Economical Runge-Kutta methods

For the numerical solution of the Cauchy problem, this paper presents an elegant idea of saving one function call for a special class of Runge-Kutt methods by using information from the previous step. The stability analysis and practical error estimation by embedded formulas of order (3,2), (5,4) are studied. Numerical examples are, also, considered, which proved that these methods are in competition with the best methods now exist.