# 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?

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Here you can see the derivation of those values. 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. Regards – Amzoti Jun 3 '13 at 13:18
I am guessing that I should look into the work of Butcher if I want to understand them huh? – Sosi Jun 3 '13 at 15:42
It is really just a short-form notation. He wrote a book and here are his notes from it (you can see his book title here): math.auckland.ac.nz/~butcher/ODE-book-2008/Tutorials/…. Regards – Amzoti Jun 3 '13 at 16:02
Ok, those notes are really cool. Thanks! – Sosi Jun 3 '13 at 16:31
You are very welcome! 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. Regards – Amzoti Jun 3 '13 at 16:37

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.
• 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.
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Nice post! and a nice green $\checkmark$, too ;-) – amWhy Jun 4 '13 at 0:15
I just wish I had more of your posts to read! I'll have to revisit a handful of your answers to help refresh some things, and to get some references! – amWhy Jun 4 '13 at 1:14
Looks inviting. Yes, it would be helpful to know what is meant there by homogenous. Hopefully, you'll get a response! – amWhy Jun 4 '13 at 1:18