zbMATH — the first resource for mathematics

Runge-Kutta interpolants based on values from two successive integration steps. (English) Zbl 0695.65046
The explicit Runge-Kutta method is one of the most popular techniques for solving non-stiff initial value problems of the form $$y'(x)=f(x,y(x))$$, $$x\geq x_ 0$$, $$y\in {\mathbb{R}}^ 4$$, $$y(x_ 0)=y_ 0$$. To get an efficient method for problems requiring dense output, one constructs an interpolant based on sufficient number of approximations $$y_ n$$ to $$y(x_ n)$$, $$x_ n=x_{n-1}+h_{n-1}$$, $$n\geq 1$$ and the corresponding derivatives $$y'_ n.$$
In the last few years several authors have been working on the idea of producing interpolants for Runge-Kutta methods. The authors of the present note deal with the construction of various new interpolants based on values of the solution and its derivative from two successive integration steps. They also make contributions to quantify the effect of variable step size on the magnitude of the error of the constructed interpolants. There are numerical tests for the efficiency of the authors’ results.
Reviewer: H.Ade

MSC:
 65L05 Numerical methods for initial value problems 34A34 Nonlinear ordinary differential equations and systems, general theory
dverk; nag
Full Text:
References:
 [1] Dormand, J. R. and Prince, P. J. ”A family of embedded Runge-Kutta formulae” Journal of Comput. and Applied Mathematics, Vol. 6, No. 1, pp. 19–26, 1980. · Zbl 0448.65045 · doi:10.1016/0771-050X(80)90013-3 [2] Enright, W. H., Jackson, K. R., Norsett, S. P., and Thomsen, P. G. ”Interpolants for Runge-Kutta formulas” ACM Trans. Math. Soft., Vol. 12, pp. 193–218, 1986. · Zbl 0617.65068 · doi:10.1145/7921.7923 [3] Fehlberg, E. ”Low order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems”, NASA TR-315, 1969. [4] Fine, J. M., ”Interpolants for Runge-Kutta-Nystrom methods”, Computing, Vol. 39, No. 1, pp. 27–42, 1987. · Zbl 0609.65051 · doi:10.1007/BF02307711 [5] Gladwell, I., ”Initial Value Routines in the NAG Library”, ACM Trans. Math. Soft., Vol. 5 pp. 386–400, 1979. · Zbl 0432.65041 · doi:10.1145/355853.355856 [6] Gladwell, I., Shampine, L. F., Baca, L. S. and Brankin, R. W., ”Practical aspects of interpolation in Runge-Kutta codes”, Rep. 102, Math. Dept. University of Manchester, Manchester, U.K., 1985. · Zbl 0621.65067 [7] Horn, M. K. ”Scaled Runge-Kutta Algorithms for handling dense output” Rept. DEVLR-FB81-13, DEVLR, Oberpfattenhonfen, F. R., 1981. [8] Horn, M. K. ”Fourth and fifth order scaled Runge-Kutta Algorithms for treating dense output”, SIAM J. Num. Anal., Vol 20, pp. 558–568, 1983. · Zbl 0511.65048 · doi:10.1137/0720036 [9] Hull, T. E., Enright, W. H., Fellen, B. M. and Sedgwick, A. E., ”Comparing numerical methods for ordinary differential equations,” SIAM J. Numer. Anal., Vol. 9, pp. 603–637, 1972. · Zbl 0221.65115 · doi:10.1137/0709052 [10] Hull, T. E., Enright, W. H. and Jackson, K. R., ”User’s guide for DVERK A subroutine for solving nonstiff ODE’S”, Rep. 100, Dept. of Computer Science, Univ. of Toronto, Canada, 1976. [11] Papageorgiou, G. and Tsitouras, Ch., ”Scaled Runge-Kutta-Nystrom methods for the second order differential equationy”=f(x, y)”, Intern. J. of Computer Mathematics, 1989, (In Press). · Zbl 0679.65051 [12] Shampine, L. F., ”Interpolation for Runge-Kutta methods”, SIAM J. Num. Anal., Vol. 22, pp. 1014–1027, 1985. · Zbl 0592.65041 · doi:10.1137/0722060 [13] Shampine, L. F., ”Some practical Runge-Kutta Formulas” Math. of Comput. Vol. 46, pp. 135–150, 1986. · Zbl 0594.65046 · doi:10.1090/S0025-5718-1986-0815836-3 [14] Tsitouras, Ch. and Papageorgiou, G., ”New interpolants for Runge-Kutta Algorithms using second derivatives”, Accepted for publication in the Intern. J. of. Computer Mathematics. · Zbl 0749.65051
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.