Derivation of Runge-Kutta methods

The general R-stage Runge-kutta method is defined by

(3.11)

where

(3.12)

with

(3.13)

(3.14)

and

(3.15)

Note: Here R-stage Runge-kutta method involves R function evaluations per step. Each of the functions may be interpreted as an approximation to the derivative and the function as a weighted mean of these approximations. We may also note that consistency demands that (The notion of consistency would be defined later; it means that the order of the method is at least one). The goal is to determine values for the parameters and such that the order of the method is as high as possible.