(pdf version of the abstract) (slides)

Sixth order methods were derived by Hut’a [5] with eight stages and by Butcher [B64] with seven stages. Eighth order methods, with 11 stages were derived by Curtis [2] and by Cooper and Verner [3].

A common pattern is proposed, for possible order \(2n\) methods, based on the methods in [3] with \(1+\tfrac12 n(n+1)\) stages. For these methods, the abscissae are located at selections of the zeros of the shifted Lobatto polynomial of degree \(n+1\).

Methods with this design give optimal number of stages for orders 2, 4, 6 and 8.

This talk contains an approach to the analysis and derivation of these methods based on a sequence of diagonal blocks. The methods of [3] are derived in a new way but an attempt to find related methods of order 10 was unsuccessful. Hence, the famous method of Hairer [4], with 17 stages, seems to be optimal.

One of the tools used in this work has been a program written by Jim Verner and myself in January 1970 and recalled in [1]. I gratefully acknowledge this collaboration and the countless other ways we have shared ideas and supported each other over the years.

References

1. J. C. Butcher and J. H; Verner, Our first B-series program, https://jcbutcher.com/1970 (1970).

2. A. R. Curtis, An eighth order Runge–Kutta process with eleven function evaluations per step, Numer. Math., 16 (1970) 268–277.

3. G. J. Cooper and J. H. Verner, Some explicit Runge–Kutta methods of high order, SIAM J. Numer. Anal., 9 (1972) 389–405.

4. E. Hairer, A Runge–Kutta method of order 10, J. Inst. Maths. Applics., 21 (1978), 47–59.

5. A. Hut’a, Une amélioration de la méthode de Runge–Kutta–Nyström pour la résolution numérique des équations différentielles du premier ordre. Acta Fac. Nat. Univ. Comenian. Math. 1 (1956) 201–224.