Runge–Kutta 11th order method

[Helgart]

Hi! I'm looking for Runge–Kutta's 11th order method. Maybe somebody knows where I can find it? I’m tired to search Google for hours... %)

[Bignose]

I have never heard of the names being specifically mentioned with an 11th order method. The common ones are 2nd order and 4th order. Usually when a 5th or 6th order are mentioned, there is another name associated with it, like "6th order Runga-Kutta-Nystrom".

 

That said, there should be formulas wherein you can derive any order method. They are based on Taylor series and the like, just determine how many terms you keep before truncating the series. 11th order seems very high, what application are you trying to do where a 4th or even 6th order isn't sufficient?

[Helgart]

It's not for app, something like academic interest. As I know, this method exists, but I can't find it.

[ajb]

Like Bignose says, 11th order seems very high to us. I am not a an expert in numerical analysis.

 

However, you say that it is for academic interest. I am fairly sure that the n-th order RK exists (we mean the method that has the error per step [math]h^{n}[/math] with [math]h[/math] being the step size?). Though I am not sure what gain would come from simply studying it. Well, I was told that one key to success as a mathematician is to calculate things to the highest order any sane person would, then go one higher!

[D H]

That said, there should be formulas wherein you can derive any order method.

Yes and no. Euler's method is essentially first order Runge-Kutta. Consider second-order Runge-Kutta, one step above Euler.

 

Two commonly used second-order Runge-Kutta integrators are the midpoint method and Heun's method. The midpoint method is

 

[math]y_{n+1} = y_n +

\Delta xf(x_n+\Delta x/2, y_n+

\Delta xf(x_n,y_n)/2)[/math]

 

The Butcher's tableau for the midpoint method is

 

 0  |
1/2 | 1/2 
----+---------
   |  0    1

 

Heun's method is given by

 

[math]y_{n+1} = y_n +

\left(f(x_n,y_n) +\frac 1 2 \Delta x

f(x_n+\Delta x,y_n+\Delta xf(x_n,y_n))\right)[/math]

 

The Butcher's tableau for Heun's method is

 

 0  |
1  |  1
----+---------
   | 1/2  1/2

 

Neither is optimal in the sense of minimizing truncation error. The second-order Runge-Kutta technique that minimizes truncation error is given by the tableau

 

 0  |
2/3 | 2/3 
----+---------
   | 1/4  3/4

 

Now, if second-order Runge-Kutta can offer so many choices (there are an infinite number), you can just imagine how messy higher order techniques can get. This article at wolfram.com describes some of the issues: http://reference.wolfram.com/legacy/v5_2/Add-onsLinks/StandardPackages/NumericalMath/Butcher.html.

 

That said, here are some links describing 10(8), 12(10) and 14(12) Runge-Kutta integrators:

 

http://www.mathworks.com/matlabcentral/fileexchange/25291-rkn1210-a-12th10th-order-runge-kutta-nystrom-integrator

http://sce.uhcl.edu/rungekutta/

http://sce.uhcl.edu/rungekutta/GlascowRK.ppt (powerpoint)

[Dave]

11th order Runge-Kutta is going a bit far. Most people only use up to around 4th order. Beyond that, you should be using a spectral type method if you want higher order accuracy.