Abstract
In this paper we apply the theory for implicit Runge-Kutta methods presented by Stetter to a number of subclasses of methods that have recently been discussed in the literature. We first show how each of these classes can be expressed within this theoretical framework and from this we are able to establish a number of relationships among these classes. In addition to improving the current state of understanding of these methods, their expression within this theoretical framework makes it possible for us to obtain results giving general forms for their stability functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. M. G. van Bokhoven,Efficient higher order implicit one-step methods for integration of stiff differential equations, BIT 20 (1980), 34–43.
J. C. Butcher,Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50–64.
id.,, Math. Comp. 26 (1972), 79–106.
id.,, BIT 24 (1984), 425–440.
J. R. Cash,A class of implicit Runge-Kutta methods for the numerical integration of stiff ordinary differential equations, J. Assoc. Comput. Mach. 22 (1975), 504–511.
id.,, in Proc. International Conference on Stiff Computation, Utah, R. C. Aikens, ed., Oxford University Press, London (1982).
J. R. Cash and D. R. Moore,A high order method for the numerical solution of two-point boundary value problems, BIT 20 (1980), 44–52.
J. R. Cash and A. Singhal,Mono-implicit Runge-Kutta formulae for the numerical integration of stiff differential systems, IMA J. Numer. Anal. 2 (1982), 211–227.
id.,, BIT 22 (1982), 184–199.
W. H. Enright and P. H. Muir,Efficient classes of Runge-Kutta methods for two-point boundary value problems, Computing 37 (1986), 315–334.
S. Gupta,An adaptive boundary value Runge-Kutta solver for first order boundary value problems, SIAM J. Numer, Anal. 22 (1985), 114–126.
W. H. Hundsdorfer and M. N. Spijker,A note on B-stability of Runge-Kutta methods, Numer. Math. 36 (1981), 319–331.
P. H. Muir,Implicit Runge-Kutta methods for two-point boundary value problems, Ph.D. Thesis, Department of Computer Science, University of Toronto; Also Department of Computer Science, Tech. Rep. 175/84 (1984), University of Toronto.
R. Scherer and H. Türke,Reflected and transposed methods, BIT 23 (1983), 262–266.
J. Sherman and W. J. Morrison,Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix, Ann. Math. Statist. 20 (1949), 621.
H. J. Stetter,Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, New York (1973).
Author information
Authors and Affiliations
Additional information
This work was supported by the Natural Sciences and Engineering Research Council of Canada.
Rights and permissions
About this article
Cite this article
Muir, P.H., Enright, W.H. Relationships among some classes of implicit Runge-Kutta methods and their stability functions. BIT 27, 403–423 (1987). https://doi.org/10.1007/BF01933734
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01933734