We have seen that the Taylor series method has desirable features, particularly
in its ability to keep the errors small, but that it also has the strong
disadvantage of requiring the evaluation of higher derivatives of the
function *f*(*x*,*y*). In the Taylor series method, each of these higher order
derivatives is evaluated at the point *x*_{i} at the beginning of the
step, in order to evaluate *y*(*x*_{i+1}) at the end of the step. We observed
that the Euler method could be improved by computing the function *f*(*x*,*y*)
at a predicted point at the far end of the step in *x*. The Runge-Kutta
approach is to aim for the desirable features of the Taylor series method,
but with the replacement of the requirement for the evaluation of higher
order derivatives with the requirement to evaluate *f*(*x*,*y*) at some
points within the step *x*_{i} to *x*_{i+1}. Since it is not initially known
at which points in the interval these evaluations should be done, it is
possible to choose these points in such a way that the result is consistent
with the Taylor series solution to some particular, which we shall call the
order of the Runge-Kutta method.

We begin with a simple case, since the derivation is straightforward, and
while this case is not particularly useful in practice, it is useful for
understanding the Runge-Kutta methods. We begin by writing the Taylor
series for the solution *y*(*x*) in the form:

We have used the differential equation to evaluate , but it remains to evaluate . Using the earlier result for the differentiation of a function like

where the subscript

and the Taylor series becomes:

The Runge-Kutta method assumes that the correct value of the slope
over the step can be written as a linear combination of the function
*f*(*x*,*y*) evaluated at certain points in the step. In the method of order
2 this results in writing the iteration step in the form:

where:

The constants

This can then be used in the Runge-Kutta formula for

On comparing this with the direct Taylor series for

We thus have three conditions on the four constants such that the direct Taylor series and the Runge-Kutta formula will agree to second order in

There are a number of interesting choices we can make, since we have only
three conditions on the four constants *A*, *B*, *P*, and *Q*. If we
choose *A* = 1/2, we have *B* = 1/2 and *P* = *Q* = 1, which leads
to the Runge-Kutta formula:

which is just the Heun method encountered earlier.

The choice of *A* = 0 leads to *B* = 1 and
,
so that the
Runge-Kutta formula takes the form:

which is a new method related to the Euler method, sometimes called the