Geodesic
A nine-parametric family of embedded methods of sixth order
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 4, pp. 449-468

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In the paper an effective explicit Runge — Kutta type method of the sixth order with an embedded error estimator of order four is presented. The method is applied to the systems that can be structurally partitioned into three subsystems. Its computational scheme effectively uses the structural properties. However this leads to much larger systems of order conditions. These nonlinear conditions and the algorithm of finding a solution with nine free parameters are presented. A certain computational scheme is written down and a numerical comparison to Dormand — Prince pairs of orders 5 and 6 is performed.
Keywords: Runge — Kutta methods, partitioned systems, order conditions, simplifying conditions. 
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     author = {I. V. Olemskoy and A. S. Eremin and O. S. Firyulina},
     title = {A nine-parametric family of embedded methods of sixth order},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {449--468},
     publisher = {mathdoc},
     volume = {19},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2023_19_4_a2/}
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I. V. Olemskoy; A. S. Eremin; O. S. Firyulina. A nine-parametric family of embedded methods of sixth order. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 4, pp. 449-468. http://geodesic.mathdoc.fr/item/VSPUI_2023_19_4_a2/