Abstract
An efficient way of implementing Implicit Runge-Kutta Methods was proposed by Butcher [3]. He showed that the most efficient methods when using this implementation are those whose characteristic polynomial of the Runge-Kutta matrix has a single reals-fold zero. In this paper we will construct such a family of methods and give some results concerning their maximum attainable order and stability properties. Some consideration is also given to showing how these methods can be efficiently implemented and, in particular, how local error estimates can be obtained by the use of embedding techniques.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. C. Butcher,Implicit Runge-Kutta Processes, Math. Comp. 18 (1964), 50–64.
J. C. Butcher,Chapter 5, Modern Numerical Methods for Ordinary Differential Equations, edited by G. Hall and J. M. Watt, Clarendon Press, Oxford 1976.
J. C. Butcher,On the Implementation of Runge-Kutta Methods, BIT 16 (1976), 237–240.
J. C. Butcher,A Transformed Implicit Runge-Kutta Method, Mathematics Department, University of Auckland, Report Series No. 111.
S. P. Nørsett,Multiple Padé approximations to the Exponential Function, Mathematics Department, University of Trondheim, Report No. 4/74.
S. P. Nørsett,Semi-Explicit Runge-Kutta Methods, Mathematics Department, University of Trondheim, Report No. 6/74.
S. P. Nørsett,Runge-Kutta Methods with a multiple real eigenvalue only, BIT 16 (1976), 388–393.
S. P. Nørsett and A. Wolfbrandt,Attainable order of rational approximations to the exponential function with only real poles, BIT 17 (1977), 200–208.
J. V. Uspensky,Theory of Equations, McGraw-Hill, New York, 1948.
J. H. Verner,Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error, Mathematics Department, University of Auckland, Report Series No. 92.
A. Wolfbrandt,A Study of Rosenbrock Processes with respect to order conditions and Stiff Stability, Department of Computer Sciences, University of Göteborg, Sweden.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Burrage, K. A special family of Runge-Kutta methods for solving stiff differential equations. BIT 18, 22–41 (1978). https://doi.org/10.1007/BF01947741
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01947741