Runge–Kutta–Fehlberg method

In mathematics, the Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the class of Runge–Kutta methods. The Runge–Kutta–Fehlberg method uses an O(h⁴) method together with an O(h⁵) method that uses all of the points of the O(h⁴) method, and hence is often referred to as an RKF45 method. Similar schemes with different orders have since been developed. By performing one extra calculation than would be required for an RK5 method, the error in the solution can be estimated and controlled and an appropriate step size can be determined automatically, making this method efficient for ordinary problems of automated numerical integration of ordinary differential equations.

The Butcher tableau is:

	0
	1/4	1/4
	3/8	3/32	9/32
	12/13	1932/2197	−7200/2197	7296/2197
	1	439/216	−8	3680/513	−845/4104
	1/2	-8/27	2	−3544/2565	1859/4104	−11/40
		25/216	0	1408/2565	2197/4104	−1/5	0
		16/135	0	6656/12825	28561/56430	−9/50	2/55

The first row of coefficients gives the fourth-order accurate method, and the second row gives the fifth-order accurate method.

References

• Erwin Fehlberg (1969). Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. NASA Technical Report 315.
• Erwin Fehlberg (1970). "Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme," Computing (Arch. Elektron. Rechnen), vol. 6, pp. 61–71.
• Ernst Hairer, Syvert Nørsett, and Gerhard Wanner (1993). Solving Ordinary Differential Equations I: Nonstiff Problems, second edition, Springer-Verlag, Berlin. ISBN 3-540-56670-8.

External links

• RKF45 computer programs (see also [1])