A Runge-Kutta method performs numerical integration of ordinary differential equations
by combining several values of the derivative within
each step. One second-order formula uses a trial step
at the midpoint of an interval
to cancel out lower-order error terms:
(where 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 the numerical solution of ordinary
differential equations when combined with an intelligent adaptive step-size routine.
The coefficients of a general Runge-Kutta method can be summarized in a Butcher
tableau .
See also Adams' Method ,
Butcher Tableau ,
Gill's Method ,
Milne's
Method ,
Ordinary Differential Equation ,
Rosenbrock Methods
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References Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Arfken, G. Mathematical
Methods for Physicists, 3rd ed. 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,
W. "Beitrag zur näherungsweisen Integration totaler Differentialgleichungen."
Z. Math. Phys. 46 , 435-453, 1901. Lambert, J. D. and
Lambert, D. Ch. 5 in Numerical
Methods for Ordinary Differential Systems: The Initial Value Problem. Lindelöf, E. "Remarques sur l'intégration
numérique des équations différentielles ordinaires." Acta
Soc. Sci. Fenn. Ser. A 2 , No. 13, 3-21, 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. Runge, C. "Ueber
die numerische Auflösung von Differentialgleichungen." Math. Ann. 46 ,
167-178, 1895. https://doi.org/10.1007/BF01446807 . Referenced
on Wolfram|Alpha Runge-Kutta Method
Cite this as:
Weisstein, Eric W. "Runge-Kutta Method."
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Subject classifications
is a Landau symbol), sometimes known as RK2, and
the fourth-order formula is
