Butcher's Tableau and Runge-Kutta 4 I have been reading about RK4 and one thing that I am not able to understand is how to determine the coefficients to use?
I found extensive information about Butcher's table (see here and here for instance). Though some properties must be satisfied (such as $\sum_{i=1}^{s} b_{i}=1$). Yet, how to exactly get the values of each coefficient is something that I don't understand and can't find information. Many sources say that the coefficients are some free parameters that must satisfy some Taylor expansion: but how, specifically?
 A: It is interesting to note that the Runge - Kutta method is the Midpoint (order two), Modified Euler or Huen, order three and then we step up to RK4.
It is also worth noting that the computing accuracy does not improve significantly for fifth or higher order RK methods, but the computational complexity increases rapidly.
Here are some relevant notes per your question and request.


*

*You can see the derivation here for some of those values (from Lecture 9 of NPTEL's Numerical solution of ODEs - Sept 2011). 

*Normally, you just look up all of these coefficients and algorithm as it become very tedious algebra and are not deriving those. 

*The Butcher table is just a shorthand for those. He wrote a book and here are his notes from it in which he discusses this shorthand.

*Please note that there are even newer variants of RK, like Runge-Kutta-Merson (RKM), an improvement over RK4, and Runge-Kutta-Fehlberg (RKF) and an improved RKF variant called the Cash-Karp-Runge-Kutta (CKRK) method.

*You might also be interested in this other set of very nice notes.

