Nonlinear stability and D-convergence of Runge-Kutta methods for delay differential equations. (English) Zbl 0904.65082

Stability and convergence of Runge-Kutta methods with Lagrangian interpolation applied to nonlinear ordinary delay differential equations are studied. The new concepts of strong algebraic stability, GDN-stability and D-convergence are introduced to facilitate the analysis. A correspondence between certain stability properties of Runge-Kutta methods applied to ordinary differential equations and relevant stability properties of the same methods enhanced by Lagrangian interpolation and applied to delay differential equations is established.


65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34K05 General theory of functional-differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
Full Text: DOI


[1] Aiguo, X., On the order of B-convergence of Runge-Kutta methods, Natural Sci. J. Xiangtan Univ., 14, 2, 16-19 (1992) · Zbl 0782.65097
[2] Bellen, A.; Zennaro, M., Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. Numer. Math., 9, 321-346 (1992) · Zbl 0749.65042
[3] Burrage, K., High order algebraically stable Runge-Kutta method, BIT, 18, 373-383 (1978) · Zbl 0401.65049
[4] Burrage, K.; Butcher, J. C., Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal., 16, 1, 46-57 (1979) · Zbl 0396.65043
[5] Dahlquist, G., G-stability is equivalent to A-stability, BIT, 18, 384-401 (1978) · Zbl 0413.65057
[6] Dekker, K.; Verwer, J. G., Stability of Runge-Kutta Method for Stiff Nonlinear Differential Equations (1984), North-Holland: North-Holland Amsterdam · Zbl 0571.65057
[7] Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations (1971), Prentice_Hall: Prentice_Hall Englewood Cliffs, NJ · Zbl 0217.21701
[8] Henrici, P., Discrete Variable Methods in Ordinary Differential Equations (1962), Wiley: Wiley New York · Zbl 0112.34901
[9] in’t Hout, K. J., A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations, BIT, 32, 634-649 (1992) · Zbl 0765.65069
[10] in’t Hout, K. J., Stability analysis of Runge-Kutta methods for systems of delay differential equations, IMA J. Numer. Anal., 17, 17-27 (1997) · Zbl 0867.65046
[11] in’t Hout, K. J.; Spijker, M. N., Stability analysis of numerical methods for DDEs, Numer. Math., 59, 807-814 (1991) · Zbl 0724.65084
[12] Jackiewicz, Z., One step methods of any order for neutral functional equations, SIAM J. Numer. Anal., 21, 486-511 (1984) · Zbl 0562.65056
[13] Torelli, L., Stability of numerical methods for delay differential equations, J. Comput. Appl. Math., 25, 15-26 (1989) · Zbl 0664.65073
[14] Zennaro, M., P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math., 49, 305-318 (1986) · Zbl 0598.65056
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.