Misha Stepanov | publications | Electron. Trans. Numer. Anal. (2025)

M. Stepanov, On Runge–Kutta methods of order 10, Electronic Transactions on Numerical Analysis 63, 609–626 (2025).

DOI: 10.1553/etna_vol63s609
arXiv: 2504.17329
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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.