Volume 63, pp. 609-626, 2025.

On Runge-Kutta methods of order 10

Misha Stepanov

Abstract

An explicit $s$-stage Runge–Kutta method of order 10 is determined by $s (s + 1) / 2$ parameters that must satisfy a non-linear algebraic system of $1205$ equations. In the literature, solutions for the cases $s = 18$ [A. R. Curtis, J. Inst. Math. Appl., 16 (1975), pp. 35–55] and $s = 17$ [E. Hairer, J. Inst. Math. Appl., 21 (1978), pp. 47–59] were analytically derived, while that for $s=16$ [D. K. Zhang, Numer. Algorithms, 96 (2024), pp. 1243–1267] was found by numerical search. In the present paper, a family of methods with $s = 15$ is derived.

Full Text (PDF) [829 KB], BibTeX , DOI: 10.1553/etna_vol63s609

Key words

minimal number of stages, explicit Runge–Kutta methods

AMS subject classifications

65L05, 65L06