A polyvalent Runge-Kutta triple. (English) Zbl 0818.65066
A Runge-Kutta triple is derived consisting of a main formula of order 7, a formula of order 5 for local error estimation and a continuous extension formula of order 5. A family of such triples depending on 6 parameters is considered. The criteria used in selecting the method include accuracy and stability of the main formula and measures of the quality of the local error estimate and the continuous extension. Testing against comparable methods is reported for some delay equations and for the DETEST set of problems.
Reviewer: G.Hall (Manchester)
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 34K05 | General theory of functional-differential equations |
References:
| [1] | Ascher, U. M.; Mattheij, R. M.M.; Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, (Prentice Hall Series in Computational Mathematics (1988), Prentice Hall: Prentice Hall Englewood Cliffs, NJ) · Zbl 0666.65056 |
| [2] | Bogacki, P.; Shampine, L. F., An efficient Runge-Kutta (4,5) pair, (SMU Math Rept. 89-20 (1991), Mathematics Department, Southern Methodist University) · Zbl 0857.65077 |
| [3] | Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations (Runge-Kutta and General LinearMethods) (1987), Wiley: Wiley New York · Zbl 0616.65072 |
| [4] | Calvo, M.; Montijano, J. I.; Rández, L., A fifth order interpolant for the Dormand and Prince Runge Kutta method, J. Comput. Appl. Math., 29, 91-100 (1990) · Zbl 0687.65078 |
| [5] | Calvo, M.; Montijano, J. I.; Rández, L., A new embedded pair of Runge-Kutta formulas of orders 5 and 6, Comput. Math. Appl., 20, 15-24 (1990) · Zbl 0712.65070 |
| [6] | Dormand, J. R.; Prince, P. J., A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6, 19-26 (1980) · Zbl 0448.65045 |
| [7] | Dormand, J. R.; Prince, P. J., Runge-Kutta triples, Comput. Math. Appl., 12A, 1007-1017 (1986) · Zbl 0618.65059 |
| [8] | Enright, W. H.; Pryce, J. D., Two FORTRAN packages for assessing initial value methods, ACM Trans. Math. Software, 13, 1-27 (1987) · Zbl 0617.65069 |
| [9] | Enright, W. H.; Jackson, K. R.; Norsett, S. P.; Thomsen, P. G., Interpolants for Runge-Kutta formulas, ACM Trans. Math. Software, 12, 193-218 (1986) · Zbl 0617.65068 |
| [10] | Fehlberg, E., Classical fifth, sixth, seventh and eight order Runge-Kutta formulas with stepsize control, NASA TR R-287 (1968) |
| [11] | Fehlberg, E., Low order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems, NASA TR R-315 (1969) |
| [12] | Gladwell, I.; Shampine, L. F.; Baca, L. S.; Brankin, R. W., Practical aspects of interpolation in Runge-Kutta codes, SIAM J. Sci. Statist. Comput., 8, 322-341 (1987) · Zbl 0621.65067 |
| [13] | Grande, T.; Montijano, J. I.; Rández, L., A code for solving differential equations with several constant delays, Helsinki. Helsinki, 1990 Conference on Computational Ordinary Differential Equations (1990), Presented to the |
| [14] | Hull, T. E.; Enright, W. H.; Fellen, B. M.; Sedgwick, A. E., Comparing numerical methods for ODE’s, SIAM J. Numer. Anal., 9, 603-637 (1972) · Zbl 0221.65115 |
| [15] | Hull, T. E.; Enright, W. H.; Jackson, K. R., User’s guide for DVERK—a subroutine for solving nonstiff ODE’s (1976), Department of Computer Science, University of Toronto: Department of Computer Science, University of Toronto Toronto, Ont, Report 100 |
| [16] | Jorge, J. C.; Rández, L., Un algoritmo para el cáculo de las condiciones de orden \(p\), con \(p\) arbitrario para un método Runge-Kutta (1989), Publicaciones del Seminario Garciá de Galdeano. Sección 1. Universidad de Zaragoza |
| [17] | Oberle, H. J.; Pesch, H. J., Numerical treatment of delay differential equations by Hermite interpolation, Numer. Math., 37, 235-255 (1981) · Zbl 0469.65057 |
| [18] | Prince, P. J.; Dormand, J. R., High order embedded Runge-Kutta formulae, J. Comput. Appl. Math., 7, 67-75 (1981) · Zbl 0449.65048 |
| [19] | Rández, L., Improving the efficiency of the multiple shooting technique, Comput. Math. Appl., 24, 127-132 (1992) · Zbl 0766.65063 |
| [20] | Shampine, L. F., Interpolation for Runge-Kutta methods, SIAM J. Numer. Anal., 22, 1014-1027 (1985) · Zbl 0592.65041 |
| [21] | Shampine, L. F., Some practical Runge-Kutta formulas, Math. Comp., 173, 135-150 (1986) · Zbl 0594.65046 |
| [22] | Shampine, L. F.; Gladwell, I.; Brankin, R. W., Reliable solution of special event location problems for ODEs, ACM Trans. Math. Software, 17, 11-25 (1991) · Zbl 0900.65208 |
| [23] | Verner, J. H., Explicit Runge-Kutta methods with estimates of the local truncation error, SIAM J. Numer. Anal., 4, 772-790 (1978) · Zbl 0403.65029 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
