Numerical Solutions : Examining the Runge-Kutta Felberg Method

By Eric Rane

The Runge-Kutta Felberg method is a specialized method of the 4th order Runge-Kutta method . The 4th order Runge-Kutta method is based on the same concepts as the 2nd order Runge-Kutta method. These equations are derived from the nth Taylor Polynomial. 2nd order Runge-Kutta uses a 2nd order Taylor Polynomial, and 4th order Runge-Kutta uses (you guessed it) a 4th order Taylor Polynomial. The algorithm for evaluating a differential equation using Runga-Kutta 4 looks like this: Where f(t,y) = y'(t) This is derived from a 2nd order Taylor polynomial Notice the similarities in the coefficients on the Taylor polynomial. They are directly related. This method is much more accurate than Euler's method, however, it is limited in that when there are large amounts of change in a differential equation, it keeps the same step size ((N-1)/t) A better approach for this is a method called the Runge-Kutta-Felberg method. The Runge-Kutta-Felberg is much more accurate because it uses the fifth term of a 5th order Taylor polynomial (instead of a 4th order) to calculate the error term (every Taylor has a n error term bound by the (n+1)th term). This works because when a function starts changing too much to be accurate, it recalculates the step value automatically, and when the function is changing very little or not at all, the step value increases, increasing the speed of the calculations. This is the code snippet I used in numSolve's rkfelberg.dll For this example, I used the standard error control function in the rkfelberg algorithm, however, it was noted by my sources that error control function is normaly changed to make the increased calculation overhead worthwhile. If you have DirectX 5 installed on your machine, you can download the program and run it your computer. Please read the readme.txt file before running it. The distrabution zip also comes with the source. I don't mind if you use the source for your own projects, but please give me credit. Get source.zip. Bibliography Abrash, Michael Michael Abrash's Graphics Programming Black Book: Special Edition Scottsdale, Arizona: Coriolis 1997 Burden, L. Richard and J. Douglas Faires Numerical Analysis 6th ed. London: Brooks/Cole 1997 and many thanks from Dave Arnold and Dustin Villigran for helping me through everything.