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I tried to go to the primary sources in order to understand how to use Butcher tables to simplify the algebra I need to do when using Taylor series to find the order of accuracy of a scheme, for instance.

However, maybe because of a lack of relevant background, I found it particularly tough to understand how to utilize Butcher tables from Butcher's book.

Are there good, relatively self-contained (i.e. minimum prerequisites) books or tutorials that cover the necessary mathematics I need to utilize Butcher tables?

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  • $\begingroup$ I am wondering why you ask for Butcher tableaus specifically. At the end of the day they are only a way to denote a specific Runge–Kutta method. If you want to understand how to derive a Runge–Kutta method (and thus arrive at a Butcher tableau), I recommend this answer to a question on Math.SE. It sadly is the only good explanation of Runge–Kutta methods, that I am aware of. $\endgroup$ – Wrzlprmft Mar 25 '15 at 12:16
  • $\begingroup$ @Wrzlprmft I ask specifically for Butcher tables because often times when you have a Runge-Kutta method and you want to do an accuracy analysis for it, a Butcher table is useful in organizing all the Taylor series terms and seeing which one cancels which...or at least, that's what I have been led to believe? $\endgroup$ – user89 Mar 25 '15 at 18:31
  • $\begingroup$ I remain interested in the answer to this question! $\endgroup$ – user89 May 1 '15 at 18:19
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It sounds like there are two things you might want to "use" them for:

  1. To implement a method. Any reference will give you a clear algorithmic description that should make this easy.
  2. To check the order of a method. I'm not sure why you need to do this, but it's just a matter of looking up the order conditions (equations) and plugging in numbers.

You can find both in almost any reference, though the order conditions for very high order methods are only in more specialized sources (like Butcher's book). Just to give you something concrete, I recommend chapter 7 of LeVeque's book on finite differences.

For a Python implementation of the algorithm and the order conditions (up to order 14) see my Nodepy package.

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