Fehlberg [Computing, 4 (1969), pp. 93–106] developed a family of eight-stage pairs of Runge–Kutta methods of orders 5 and 6. Subsequently, improved versions were derived independently by Butcher [“The Numerical Analysis of Ordinary Differential Equations,” John Wiley, New York, 1987] and Verner. Those obtained by Verner are only a slight generalization of Fehlberg’s methods, whereas those of Butcher have quite a different structure, and require one more stage when the high-order method is propagated. Prince and Dormand [J. Comput. Appl. Math., 7 (1981), pp. 67–76] have since constructed methods similar to, but more general than, those developed by Verner. This paper provides explicit formulas that relate the methods developed by Fehlberg, by Verner and by Dormand and Prince. In particular, each of Verner’s methods leads to a two-parameter family of Dormand–Prince methods. The formulas facilitate the study of these methods, which include the most effective known methods of orders 5 and 6. A particular set ...
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