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Background

For a study that is beyond the scope of this question, I applied all of GSL’s adaptive Runge–Kutta methods to a certain problem. This includes a Runge–Kutta method of 2nd and 3rd order, denoted as gsl_odeiv2_step_rk2. It is arguably a rather bad method, but this makes the method particularly intersting for my application, about which I am writing a paper now. For this purpose, I am interested in further references about this method, and in particular about its name and origin.

The method

The method is mainly what Wikpedia calls Kutta’s third-order method and uses the midpoint method or modified Euler method for error estimation. The Butcher tableau is:

$$\begin{array}{c|ccc} 0 \\ \frac{1}{2} & \frac{1}{2} \\ 1 & -1 & 2 \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ & 0 & 1 \end{array}$$

My research so far

  • In the GSL’s source code, this method is called Euler–Cauchy, however all other sources I could find apply this name to the classical Euler method, so I am skeptical about this.
  • The source code further references Abramowitz and Stegun for the component methods. The component methods can indeed be found there but is neither equipped with a name nor a reference.
  • No other list of Runge–Kutta methods I could find contains the method as a whole. The 2nd-order method is usually called midpoint method. The only source I could find that mentions the 3rd-order method is Wikipedia, which calls it Kutta’s third-order method, but lists no source to which I have easy access.
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  • $\begingroup$ The second order method is not the implicit midpoint method. It is sometimes called the improved Euler method and sometimes Heun's method. $\endgroup$ – Dr. Lutz Lehmann Apr 29 '15 at 15:02
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    $\begingroup$ @LutzL: I didn’t say that it’s the implicit midpoint method; it’s the explicit midpoint method. As for the other names, I did find it to be called improved Euler but never Heun’s method. The latter is used for the method with the following Butcher tableau in all resources I could find so far: $$\begin{array}{c|cc} 0 \\ 1 & 1 \\\hline & \frac{1}{2} & \frac{1}{2}\end{array}$$ $\endgroup$ – Wrzlprmft Apr 29 '15 at 15:43
  • $\begingroup$ Yes. I misremembered the wikipedia page on Heun's method, too much emphasis on the section there about the Ralston method. $\endgroup$ – Dr. Lutz Lehmann Apr 29 '15 at 17:03
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John Butcher's Numerical Methods for Ordinary Differential Equations lists your 3rd-order method as (233f) on page 95, though it gives no attribution. The 2nd-order method is of course well-known as you say. I know of no reference that suggests pairing them, but it is a fairly obvious embedded error estimator for this particular 3rd-order method.

The 3rd-order method is not mentioned in the book of Hairer, Wanner, & Norsett, nor in Butcher's paper on the history of Runge-Kutta methods.

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Only a partial answer:

I managed to take a look into Kutta’s Beitrag zur näherungweisen Integration totaler Differentialgleichungen, which does indeed mention and derive the 3rd-order method, giving no other source than relating it to Simpson’s rule. Thus I deem it appropriate to call it Kutta’s third-order method.

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  • $\begingroup$ Nice. Did you find an electronic copy? $\endgroup$ – David Ketcheson Apr 28 '15 at 4:05
  • $\begingroup$ No, I used my university library’s service of summoning the original journal from the archives to a room in which I could read it (but nothing more). $\endgroup$ – Wrzlprmft Apr 28 '15 at 6:32

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