# zbMATH — the first resource for mathematics

A new embedded pair of Runge-Kutta formulas of orders 5 and 6. (English) Zbl 0712.65070
Authors’ summary: A new pair of embedded Runge-Kutta (RK) formulas of orders 5 and 6 is presented. It is derived from a family of RK methods depending on eight parameters by using certain measures of accuracy and stability. Numerical tests comparing its efficiency to other formulas of the same in current use are presented. With an extra function evaluation per step, a $$C^ 1$$-continuous interpolant of order 5 can be obtained.
Reviewer: I.Dvořák

##### MSC:
 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L05 Numerical methods for initial value problems 34A34 Nonlinear ordinary differential equations and systems, general theory
##### Software:
DEPAC; dverk; NSDTST; STDTST
Full Text:
##### References:
 [1] Shampine, L.F.; Watts, H.A., Practical solution of ordinary differential equations by Runge-Kutta methods, () · Zbl 0221.65117 [2] Shampine, L.F.; Watts, H.A., DEPAC—design of a user oriented package of ODE solvers, () · Zbl 0407.68036 [3] Fehlberg, E., Classical fifth, sixth, seventh and eighth order Runge-Kutta formulas with stepwise control, NASA report TR R-287, (1968) [4] Fehlberg, E., Low order classical Runge-Kutta formulas with stepwise control and their application to some heat transfer problems, NASA report TR R-315, (1969) [5] Hull, T.E.; Enright, W.H.; Jackson, K.R., User’s guide for DVERK—a subroutine for solving nonstiff ODE’s, () · Zbl 0391.65030 [6] Verner, J.H., Explicit Runge-Kutta methods with estimates of the local truncation error, SIAM jl numer. analysis, 4, 772-790, (1978) · Zbl 0403.65029 [7] Dormand, J.R.; Prince, P.J., A family of embedded Runge-Kutta formulae, J. comput. appl. math., 6, (1980) · Zbl 0448.65045 [8] Prince, P.J.; Dormand, J.R., High order embedded Runge-Kutta formulae, J. comput. appl. math., 7, 67-75, (1981) · Zbl 0449.65048 [9] M. Calvo, J.I. Montijano and L. Randez, A fifth order interpolant for the Dormand and Prince Runge-Kutta methodJ. Comput. appl. Math. (in press). · Zbl 0687.65078 [10] Enright, W.H.; Jackson, K.R.; Norsett, S.P.; Thomsen, P.G., Interpolants for Runge-Kutta formulas, ACM trans. math. softw., 12, 193-218, (1986) · Zbl 0617.65068 [11] Gladwell, I.; Shampine, L.F.; Baca, L.S.; Brankin, R.W., Practical aspects of interpolation in Runge-Kutta codes, SIAM jl sci. stat. comput., 8, 322-341, (1987) · Zbl 0621.65067 [12] Shampine, L.F., Interpolation for Runge-Kutta methods, SIAM jl numer. analysis, 22, 1014-1027, (1985) · Zbl 0592.65041 [13] Shampine, L.F., Some practical Runge-Kutta formulas, Math. comput., 173, 135-150, (1986) · Zbl 0594.65046 [14] Enright, W.H.; Pryce, J.D., Two FORTRAN packages for assessing initial value methods, ACM toms, 13, 1-27, (1987) · Zbl 0617.65069 [15] Hull, T.E.; Enright, W.H.; Fellen, B.M.; Sedgwick, A.E., Comparing numerical methods for ODE’s, SIAM jl numer. analysis, 9, 603-637, (1972) · Zbl 0221.65115 [16] Dormand, J.R.; Prince, P.J., A reconsideration of some embedded Runge-Kutta formulae, J. comput. appl. math., 15, 203-211, (1986) · Zbl 0602.65046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.