Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Implicit Runge-Kutta formulae with built-in estimates of the accumulated truncation error

Implizite Runge-Kutta-Formeln mit eingebauter Schätzung des globalen Diskretisierungsfehlers

  • 42 Accesses

  • 3 Citations

Abstract

Explicit Runge-Kutta formulae with built-in estimates of the accumulated truncation error are well known. A method is presented for developingA-stable and stronglyA-stable Runge-Kutta algorithms with built-in estimates of the accumulated truncation error.

Zusammenfassung

Explizite Runge-Kutta-Formeln mit eingebauter Schätzung des globalen Diskretisierungsfehlers sind wohlbekannt. Es wird ein Vorgehen dargestellt,A-stabile und starkA-stabile Runge-Kutta-Formeln mit eingebauter Schätzung des globalen Diskretisierungsfehlers zu gewinnen.

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

References

  1. [1]

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

  2. [2]

    Chipman, F. H.: A stable Runge-Kutta processes. BIT11, 384–388 (1971).

  3. [3]

    Dormand, J. R., Duckers, R. R., Prince, P. J.: Global error estimation with Runge-Kutta methods. IMA Journal of Numerical Analysis4, 169–184 (1984).

  4. [4]

    Enright, W. H., Hull, T. E., Lindberg, B.: Comparing numerical methods for stiff systems of odes.

  5. [5]

    Fehlberg, E.: Some old and new Runge-Kutta formulas with stepsize control and their error coefficients. Computing34, 265–270 (1985).

  6. [6]

    Grigorieff, R. D.: Numerik gewöhnlicher Differentialgleichungen 1. Stuttgart: Verlag B. G. Teubner 1972.

  7. [7]

    Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. New York: John Wiley and Sons 1963.

  8. [8]

    Lapidus, L., Seinfeld, J. H.: Numerical Solution of Ordinary Differential Equations. New York: Academic Press 1971.

  9. [9]

    Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Quart. Appl. Math.2, 164–168 (1944).

  10. [10]

    Marquardt, D. W.: An algorithm for least-squares estimation of nonlinear parameters. J. SIAM Appl. Math.11, 431–441 (1963).

  11. [11]

    Merluzzi, P. J.: A new class of Runge-Kutta integration processes, with improved error estimates, for automatic simulation. Ph. D. Thesis, Case Western Reserve University, 1974.

  12. [12]

    Merluzzi, P., Brosilow, C.: Runge-Kutta integration algorithms with built-in estimates of the accumulated truncation error. Computing20, 1–16 (1978).

  13. [13]

    Schmid, F. J.: Runge-Kutta-Verfahren mit eingebauter Schätzung des globalen Diskretisierungsfehlers. Diplomarbeit, Mathematisches Institut, Universität München.

  14. [14]

    Shampine, L. F., Watts, H. A.: Global error estimation for ordinary differential equations. ACM Transactions on Math. Software2, 172–186 (1976).

  15. [15]

    Stetter, H. J.: Local estimation of the global discretization error. SIAM J. Numer. Anal.8, 512–523 (1971).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Richert, W.R. Implicit Runge-Kutta formulae with built-in estimates of the accumulated truncation error. Computing 39, 353–362 (1987). https://doi.org/10.1007/BF02239977

Download citation

Key words

  • Runge-Kutta-integration
  • truncation error
  • error estimates
  • accumulated error
  • true error
  • A-stability