# Looking for references on this adaptive Runge–Kutta method (GSL’s rk2)

## 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.
• The second order method is not the implicit midpoint method. It is sometimes called the improved Euler method and sometimes Heun's method. Commented Apr 29, 2015 at 15:02
• @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: $$\begin{array}{c|cc} 0 \\ 1 & 1 \\\hline & \frac{1}{2} & \frac{1}{2}\end{array}$$ Commented Apr 29, 2015 at 15:43
• Yes. I misremembered the wikipedia page on Heun's method, too much emphasis on the section there about the Ralston method. Commented Apr 29, 2015 at 17:03