Runge-Kutta methods are designed to approximate Taylor series methods, but have the advantage of not requiring explicit evaluations of the derivatives of . The basic idea is to use a linear combination of values of to approximate . This linear combination is matched up as closely as possibly with a Taylor series for to obtain methods of the highest possible order p. Euler's method is an example using one function evaluation.
We illustrate the development of Runge-Kutta formulas by deriving a method using two evaluations of per step; the technique employed in the derivation extends easily to the development of all Runge-Kutta type formulas. Given values , , choose values , and constants , so as to match
with the Taylor expansion,
as closely as possible. In what follows all arguments of f and its derivatives will be suppressed when they are evaluated at . It will also be convenient to express , as