Runge-Kutta Methods

where

That is a lot of parameters.
Some necessary conditions for convergence are:

Best order of Runge-Kutta method is .

Order

Consider a 2-stage Runge-Kutta method:

Now Taylor expand :

Now noting that , we find that the thing in brackets is just .

Finally we find:

Hence we have order at least when and .
We have arrived at the Trapezoidal rule (ODEs).

Trying , we find that a higher order is impossible with this method.
Sometimes this doesn’t work!! So we would need to expand to higher order…