A Method for Improving the Embedded Runge Kutta Fehlberg 4(5)

In this paper, we introduce a method for improving
the embedded Runge-Kutta-Fehlberg4(5) method. At each integration
step, the proposed method is comprised of two equations for the
solution and the error, respectively. These solution and error are
obtained by solving an initial value problem whose solution has the
information of the error at each integration step. The constructed algorithm
controls both the error and the time step size simultaneously and
possesses a good performance in the computational cost compared to
the original method. For the assessment of the effectiveness, EULR
problem is numerically solved.

• Sunyoung Bu
• Wonkyu Chung
• Philsu Kim

• Embedded Runge-Kutta-Fehlberg method
• Initial value problem.

[1] J. R. DORMAND, AND P. J. PRINCE, High order embedded Runge-Kutta
formulae, J. Comp. and Applied Math., 7(1) (1981) pp. 67–75.
[2] E. FEHLBERG, Classical fifth-, sixth-, seventh-, and eighth-order Runge-
Kutta formulas with stepsize control, NASA; for sale by the Clearinghouse
for Federal Scientific and Technical Information, Springfield, VA, 1968.
[3] K. GUSTAFSSON, Control-theoretic techniques for stepsize selection in
implicit Runge-Kutta methods, ACM Trans. Math. Software, 20(4) (1994),
pp. 496–517.
[4] E. HAIRER, S. P. NORSETT, AND G. WANNER, Solving ordinary
differential equations. I Nonstiff, Springer Series in Computational
Mathematics, Springer, 1993.
[5] E. HAIRER, AND G. WANNER, Solving ordinary differential equations.
II Stiff and Differential-Algebraic Problems, Springer Series in
Computational Mathematics, Springer, 1996.
[6] C. JOHNSON, Error estimates and adaptive time-step control for a class
of one-step methods for stiff ordinary differential equations, SIAM J.
Numer. Anal., 25(4) (1988), pp. 908–926.
[7] D. KAVETSKI, P. BINNING, AND S. W. SLOAN, Adaptive time stepping
and error control in a mass conservative numerical solution of the mixed
form of Richards equation, Advances in Water Resources, 24 (2001), pp.
595–605.
[8] L. F. SHAMPINE, Vectorized solution of ODEs in MATLAB, Scalable
Comput.: Pract. Experience, 10. (2010), pp. 337–345.