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To obtain high-order integration methods for ordinary differential equatic which combine to some extent the advantage of Runge-Kutta methods on one hand and line multistep methods on the other, the use of “modified multistep” or “hybrid” method has been proposed by various researchers. In this paper formulas are derived for method which use one extra intermediate point than in the previously published methods so that there are analogues of the fourth-order Runge-Kutta method. A five-stage method of order 7 is already given.

A Multistep Generalization of Runge-Kutta Methods With Four or Five Stages