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.
  • $\begingroup$ The second order method is not the implicit midpoint method. It is sometimes called the improved Euler method and sometimes Heun's method. $\endgroup$ – Lutz Lehmann Apr 29 '15 at 15:02
  • 1
    $\begingroup$ @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. $\endgroup$ – Lutz Lehmann Apr 29 '15 at 17:03

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.

| cite | improve this answer | |

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.

| cite | improve this answer | |
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.