zbMATH — the first resource for mathematics

Interpolating Runge-Kutta methods for vanishing delay differential equations. (English) Zbl 0843.65052
The authors describe a new method which modifies continuous Runge-Kutta (RK) codes so as to be applicable to the approximation of the solution of differential delay problems which have a delay which vanishes. For a delay function which vanishes during a step \([t_n, t_{n+ 1}]\) of the calculation the new algorithm uses values from the preceding step \([t_{n- 1}, t_n]\) to generate interpolants needed to construct the stage values for the \(n\)th step. The order of accuracy of the modified method is designed to be equal to that of the original method.
The authors have written codes DDRK6N, DDVSS6 based on the new algorithm which are sixth-order RK codes from the Verner class [c.f. J. H. Verner, SIAM J. Numer. Anal. 30, No. 5, 1446-1466 (1993; Zbl 0787.65047)]. Results obtained by application of these codes to three test problems indicate that the new codes are more accurate than existing codes.

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems
34K05 General theory of functional-differential equations
Full Text: DOI
[1] Calvo, M., Montijano, J. I., Rández, L.: A fifth order interpolant for the Dormand and Prince Runge-Kutta method. J. Comp. Appl. Math.29, 91–100 (1990). · Zbl 0687.65078 · doi:10.1016/0377-0427(90)90198-9
[2] Enright, W. H.: A new error-control for initial value solvers. Appl. Math. Comput.31, 288–301 (1989). · Zbl 0674.65060 · doi:10.1016/0096-3003(89)90123-9
[3] Enright, W. H.: The relative efficiency of alternative defect control schemes for high order continuous Runge-Kutta formulas. SIAM J. Numer. Anal.30, 1419–1445 (1993). · Zbl 0787.65046 · doi:10.1137/0730074
[4] Enright, W. H., Jackson, K. R., Nørsett, Thomsen, P. G.: Interpolants for Runge-Kutta formulas. ACM Trans. Math. Software12, 193–218 (1986). · Zbl 0617.65068 · doi:10.1145/7921.7923
[5] Enright, W. H., Hu, M.: Interpolating Runge-Kutta methods for vanishing delay differential equations. Rep. 292, Dept. of Computer Science, Univ. of Toronto, Canada, 1994. · Zbl 0843.65052
[6] Hayashi, H., Enright, W. H.: A new algorithm for vanishing delay problems, CAMS annual meeting, May 30–June 2, 1993, York University (invited oral presentation).
[7] Horn, M. K.: Fourth- and fifty-order, scaled Runge-Kutta algorithms for treating dense output. SIAM J. Numer. Anal.20, 558–568 (1983). · Zbl 0511.65048 · doi:10.1137/0720036
[8] Hull, T. E., Enright, W. H., Jackson, K. R.: User’s guide for DVERK – A subroutine for solving nonstiff ODE’s. Rep. 100, Dept. of Computer Science, Univ. of Toronto, 1976.
[9] Karoui, A., Vaillancourt, R.: A numerical method for vanishing-lag delay differential equations. Private communication, 1993. · Zbl 0834.65054
[10] Neves, K. W.: Automatic integration of functional differential equations: An approach, ACM Trans. Math. Software1, 357–368 (1986). · Zbl 0315.65045 · doi:10.1145/355656.355661
[11] Neves, K. W., Thomson, S.: Solution of systems of functional differential equations with state dependent delays. Technical Report TR-92-003, Computer Science, Radford University, 1992.
[12] Owren, B., Zennaro, M.: Derivation of efficient, continuous, explicit Runge-Kutta methods. SIAM J. Sci. Stat. Comput.13, 1488–1501 (1992). · Zbl 0760.65073 · doi:10.1137/0913084
[13] Paul, C. A. H.: Developing a delay differential equation solver. Appl. Numer. Math.9, 403–414 (1992). · Zbl 0779.65043 · doi:10.1016/0168-9274(92)90030-H
[14] Sharp, P. W., Smart, E.: Private communication (1990).
[15] Tavernini, L.: One-step methods for the numerical solution of Volterra functional differential equations. SIAM J. Numer. Anal.8, 786–795 (1971). · Zbl 0231.65070 · doi:10.1137/0708072
[16] Verner: Differentiable interpolants for high-order Runge-Kutta methods. SIAM J. Numer. Anal.30, 1446–1466 (1993). · Zbl 0787.65047 · doi:10.1137/0730075
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.