Order barriers for continuous explicit Runge-Kutta methods

Owren, Brynjulf; Zennaro, Marino

doi:10.1090/S0025-5718-1991-1068811-2

Order barriers for continuous explicit Runge-Kutta methods
HTML articles powered by AMS MathViewer

by Brynjulf Owren and Marino Zennaro PDF

Math. Comp. 56 (1991), 645-661 Request permission

Abstract:

In this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations, the need for which occurs frequently in applications. Whereas most of the commonly used multi-step methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations, this is not always the case for one-step methods. We consider the class of explicit Runge-Kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given uniform order p. These bounds are similar to the Butcher barriers known for the discrete case, and are derived up to order $p = 5$. As far as we know, the examples we present of 8-stage continuous Runge-Kutta methods of uniform order 5 are the first of their kind.

References

Alfredo Bellen and Marino Zennaro, Stability properties of interpolants for Runge-Kutta methods, SIAM J. Numer. Anal. 25 (1988), no. 2, 411–432. MR 933733, DOI 10.1137/0725028
J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc. 3 (1963), 185–201. MR 0152129
J. C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50–64. MR 159424, DOI 10.1090/S0025-5718-1964-0159424-9
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
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. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987. Nonstiff problems. MR 868663, DOI 10.1007/978-3-662-12607-3
M. K. Horn, Fourth- and fifth-order, scaled Runge-Kutta algorithms for treating dense output, SIAM J. Numer. Anal. 20 (1983), no. 3, 558–568. MR 701096, DOI 10.1137/0720036
S. P. Nørsett and G. Wanner, Perturbed collocation and Runge-Kutta methods, Numer. Math. 38 (1981/82), no. 2, 193–208. MR 638444, DOI 10.1007/BF01397089
Brynjulf Owren and Marino Zennaro, Continuous explicit Runge-Kutta methods, Computational ordinary differential equations (London, 1989) Inst. Math. Appl. Conf. Ser. New Ser., vol. 39, Oxford Univ. Press, New York, 1992, pp. 97–105. MR 1387127
Lawrence F. Shampine, Interpolation for Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), no. 5, 1014–1027. MR 799125, DOI 10.1137/0722060
Marino Zennaro, One-step collocation: uniform superconvergence, predictor-corrector method, local error estimate, SIAM J. Numer. Anal. 22 (1985), no. 6, 1135–1152. MR 811188, DOI 10.1137/0722068
M. Zennaro, Natural continuous extensions of Runge-Kutta methods, Math. Comp. 46 (1986), no. 173, 119–133. MR 815835, DOI 10.1090/S0025-5718-1986-0815835-1
Marino Zennaro, Natural Runge-Kutta and projection methods, Numer. Math. 53 (1988), no. 4, 423–438. MR 951323, DOI 10.1007/BF01396327