Stability analysis of Runge-Kutta methods for non-linear delay differential equations

BIT Numerical Mathematics

Volumn 39, Issue 2, 1999, Pages 270-280

  Chengming, Huang ^a,b   Hongyuan, Fu ^b   Shoufu, Li ^a   Guangnan, Chen ^b  

^a XIANGTAN UNIVERSITY   (China)

^b RESEARCH CENTER OF LASER FUSION   (China)

Author keywords

Delay differential equations; Runge Kutta methods, nonlinear stability

Indexed keywords

EID: 0001129731     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022341929651     Document Type: Article

References (21)

1. V. K. Barwell, Special stability problems for functional equations, BIT, 15 (1975), pp. 130-135.
2. A. Bellen and M. Zennaro, Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. Numer. Math., 9 (1992), pp. 321-346.
3. K. Burrage and J. C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 16 (1979), pp. 46-57.
4. K. Burrage and J. C. Butcher, Non-linear stability of a general class of differential equation methods, BIT 20 (1980), pp. 185-203.
5. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, John Wiley, New York, 1987.
6. G. J. Cooper, A generalization of algebraic stability for Runge-Kutta methods, IMA J. Numer. Anal., 4 (1984), pp. 427-440.
7. K. Dekker and J. G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, CWI Monographs 2, North-Holland, Amsterdam, 1984.
8. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, Berlin, 1991.
9. C. Huang, S. Li, H. Fu, and G. Chen, Stability and error analysis of one-leg methods for nonlinear delay differential equations, Research Report, ICAM, Xiangtan University, 1997, to appear in J. Comput. Appl. Math. 102 (1999).
10. K. J. in' t Hout, A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations, BIT, 32 (1992), pp. 634-649.
11. K. J. in' t Hout, Stability analysis of Runge-Kutta methods for systems of delay differential equations, IMA J. Numer. Anal., 17 (1997), pp. 17-27.
12. K. J. in't Hout, A note on unconditional maximum norm contractivity of diagonally split Runge-Kutta methods, SIAM J. Numer. Anal., 33 (1996), pp. 1125-1134.
13. K. J. in' t Hout and M. N. Spijker, Stability analysis of numerical methods for delay differential equations, Numer. Math., 59 (1991), pp. 807-814.
14. S. Li, Stability criteria for Runge-Kutta methods, Natur. Sci. J. Xiangtan Univ., 9 (1987), pp. 11-17.
15. S. Li, Theory of Computaional Methods for Stiff Differential Equations, Hunan Science and Technology Publisher, Changsha, 1997.
16. M. Z. Liu and M. N. Spijker, The stability of the θ-methods in the numerical solution of delay differential equations, IMA J. Numer. Anal., 10 (1990), pp. 31-48.
17. L. Torelli, Stability of numerical methods for delay differential equations, J. Comput. Appl. Math., 25 (1989), pp. 15-26.
18. D. S. Watanabe and M. G. Roth, The stability of difference formulas for delay differential equations, SIAM J. Numer. Anal., 22 (1985), pp. 132-145.
19. M. Zennaro, P-stability of Runge-Kutta methods for delay differential equations, Numer. Math., 49 (1986), pp. 305-318.
20. M. Zennaro, Contractivity of Runge-Kutta methods with respect to forcing terms, Appl. Numer. Math., 10 (1993), pp. 321-345.
21. M. Zennaro, Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations, Numer. Math., 77 (1997), pp. 549-563.