Runge-Kutta Method
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
(where 
is a Landau
symbol), sometimes known as RK2, and the fourth-order formula is
(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.
SEE ALSO: Adams' Method,
Gill's Method,
Milne's Method,
Ordinary
Differential Equation,
Rosenbrock Methods
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.
Referenced on Wolfram|Alpha:
Runge-Kutta Method
CITE THIS AS:
Weisstein, Eric W. "Runge-Kutta Method."
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