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 it reuses the same intermediate calculations to produce two estimates of different accuracy, allowing for automatic error estimation. 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]

Coefficients for RK4(5), Formula 2 (Fehlberg)^[2]
κ	α_κ	β_κλ					c_κ (4th order)	ĉ_κ (5th order)
κ	α_κ	λ=0	λ=1	λ=2	λ=3	λ=4	c_κ (4th order)	ĉ_κ (5th order)
0	0	0					25/216	16/135
1	1/4	1/4					0	0
2	3/8	3/32	9/32				1408/2565	6656/12825
3	12/13	1932/2197	−7200/2197	7296/2197			2197/4104	28561/56430
4	1	439/216	−8	3680/513	−845/4104		−1/5	−9/50
5	1/2	−8/27	2	−3544/2565	1859/4104	−11/40		2/55

Implementing an RK4(5) Algorithm

The coefficients found by Fehlberg for Formula 1 (derivation with his parameter α₂=1/3) are given in the table below. Note that while Fehlberg's original tables begin indexing at 0, the standard mathematical convention for Runge–Kutta methods established by Butcher uses coefficients denoted as $a_{ij}$ with $i=1,2,\ldots ,s$ and $j=1,2,\ldots ,s$ . Therefore, when implementing these methods in programming languages that use 0-based array indexing, there is a shift between the mathematical notation and array indices: what appears as $k_{1}$ in the mathematical formulation corresponds to index 0 in the implementation arrays.^[3]

Coefficients for RK4(5), Formula 1 (Fehlberg)^[2]
κ	α_κ	β_κλ					c_κ (4th order)	ĉ_κ (5th order)
κ	α_κ	λ=0	λ=1	λ=2	λ=3	λ=4	c_κ (4th order)	ĉ_κ (5th order)
0	0	0					1/9	47/450
1	2/9	2/9					0	0
2	1/3	1/12	1/4				9/20	12/25
3	3/4	69/128	−243/128	135/64			16/45	32/225
4	1	-17/12	27/4	-27/5	16/15		1/12	1/30
5	5/6	65/432	-5/16	13/16	4/27	5/144		6/25

Fehlberg^[4] outlines a solution to solving a system of n differential equations of the form: ${\frac {dy_{i}}{dx}}=f_{i}(x,y_{1},y_{2},\ldots ,y_{n}),i=1,2,\ldots ,n$ to iterative solve for $y_{i}(x+h),i=1,2,\ldots ,n$ where h is an adaptive stepsize to be determined algorithmically:

The solution is the weighted average of six increments, where each increment is the product of the size of the interval, ${\textstyle h}$ , and an estimated slope specified by function f on the right-hand side of the differential equation.

${\begin{aligned}k_{1}&=h\cdot f(x+A(1)\cdot h,y)\\k_{2}&=h\cdot f(x+A(2)\cdot h,y+B(2,1)\cdot k_{1})\\k_{3}&=h\cdot f(x+A(3)\cdot h,y+B(3,1)\cdot k_{1}+B(3,2)\cdot k_{2})\\k_{4}&=h\cdot f(x+A(4)\cdot h,y+B(4,1)\cdot k_{1}+B(4,2)\cdot k_{2}+B(4,3)\cdot k_{3})\\k_{5}&=h\cdot f(x+A(5)\cdot h,y+B(5,1)\cdot k_{1}+B(5,2)\cdot k_{2}+B(5,3)\cdot k_{3}+B(5,4)\cdot k_{4})\\k_{6}&=h\cdot f(x+A(6)\cdot h,y+B(6,1)\cdot k_{1}+B(6,2)\cdot k_{2}+B(6,3)\cdot k_{3}+B(6,4)\cdot k_{4}+B(6,5)\cdot k_{5})\end{aligned}}$

Then the weighted average is:^[2]

$y(x+h)=y(x)+{\hat {c}}_{1}k_{1}+{\hat {c}}_{2}k_{2}+{\hat {c}}_{3}k_{3}+{\hat {c}}_{4}k_{4}+{\hat {c}}_{5}k_{5}+{\hat {c}}_{6}k_{6}$ The estimate of the truncation error is:^[2] $\mathrm {TE} =\left|({\hat {c}}_{1}-c_{1})k_{1}+({\hat {c}}_{2}-c_{2})k_{2}+({\hat {c}}_{3}-c_{3})k_{3}+({\hat {c}}_{4}-c_{4})k_{4}+({\hat {c}}_{5}-c_{5})k_{5}+({\hat {c}}_{6}-c_{6})k_{6}\right|$

${\text{where }}c_{i}{\text{ and }}{\hat {c}}_{i}{\text{ are the 4th- and 5th-order weights from the embedded Runge–Kutta pair, and }}k_{i}{\text{ are the stage derivatives.}}$

At the completion of the step, a new stepsize is calculated:^[5]

$h_{\text{new}}=0.9\cdot h\cdot \left({\frac {\varepsilon }{TE}}\right)^{1/5}$

If ${\textstyle \mathrm {TE} >\varepsilon }$ , then replace ${\textstyle h}$ with ${\textstyle h_{\text{new}}}$ and repeat the step. If ${\textstyle TE\leqslant \varepsilon }$ , then the step is completed. Replace ${\textstyle h}$ with ${\textstyle h_{\text{new}}}$ for the next step.

The coefficients found by Fehlberg for Formula 2 (derivation with his parameter α₂ = 3/8) are given in the table below:

Coefficients for RK4(5), Formula 2 (Fehlberg)^[2]
κ	α_κ	β_κλ					c_κ (4th order)	ĉ_κ (5th order)
κ	α_κ	λ=0	λ=1	λ=2	λ=3	λ=4	c_κ (4th order)	ĉ_κ (5th order)
0	0	0					25/216	16/135
1	1/4	1/4					0	0
2	3/8	3/32	9/32				1408/2565	6656/12825
3	12/13	1932/2197	-7200/2197	7296/2197			2197/4104	28561/56430
4	1	439/216	-8	3680/513	-845/4104		-1/5	-9/50
5	1/2	-8/27	2	-3544/2565	1859/4104	-11/40		2/55

In another table in Fehlberg,^[2] coefficients for an RKF4(5) derived by D. Sarafyan are given:

Coefficients for Sarafyan's RK4(5), Table IV in Fehlberg^[2]
κ	α_κ	β_κλ					c_κ (4th order)	ĉ_κ (5th order)
κ	α_κ	λ=0	λ=1	λ=2	λ=3	λ=4	c_κ (4th order)	ĉ_κ (5th order)
0	0	0					1/6	1/24
1	1/2	1/2					0	0
2	1/2	1/4	1/4	1/4			2/3	0
3	1	0	-1	2			1/6	5/48
4	2/3	7/27	10/27	0	1/27			27/56
5	1/5	28/625	-1/5	546/625	54/625	-378/625		125/336

Notes

^ 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.
^ ^a ^b ^c ^d ^e ^f ^g ^h Fehlberg, Erwin. "Low-Order Classical Runge-Kutta Formulas With Stepsize Control And Their Application To Some Heat Transfer Problems" (PDF). nasa.gov. National Aeronautics And Space Administration. Retrieved 1 August 2025.
^ Butcher, John C. "Introduction to Runge–Kutta methods" (PDF). University of Auckland. Retrieved 2025-08-01.
^ Hairer, Nørsett & Wanner (1993, p. 177) refer to Fehlberg (1969)
^ Gurevich, Svetlana (2017). "Appendix A Runge-Kutta Methods" (PDF). Munster Institute for Theoretical Physics. pp. 8–11. Retrieved 4 March 2022.

References

Fehlberg, Erwin (1968) Classical fifth-, sixth-, seventh-, and eighth-order Runge-Kutta formulas with stepsize control. NASA Technical Report 287. https://ntrs.nasa.gov/api/citations/19680027281/downloads/19680027281.pdf
Fehlberg, Erwin (1969) Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems. Vol. 315. National aeronautics and space administration.
Fehlberg, Erwin (1969). "Klassische Runge-Kutta-Nystrom-Formeln funfter und siebenter Ordnung mit Schrittweiten-Kontrolle". Computing. 4: 93–106. doi:10.1007/BF02234758. S2CID 38715401.
Fehlberg, Erwin (1970) Some experimental results concerning the error propagation in Runge-Kutta type integration formulas. NASA Technical Report R-352. https://ntrs.nasa.gov/api/citations/19700031412/downloads/19700031412.pdf
Fehlberg, Erwin (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
Hairer, Ernst; Nørsett, Syvert; Wanner, Gerhard (1993). Solving Ordinary Differential Equations I: Nonstiff Problems (Second ed.). Berlin: Springer-Verlag. ISBN 3-540-56670-8.
Sarafyan, Diran (1966) Error Estimation for Runge-Kutta Methods Through Pseudo-Iterative Formulas. Technical Report No. 14, Louisiana State University in New Orleans, May 1966.

Fehlberg, E (1958). "Eine Methode zur Fehlerverkleinerung beim Runge-Kutta-Verfahren". Zeitschrift für Angewandte Mathematik und Mechanik. 38 (11/12): 421–426. Bibcode:1958ZaMM...38..421F. doi:10.1002/zamm.19580381102.
Fehlberg, E (1964). "New high-order Runge-Kutta formulas with step size control for systems of first and second-order differential equations". Zeitschrift für Angewandte Mathematik und Mechanik. 44 (S1): T17 – T29. doi:10.1002/zamm.19640441310.
Fehlberg, E (1972). "Klassische Runge-Kutta-Nystrom-Formeln mit Schrittweiten-Kontrolle fur Differentialgleichungen x.. = f(t,x)". Computing. 10: 305–315. doi:10.1007/BF02242243. S2CID 37369149.
Fehlberg, E (1975). "Klassische Runge-Kutta-Nystrom-Formeln mit Schrittweiten-Kontrolle fur Differentialgleichungen x.. = f(t,x,x.)". Computing. 14: 371–387. doi:10.1007/BF02253548. S2CID 30533090.
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. doi:10.1016/0898-1221(93)90303-D..
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. 1 (14): 105–112. Bibcode:2008IJSTQ..14..105H. doi:10.1109/JSTQE.2007.913971. S2CID 13069335..
Simos, T. E. (1995). "Modified Runge–Kutta–Fehlberg methods for periodic initial-value problems". Japan Journal of Industrial and Applied Mathematics. 12 (1): 109. doi:10.1007/BF03167384. S2CID 120146558..
Sarafyan, D. (1994). "Approximate Solution of Ordinary Differential Equations and Their Systems Through Discrete and Continuous Embedded Runge-Kutta Formulae and Upgrading Their Order". Computers & Mathematics with Applications. 28 (10–12): 353–384. doi:10.1016/0898-1221(94)00201-0.
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. doi:10.1016/j.aej.2015.12.026.

[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.

[FehlbergNASA-2] ^ ^a ^b ^c ^d ^e ^f ^g ^h Fehlberg, Erwin. "Low-Order Classical Runge-Kutta Formulas With Stepsize Control And Their Application To Some Heat Transfer Problems" (PDF). nasa.gov. National Aeronautics And Space Administration. Retrieved 1 August 2025.

[3] Butcher, John C. "Introduction to Runge–Kutta methods" (PDF). University of Auckland. Retrieved 2025-08-01.

[:0-4] Hairer, Nørsett & Wanner (1993, p. 177) refer to Fehlberg (1969)

[5] Gurevich, Svetlana (2017). "Appendix A Runge-Kutta Methods" (PDF). Munster Institute for Theoretical Physics. pp. 8–11. Retrieved 4 March 2022.

[1]

[2]

[3]

[4]

[5]

Runge–Kutta–Fehlberg method

Butcher tableau for Fehlberg's 4(5) method

Implementing an RK4(5) Algorithm

See also

Notes

References

Further reading