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 large class of Runge–Kutta methods.

The novelty of Fehlberg's method is that it is an embedded method from the Runge–Kutta family, meaning that identical function evaluations are used in conjunction with each other to create methods of varying order and similar error constants. The method presented in Fehlberg's 1969 paper has been dubbed the RKF45 method, and is a method of order O(h⁴) with an error estimator of order O(h⁵).[1] By performing one extra calculation, the error in the solution can be estimated and controlled by using the higher-order embedded method that allows for an adaptive stepsize to be determined automatically.

Butcher tableau for Fehlberg's 4(5) method

Any Runge–Kutta method is uniquely identified by its Butcher tableau. The embedded pair proposed by Fehlberg[2]

	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
		16/135	0	6656/12825	28561/56430	−9/50	2/55
		25/216	0	1408/2565	2197/4104	−1/5	0

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

See also

• List of Runge–Kutta methods
• Numerical methods for ordinary differential equations
• Runge–Kutta methods

Notes

1. According to Hairer et al. (1993, §II.4), the method was originally proposed in Fehlberg (1969); Fehlberg (1970) is an extract of the latter publication.
2. Hairer, Nørsett & Wanner (1993, p. 177) harvtxt error: no target: CITEREFHairerNørsettWanner1993 (help) refer to Fehlberg (1969) harvtxt error: no target: CITEREFFehlberg1969 (help)

References

• Free software implementation in GNU Octave: http://octave.sourceforge.net/odepkg/function/ode45.html
• 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. doi:10.1007/BF02241732
• 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.

Further reading

• Simos, T. E. (1993). A Runge-Kutta Fehlberg method with phase-lag of order infinity for initial-value problems with oscillating solution. Computers & Mathematics with Applications, 25(6), 95-101.
• Handapangoda, C. C., Premaratne, M., Yeo, L., & Friend, J. (2008). Laguerre Runge-Kutta-Fehlberg Method for Simulating Laser Pulse Propagation in Biological Tissue. IEEE Journal of Selected Topics in Quantum Electronics, 14(1), 105-112.
• Paul, S., Mondal, S. P., & Bhattacharya, P. (2016). Numerical solution of Lotka Volterra prey predator model by using Runge–Kutta–Fehlberg method and Laplace Adomian decomposition method. Alexandria Engineering Journal, 55(1), 613-617.
• Filiz, A. (2014). Numerical solution of linear Volterra integro-differential equation using Runge-Kutta-Fehlberg method. Applied and Computational Mathematics, 3(1), 9-14.
• Simos, T. E. (1995). Modified Runge-Kutta-Fehlberg methods for periodic initial-value problems. Japan journal of industrial and applied mathematics, 12(1), 109.