Skip to main content
SpringerLink
Log in

Rational Runge-Kutta methods for solving systems of ordinary differential equations

Rationale Runge-Kutta Methoden für systeme gewöhnlicher Differentialgleichungen

  • Published:
Computing Aims and scope Submit manuscript

Abstract

Some nonlinear methods for solving single ordinary differiential equations are generalized to solve systems of equations. To perform this, a new vector product, compatible with the Samelson inverse of a vector, is defined. Conditions for a given order are derived.

Zusammenfassung

Einige nichtlineare Methoden zur numerischen Lösung skalarer gewöhnlicher Differentialgleichungen werden für Systeme verallgemeinert. Zu diesem Zweck wird ein mit den Samelson-Inversen eines Vektors kompatibles Vektorprodukt definiert. Die Bedingungen für einer vorgegebene Ordnung werden hergeleitet.

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. Padé, H.: Sur la représentation approchée d'une fonction par des fractions rationelles. Am. Sci. Ecole Normale Supér.9, 1–92 (1892).

    Google Scholar 

  2. Merson, R. H.: An operational method for the study of investigration processes, Proceedings of a symposiusm on data processing and Automatic Computing Machines at Weapons Research Establishment (1957), Salisbury, Australia. Paper No. 110.

  3. Wynn, P.: Acceleration Techniques for Iterated vector and Matrix Problems. Math. Comp.16, 322 (1962).

    Google Scholar 

  4. Butcher, J. C.: Coefficients for the study of Runge-Kutta integration processes. J. Aust. Math. Soc.3, 185–201 (1963).

    Google Scholar 

  5. Butcher, J. C.: Implicit Runge-Kutta processes. Math. Comp.18, 50–64 (1964).

    Google Scholar 

  6. Scraton, R. E.: Estimation of the truncation error in Runge-Kuta and allied processes. The Computer Journal7, 245–248 (1964).

    Google Scholar 

  7. Butcher, J. C.: On the attainable order of Runge-Kutta methods. Math. Comp.19 408–417 (1965).

    Google Scholar 

  8. Lambert, J. D., Shaw, B.: On the numerical solution ofy′=f(x, y) by a class of formulae based on rational approximation. Math. Comp.19, 456–462 (1965).

    Google Scholar 

  9. Curtis, A. R.: Letter to the Editor. The Computer Journal8, 52 (1965).

    Google Scholar 

  10. Gragg, W. B.: The Padé table and its relation to certain algorithms of numerical analysis. SIAM Review14, 1–62 (1972).

    Article  Google Scholar 

  11. Lambert, J. D.: The unconvential classes of methods for stiff systems, in: Stiff Differential Equations (Willoughby, R., ed.), pp. 171–186. 1974.

  12. Lambert, J. D.: Computational methods in ordinary differential equations. London: J. Wiley 1974.

    Google Scholar 

  13. Wambecq, A.: Nonlinear methods in solving ordinary differential equations. J. Comp. Appl. Math.2, 27–33 (1976).

    Article  Google Scholar 

  14. Gear, G. W.: Numerical initial valu9e problems in ordinary differential equations, pp. 218–219. Englewood Cliffs, N. J.: Prentice-Hall 1971.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mathematics and Programming Division, K. U. Leuven, Celestijnenlaan 200 B, B-3030, Heverlee, Gelgium

    A. Wambecq

Authors
  1. A. Wambecq
    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

Wambecq, A. Rational Runge-Kutta methods for solving systems of ordinary differential equations. Computing 20, 333–342 (1978). https://doi.org/10.1007/BF02252381

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02252381

Keywords

Search

Navigation