A method of numerically 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

			(1)
			(2)
			(3)

(where  is a Landau symbol), sometimes known as RK2, and the fourth-order formula is

			(4)
			(5)
			(6)
			(7)
			(8)

(Press et al. 1992), sometimes known as RK4. 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, I. 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://lec.ugr.es/~julyan/numerics.html.

Kutta, M. W. Z. für Math. u. Phys. 46, 435, 1901.

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

Lindelöf, E. Acta Soc. Sc. Fenn. 2, 1938.

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.

Runge, C. Math. Ann. 46, 167, 1895.