Skip to main content
SpringerLink
Log in

The stability of natural Runge-Kutta methods for nonlinear delay differential equations

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any collocation method is equivalent to one of such methods. In this paper, stability properties of natural Runge-Kutta methods are studied using nonlinear DDEs which have a quadratic Liapunov functional. A discrete analogue of the functional is defined for each method, and the stability of the method is examined on the basis of this analogue. In particular, it is shown that an algebracially stable method, if it satisfies an additional condition, preserves the asymptotic properties of the original equations for every stepsize.

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. V.K. Barwell, Special stability problems for functional differential equations. BIT,15 (1975), 130–135.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Bellen, Z. Jackiewicz and M. Zennaro, Stability analysis of one-step methods for neutral delay-differential equations. Numer. Math.52 (1988), 605–619.

    Article  MATH  MathSciNet  Google Scholar 

  3. J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations. John Wiley & Sons, Chichester, 1987.

    MATH  Google Scholar 

  4. P. Constantin and C. Foias, Navier-Stokes Equations. The University of Chicago Press, Chicago, 1988.

    MATH  Google Scholar 

  5. G.J. Cooper, An algebraic condition forA-stable Runge-Kutta methods. Numerical Analysis (eds. D. F. Griffiths and G. A. Watson), Pitman Research Notes in Mathematics,140, Pitman, London, 1986, 32–46.

    Google Scholar 

  6. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Springer-Verlag, Berlin, Heidelberg, New York, 1991.

    MATH  Google Scholar 

  7. J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations. Springer-Verlag, Berlin, Heidelberg, New York, 1993.

    MATH  Google Scholar 

  8. G. Hu and T. Mitsui, Stability analysis of numerical methods for systems of neutral delay-differential equations. BIT,35 (1995), 504–515.

    Article  MATH  MathSciNet  Google Scholar 

  9. A.R. Humphries and A.M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems. SIAM J. Numer. Anal.,31 (1994), 1452–1485.

    Article  MATH  MathSciNet  Google Scholar 

  10. K.J. in’t Hout, A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations. BIT,32 (1992), 634–649.

    Article  MATH  MathSciNet  Google Scholar 

  11. K.J. in’t Hout, Stability analysis of Runge-Kutta methods for systems of delay differential equations. Preprint, Department of Mathematics and Computer Science, Leiden University, 1995.

  12. T. Koto, A stability property ofA-stable natural Runge-Kutta methods for systems of delay differential equations. BIT,34 (1994), 262–267.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Koto, Stability properties of Runge-Kutta methods for delay differential equations. Advances in Numerical Mathematics; Proceedings of the Second Japan-China Seminar on Numerical Mathematics (eds. T. Ushijima, Z.-C. Shi and T. Kako), Lecture Notes in Num. Appl. Anal.,14 (1995), 107–114, Kinokuniya, Tokyo.

  14. Y. Kuang, Delay Differential Equations. With Applications in Population Dynamics. Academic Press, San Diego, 1993.

    MATH  Google Scholar 

  15. R. Scherer and M. Müller, Algebraic conditions forA-stable Runge-Kutta methods. Computational Ordinary Differential Equations (eds. J.R. Cash and I. Gradwell), Clarendon Press, Oxford, 1992, 1–7.

    Google Scholar 

  16. R. Scherer and W. Wendler, Complete algebraic characterization ofA-stable Runge-Kutta methods. SIAM J. Numer. Anal.,31 (1994), 540–551.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Slemrod and E.F. Infante, Asymptotic stability criteria for linear systems of difference-differential equations of neutral type and their discrete analogues. J. Math. Anal. Appl.,38 (1972), 399–415.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Zennaro, Natural continuous extensions of Runge-Kutta methods. Math. Comput.,46 (1986), 119–133.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Zennaro,P-stability properties of Runge-Kutta methods for delay differential equations. Numer. Math.,49 (1986), 305–318.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Zennaro, Natural Runge-Kutta and projection methods. Numer. Math.,53 (1988), 423–438.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Zennaro, Delay differential equations: Theory and numerics. Theory and Numerics of Ordinary and Partial Differential Equations (eds. M. Ainsworth, J. Levesley, W.A. Light and M. Marletta), Oxford University Press, Oxford, 1995, 291–333.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Information Mathematics, The University of Electro-Communications, 1-5-1, Chofugaoka, 182, Chofu, Tokyo, Japan

    Toshiyuki Koto

Authors
  1. Toshiyuki Koto
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Koto, T. The stability of natural Runge-Kutta methods for nonlinear delay differential equations. Japan J. Indust. Appl. Math. 14, 111–123 (1997). https://doi.org/10.1007/BF03167314

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167314

Key words

Search

Navigation