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

$\displaystyle k_1$	$\textstyle =$	$\displaystyle h f(x_n,y_n)$
$\displaystyle k_2$	$\textstyle =$	$\displaystyle h f(x_n+{\textstyle{1\over 2}}h, y_n+{\textstyle{1\over 2}}k_1)$
$\displaystyle y_{n+1}$	$\textstyle =$	$\displaystyle y_n+k_2+{\mathcal O}(h^3),$

and the fourth-order formula is

$\displaystyle k_1$	$\textstyle =$	$\displaystyle h f(x_n,y_n)$
$\displaystyle k_2$	$\textstyle =$	$\displaystyle h f(x_n+{\textstyle{1\over 2}}h, y_n+{\textstyle{1\over 2}}k_1)$
$\displaystyle k_3$	$\textstyle =$	$\displaystyle h f(x_n+{\textstyle{1\over 2}}h, y_n+{\textstyle{1\over 2}}k_2)$
$\displaystyle k_4$	$\textstyle =$	$\displaystyle h f(x_n+h, y_n+k_3)$
$\displaystyle y_{n+1}$	$\textstyle =$	$\displaystyle y_n+{\textstyle{1\over 6}}k_1+{\textstyle{1\over 3}}k_2+{\textstyle{1\over 3}}k_3+{\textstyle{1\over 6}}k_4+{\mathcal O}(h^5).$

(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.