The Runge - Kutta method uses a modification of the Euler method to find a solution to the initial value problem
The local truncation error can be reduced by using a more accurate method for estimating
the integral than Euler's e.g., the trapezium rule:
gives a local truncation error of order 3, but it cannot be used directly as we do not know
We can estimate
using Euler's formula
then
leading to the new scheme
We can estimate the local truncation error of this scheme, using the Taylor series expansion
Substituting
and
Hence
(1)
This is only one example of the Runge – Kutta method. The general 2nd order Runge -Kutta scheme takes the form
Repeating the earlier analysis, we see that
So
Comparing this expression with (1) we get
Since we have 3 equations and 4 unknowns, there are infinitely many solutions. The most popular are:
Modied Euler:
Midpoint method:
Heun's method:
