The names of Runge and Kutta are traditionally associated with a class of methods for the numerical
integration of ordinary differential equations.
From the discussion in a previous entry (see menu under) "ODE  Euler Method",
the main reason why Euler's method has a large truncation error per step is that in evolving the solution
from x_{n} to x_{n + 1}, the method only evaluates derivatives at the beginning of the
interval, i.e., at n.
The method is, therefore, very asymmetric in regards to the
beginning and the end of the interval.
It is possible to construct a more symmetric integration method by making an Eulerlike
trial step to the midpoint of the interval, and then using the values of both x and y at the midpoint
to make the real step across the interval.
To be more specific, introducing two parameters k_{1} and k_{2}:
k_{1} = h f (x_{n}, y_{n})
k_{2} = h f (x_{n} + h/2, y_{n} + k_{1}/2)
y_{n + 1} = Y_{n} + k_{2} 
This new symmetrization cancels out the firstorder error, making the method secondorder, and
is generally known as the secondorder RungeKutta method. Euler's method can be regarded as a firstorder
RungeKutta method.
We show in the next tab our code, test and implementation of this method.
