Runge-Kutta Method

A method of integrating Ordinary Differential Equations by using a trial step at the midpoint of an interval to cancel out lower-order error terms. The second-order formula is

and the fourth-order formula is

(Press et al. 1992). This method is reasonably simple and robust and is a good general candidate for numerical solution of differential equations when combined with an intelligent adaptive step-size routine.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 896-897, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 492-493, 1985.

Cartwright, J. H. E. and Piro, O. The Dynamics of Runge-Kutta Methods.'' Int. J. Bifurcations Chaos 2, 427-449, 1992. http://formentor.uib.es/~julyan/TeX/rkpaper/root/root.html.

Lambert, J. D. and Lambert, D. Ch. 5 in Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. New York: Wiley, 1991.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Runge-Kutta Method'' and Adaptive Step Size Control for Runge-Kutta.'' §16.1 and 16.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 704-716, 1992.