<s>[INST] difference between runge kutta methods of same order I recently read about Runge-Kutta methods for solving differential equations. So far I understood the idea but up to know nobody could answer me following question:
If we consider the explicit RK methods of order 4, we get for example the classical RK method, the Gills-formula and the 3/8 rule. They all do the 'same' but what is the difference between them? Do they have different properties or do we just have those different ones for historical reasons?
Classical:
$
\begin{array}{c|cccc}
0   & 0   & 0   & 0   & 0\\
1/2 & 1/2 & 0   & 0   & 0\\
1/2 & 0   & 1/2 & 0   & 0\\
1   & 0   & 0   & 1   & 0\\
\hline
    & 1/6 & 1/3 & 1/3 & 1/6\\
\end{array}$
3/8 Rule:
$\begin{array}{c|cccc}
0   & 0   & 0   & 0   & 0\\
1/3 & 1/3 & 0   & 0   & 0\\
2/3 & -1/3   & 1 & 0   & 0\\
1   & 1   & -1   & 1   & 0\\
\hline
    & 1/8 & 3/8 & 3/8 & 1/8\\
\end{array}$
Gills-formula:
$\begin{array}{c|cccc}
0   & 0   & 0   & 0   & 0\\
1/2 & 1/2 & 0   & 0   & 0\\
1/2 & \frac{\sqrt{2}-1}{2}   & \frac{2-\sqrt{2}}{2} & 0   & 0\\
1   & 0   & -\frac{\sqrt{2}}{2}   & \frac{2+\sqrt{2}}{2}   & 0\\
\hline
    & 1/6 & \frac{2-\sqrt{2}}{2} & \frac{2+\sqrt{2}}{2} & 1/6\\
\end{array}$
 [/INST] The "classical" or Heun-Kutta method is very simple in its equations/coefficients, easy to remember and easy to casually implement.
The 3/8 method (which was introduced first in the same paper of Kutta as the "classical" method) has slightly smaller coefficients in the leading error terms of the local error.
The Gill's method, according to an ancient tome on my desk, requires a minimum of storage spaces, has the highest accuracy among the 4th order methods and has a short code sequence in its implementation. (T.E. Shoup: Applied numerical methods for the micro-computer, Prentice Hall 1984, p. 116ff)
</s>