Continuous explicit Runge-Kutta methods of order $5$
HTML articles powered by AMS MathViewer
- by J. H. Verner and M. Zennaro PDF
- Math. Comp. 64 (1995), 1123-1146 Request permission
Abstract:
A continuous explicit Runge-Kutta (CERK) method provides a continuous approximation to an initial value problem. Such a method may be obtained by appending additional stages to a discrete method, or alternatively by solving the appropriate order conditions directly. Owren and Zennaro have shown for order 5 that the latter approach yields some CERK methods that require fewer derivative evaluations than methods obtained by appending stages. In contrast, continuous methods of order 6 that require the minimum number of stages can be obtained by appending additional stages to certain discrete methods. This article begins a study to understand why this occurs. By making no assumptions to simplify solution of the order conditions, the existence of other types of CERK methods of order 5 is established. While methods of the new families may not be as good for implementation as the Owren-Zennaro methods, the structure is expected to lead to a better understanding of how to construct families of methods of higher order.References
- J. C. Butcher, The numerical analysis of ordinary differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. MR 878564
- J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6 (1980), no. 1, 19–26. MR 568599, DOI 10.1016/0771-050X(80)90013-3
- W. H. Enright, K. R. Jackson, S. P. Nørsett, and P. G. Thomsen, Interpolants for Runge-Kutta formulas, ACM Trans. Math. Software 12 (1986), no. 3, 193–218. MR 889066, DOI 10.1145/7921.7923
- E. Fehlberg, Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle, Computing (Arch. Elektron. Rechnen) 4 (1969), 93–106 (German, with English summary). MR 260179, DOI 10.1007/bf02234758
- Brynjulf Owren and Marino Zennaro, Order barriers for continuous explicit Runge-Kutta methods, Math. Comp. 56 (1991), no. 194, 645–661. MR 1068811, DOI 10.1090/S0025-5718-1991-1068811-2
- Brynjulf Owren and Marino Zennaro, Derivation of efficient, continuous, explicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput. 13 (1992), no. 6, 1488–1501. MR 1185658, DOI 10.1137/0913084
- P. J. Prince and J. R. Dormand, High order embedded Runge-Kutta formulae, J. Comput. Appl. Math. 7 (1981), no. 1, 67–75. MR 611953, DOI 10.1016/0771-050X(81)90010-3 M. Santo, Metodi continui ad un passo per la risoluzione numerica di equazioni differenziali ordinarie, Ph.D. thesis, Univ. of Udine, Italy, 1991.
- J. H. Verner, Explicit Runge-Kutta methods with estimates of the local truncation error, SIAM J. Numer. Anal. 15 (1978), no. 4, 772–790. MR 483471, DOI 10.1137/0715051
- J. H. Verner, Differentiable interpolants for high-order Runge-Kutta methods, SIAM J. Numer. Anal. 30 (1993), no. 5, 1446–1466. MR 1239830, DOI 10.1137/0730075 J.H. Verner and M. Zennaro, Continuous explicit Runge-Kutta methods of order 5, Report 1993-08, Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada (1993), 1-32.
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp. 64 (1995), 1123-1146
- MSC: Primary 65L06; Secondary 65Y20
- DOI: https://doi.org/10.1090/S0025-5718-1995-1284672-4
- MathSciNet review: 1284672