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.

This shows the computational time in real time used during a 3-body simulation evolved with the Runge-Kutta-Fehlberg method. Most of the computer time is spent when the bodies pass close by and are susceptible to numerical error.

Implementing an RK4(5) Algorithm

The coefficients found by Fehlberg for Formula 1 (derivation with his parameter α2=1/3) are given in the able below, using array indexing of base 1 instead of base 0 to be compatible with most computer languages:

COEFFICIENTS FOR RK4(5), FORMULA 1 Table II in Fehlberg[2]

K	A(K)	B(K,L)					C(K)	CH(K)	CT(K)
		L=1	L=2	L=3	L=4	L=5
1	0						1/9	47/450	-1/150
2	2/9	2/9					0	0	0
3	1/3	1/12	1/4				9/20	12/25	3/100
4	3/4	69/128	-243/128	135/64			16/45	32/225	-16/75
5	1	-17/12	27/4	-27/5	16/15		1/12	1/30	-1/20
6	5/6	65/432	-5/16	13/16	4/27	5/144		6/25	6/25

Fehlberg[2] outlines a solution to solving a system of n differential equations of the form:

${\displaystyle {\operatorname {d} \!y_{i} \over \operatorname {d} \!x}=f_{i}(x,y_{1},y_{2},...,y_{n}),i=1,2,...,y_{n}}$

to iterative solve for

${\displaystyle y_{i}(x+h),i=1,2,...,n}$

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.

${\displaystyle k_{1}=h\cdot f(x+A(1)\cdot h,y)}$

${\displaystyle k_{2}=h\cdot f(x+A(2)\cdot h,y+B(2,1)\cdot k_{1})}$

${\displaystyle k_{3}=h\cdot f(x+A(3)\cdot h,y+B(3,1)\cdot k_{1}+B(3,2)\cdot k_{2})}$

${\displaystyle k_{4}=h\cdot f(x+A(4)\cdot h,y+B(4,1)\cdot k_{1}+B(4,2)\cdot k_{2}+B(4,3)\cdot k_{3})}$

${\displaystyle k_{5}=h\cdot f(x+A(5)\cdot h,y+B(5,1)\cdot k_{1}+B(5,2)\cdot k_{2}+B(5,3)\cdot k_{3}+B(5,4)\cdot k_{4})}$

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

Then the weighted average is:

${\displaystyle y(x+h)=y(x)+CH(1)\cdot k_{1}+CH(2)\cdot k_{2}+CH(3)\cdot k_{3}+CH(4)\cdot k_{4}+CH(5)\cdot k_{5}+CH(6)\cdot k_{6}}$

The estimate of the truncation error is:

${\displaystyle TE=|CT(1)\cdot k_{1}+CT(2)\cdot k_{2}+CT(3)\cdot k_{3}+CT(4)\cdot k_{4}+CT(5)\cdot k_{5}+CT(6)\cdot k_{6}|}$

At the completion of the step, a new stepsize is calculated:

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

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

The coefficients found by Fehlberg for Formula 2 (derivation with his parameter α2=3/8) are given in the able below, using array indexing of base 1 instead of base 0 to be compatible with most computer languages:

COEFFICIENTS FOR RK4(5), FORMULA 2 Table III in Fehlberg[2]

K	A(K)	B(K,L)					C(K)	CH(K)	CT(K)
		L=1	L=2	L=3	L=4	L=5
1	0						25/216	16/135	1/360
2	1/4	1/4					0	0	0
3	3/8	3/32	9/32				1408/2565	6656/12825	-128/4275
4	12/13	1932/2197	-7200/2197	7296/2197			2197/4104	28561/56430	-2187/75240
5	1	439/216	-8	3680/513	-845/4104		-1/5	-9/50	1/50
6	1/2	-8/27	2	-3544/2565	1859/4104	-11/40		2/55	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]

K	A(K)	B(K,L)					C(K)	CH(K)	CT(K)
		L=1	L=2	L=3	L=4	L=5
1	0	0					1/6	1/24	-1/8
2	1/2	1/2					0	0	0
3	1/2	1/4	1/4				2/3	0	-2/3
4	1	0	-1	2			1/6	5/48	-1/16
5	2/3	7/27	10/27	0	1/27			27/56	27/56
6	1/5	28/625	-1/5	546/625	54/625	-378/625		125/336	125/336

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

• 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. https://ntrs.nasa.gov/api/citations/19690021375/downloads/19690021375.pdf
• Erwin Fehlberg (1968) Classical fifth-, sixth-, seventh-, and eighth-order Runge-Jutta formulas with stepsize control. NASA Technical Report 287. https://ntrs.nasa.gov/api/citations/19680027281/downloads/19680027281.pdf
• Erwin Fehlberg (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
• 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.
• Diran Sarafyan (1966) Error Estimation for Runge-Kutta Methods Through Pseudo-Iterative Formulas. Technical Report No. 14, Louisiana State University in New Orleans, May 1966.

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.
• 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 Math. Applic. Vol. 28, No. 10-12, pp. 353–384, 1994 https://core.ac.uk/download/pdf/82540775.pdf