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I have a problem about determining $a_1, a_2, k_1, k_2, \ldots a_n, k_n,\ldots$

In the general form of the Higher Order Runge-Kutta below :

$$y_{r+1}=y_r+a_1k_1+a_2k_2+\cdots+a_nk_n$$

For the convenient, i'll write it down The Runge Kutta $2^{\text{nd}}$ in my book just in case if the formula is different from your views:

$$ \begin{align} k_1&=hf(x_r,y_r)\\ k_2&=hf(x_r+p_1h,y_r+q_{11}k_1)\\ y_{r+1}&=y_r+(a_1k_1+a_2k_2) \end{align} $$

I'm not really sure, but some sources talk about Butcher Tableau, slope for each $k_n$, Taylor expansion, and rooted trees, for determining the $a_n$. And i don't really understand what their relation is, especially for Butcher Tableau, and rooted trees.

Could you explain all of these for me? Or just give me a good reference that discusses about derivation of the higher order Runge-Kutta method in detail, please?

Because, some books just skip over the derivation part.

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A good source on Butcher tableaus and B-trees would, almost canonically, be the book by Butcher himself. For a short overview, see the three sets of slides with an introduction to B-trees, demonstration of the method for up to order 4 and outlook to implicit methods, or a historical overview

You could of course also read the most original source, the introduction of Runge-Kutta methods by M. Wilhelm Kutta (1901), "Beitrag zur näherungsweisen Lösung totaler Differentialgleichungen", where the order conditions are established and solved up to order 4, with an order 5 method that turned out not to be correct for systems.

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  • $\begingroup$ Is this source containing how to derive the RK formula? $\endgroup$ – user516076 Nov 15 '19 at 3:15

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