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 via an institution to check access.
Similar content being viewed by others
References
Butcher, J. C.: Coefficients for the study of Runge-Kutta integration processes. J. Aust.Math. Soc.3, 185–201 (1963).
Chipman, F. H.: A stable Runge-Kutta processes. BIT11, 384–388 (1971).
Dormand, J. R., Duckers, R. R., Prince, P. J.: Global error estimation with Runge-Kutta methods. IMA Journal of Numerical Analysis4, 169–184 (1984).
Enright, W. H., Hull, T. E., Lindberg, B.: Comparing numerical methods for stiff systems of odes.
Fehlberg, E.: Some old and new Runge-Kutta formulas with stepsize control and their error coefficients. Computing34, 265–270 (1985).
Grigorieff, R. D.: Numerik gewöhnlicher Differentialgleichungen 1. Stuttgart: Verlag B. G. Teubner 1972.
Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. New York: John Wiley and Sons 1963.
Lapidus, L., Seinfeld, J. H.: Numerical Solution of Ordinary Differential Equations. New York: Academic Press 1971.
Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Quart. Appl. Math.2, 164–168 (1944).
Marquardt, D. W.: An algorithm for least-squares estimation of nonlinear parameters. J. SIAM Appl. Math.11, 431–441 (1963).
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.
Merluzzi, P., Brosilow, C.: Runge-Kutta integration algorithms with built-in estimates of the accumulated truncation error. Computing20, 1–16 (1978).
Schmid, F. J.: Runge-Kutta-Verfahren mit eingebauter Schätzung des globalen Diskretisierungsfehlers. Diplomarbeit, Mathematisches Institut, Universität München.
Shampine, L. F., Watts, H. A.: Global error estimation for ordinary differential equations. ACM Transactions on Math. Software2, 172–186 (1976).
Stetter, H. J.: Local estimation of the global discretization error. SIAM J. Numer. Anal.8, 512–523 (1971).
Author information
Authors and Affiliations
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02239977