Efficiency the concept effective order & combination of Runge-Kutta Methods

In this paper, Runge-Kutta (RK) methods is introduced; The conditions of order p and its equations have been investigated by calculating Fréchet derivatives and displaying them graphically as rooted trees. The concept of effective order and combination of methods are introduced; The efficiency of the effective order method is explicitly compared to the RK method with the classical order in terms of computational speed and reduction of the number of ancient conditions.

Keywords:

order , Effective order , Runge-Kutta Methods , Combination of methods