Abstract
Well-known stability concepts for Runge-Kutta methods areA-stability andB-stability. These stability properties can be characterized by algebraic conditions related to the generating matrix of the method. In this note we show, thatI-stable methods can be characterized similarly, yielding aunified description ofB-,A- andI-stability in terms of a matrixR.I-stability, although a weaker concept thanA-stability, is of some relevance in parallelizing Runge-Kutta methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. D. O. Anderson and J. B. Moore,Algebraic structure of generalized positive real matrices, SIAM J. Control v. 6, 1968, pp. 615–624.
K. Dekker and J. G. Verwer,Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North Holland, Amsterdam, New York, Oxford, 1984.
E. Hairer and G. Wanner,Algebraically stable and implementable Runge-Kutta methods of high order, SIAM, J. Numer. Anal. v. 18, 1981, pp. 1098–1108.
S. L. Keeling,On implicit Runge-Kutta methods with a stability function having distinct real poles, BIT v. 29, 1989, pp. 91–109.
M. Müller,Algebraische Stabilitätsbedingungen für Runge-Kutta-Verfahren, Dissertation, Universität Karlsruhe, 1990.
S. P. Nørsett and G. Wanner,The real-pole sandwich for rational approximations and oscillation equations, BIT v. 19, 1979, pp. 79–94.
S. P. Nørsett and A. Wolfbrandt,Attainable order of rational approximations to the exponential function with only real poles, BIT v. 17, 1977, pp. 200–208.
G. N. de Oliveira,Matrices with prescribed submatrices, Linear and Multilinear Algebra v. 12, 1982, pp. 125–138.
R. Scherer and H. Türke,Algebraic characterization of A-stable Runge-Kutta methods, Appl. Numer. Math., v. 5, 1989, pp. 133–144.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Müller, M. Algebraic characterization ofI-stable Runge-Kutta methods. BIT 31, 314–320 (1991). https://doi.org/10.1007/BF01931290
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01931290