Google Scholar 

  3. J.C. Butcher, A multistep Generalization of Runge-Kutta methods with four or five stages, J. Assoc. Comput. Mach. 14(1) (1967) 84–99.

    Google Scholar 

  4. C.W. Gear, Hybrid methods for initial value problems in ordinary differential equations, SIAM J. Numer. Anal. Ser B 2(1) (1964).

  5. W.B. Gragg and H.J. Stetter, Generalized multistep predictor-corrector methods, J. Assoc. Comput. Mach. 11(2) (1964) 188–209.

    Google Scholar 

  6. T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal. 9(4) (1972).

  7. J.J. Kohfeld and G.T. Thompson, A modification of Nordsieck's method using an “off-step” point, J. Assoc. Comput. Mach. 15(3) (1968) 390–401.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand

    J.C. Butcher

  2. Institute of Petroleum Engineering Heriot-Watt University, Edinburgh, EH14 4AS, UK

    A.E. O'Sullivan

Authors
  1. J.C. Butcher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.E. O'Sullivan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Butcher, J., O'Sullivan, A. Nordsieck Methods with an Off-Step Point. Numerical Algorithms 31, 87–101 (2002). https://doi.org/10.1023/A:1021104222126

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021104222126

Search

Navigation