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Continuous explicit Runge-Kutta methods of order $ 5$

Authors: J. H. Verner and M. Zennaro
Journal: Math. Comp. 64 (1995), 1123-1146
MSC: Primary 65L06; Secondary 65Y20
MathSciNet review: 1284672
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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.

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  • [1] J. C. Butcher, The numerical analysis of ordinary differential equations, Wiley, Chichester, 1987. MR 878564 (88d:65002)
  • [2] J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6 (1980), 19-26. MR 568599 (81g:65098)
  • [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), 193-218. MR 889066
  • [4] E. Fehlberg, Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle, Computing 4 (1969), 93-106. MR 0260179 (41:4807)
  • [5] B. Owren and M. Zennaro, Order barriers for continuous explicit Runge-Kutta methods, Math. Comp. 56 (1991), 645-661. MR 1068811 (91i:65130)
  • [6] -, Derivation of efficient continuous explicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput. 13 (1992), 1488-1501. MR 1185658 (93g:65094)
  • [7] P.J. Prince and J.R. Dormand, High order embedded Runge-Kutta formulae, J. Comput. Appl. Math. 7 (1981), 67-76. MR 611953 (82f:65080)
  • [8] M. Santo, Metodi continui ad un passo per la risoluzione numerica di equazioni differenziali ordinarie, Ph.D. thesis, Univ. of Udine, Italy, 1991.
  • [9] J.H. Verner, Explicit Runge-Kutta methods with estimates of the local truncation error, SIAM J. Numer. Anal. 15 (1978), 772-790. MR 0483471 (58:3472)
  • [10] -, Differentiable interpolants for high-order Runge-Kutta methods, SIAM J. Numer. Anal. 30 (1993), 1446-1466. MR 1239830 (94k:65100)
  • [11] 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.

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Keywords: Interpolants, continuous explicit Runge-Kutta methods
Article copyright: © Copyright 1995 American Mathematical Society

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