Stability of multistep methods for delay differential equations

Author:
Lawrence F. Wiederholt

Journal:
Math. Comp. **30** (1976), 283-290

MSC:
Primary 65Q05; Secondary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1976-0398132-4

MathSciNet review:
0398132

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The absolute and relative stability of linear multistep methods for a finite step size is studied for delay differential equations. The differential equations are assumed linear and the delays a constant integer multiple of the step size. Computable conditions for stability are developed for scalar equations. Plots of the stability regions for several common multistep methods are included. For the integration methods considered, the stability regions for delay differential equations are significantly different from the stability regions for ordinary differential equations.

**[1]**P. E. CHASE, "Stability properties of predictor-corrector methods for ordinary differential equations,"*J. Assoc. Comput. Mach.*, v. 9, 1962, pp. 457-468. MR**29**#738. MR**0163436 (29:738)****[2]**R. R. BROWN, J. D. RILEY & M. M. BENNETT, "Stability properties of Adams-Moulton type methods,"*Math. Comp.*, v. 19, 1965, pp. 90-96. MR**31**#2829. MR**0178572 (31:2829)****[3]**A. RALSTON, "Relative stability in the numerical solution of ordinary differential equations,"*SIAM Rev.*, v. 7, 1965, pp. 114-125. MR**31**#2831. MR**0178574 (31:2831)****[4]**R. W. HAMMING, "Stable predictor-corrector methods for ordinary differential equations,"*J. Assoc. Comput. Mach.*, v. 16, 1959, pp. 37-47. MR**21**#973. MR**0102179 (21:973)****[5]**W. B. GRAGG & H. J. STETTER, "Generalized multistep predictor-corrector methods,"*J. Assoc. Comput. Mach.*, v. 11, 1964, pp. 188-209. MR**28**#4680. MR**0161476 (28:4680)****[6]**R. K. BRAYTON & R. A. WILLOUGHBY, "On the numerical integration of a symmetric system of difference-differential equations of neutral type,"*J. Math. Anal. Appl.*, v. 18, 1967, pp. 182-189. MR**35**#3926. MR**0213061 (35:3926)****[7]**L. TAVERNINI,*Numerical Methods for Volterra Functional Differential Equations*, Ph. D. Thesis, University of Wisconsin, 1969.**[8]**M. A. FELDSTEIN,*Discretization Methods for Retarded Ordinary Differential Equations*, Ph. D. Thesis, University of California, Los Angeles, 1964.**[9]**M. N. SPYKER,*Stability and Convergence of Finite-Difference Methods*, Ph. D. Thesis, Centraal-Reken-Instituul, Ryksuniversiteil, Leiden, The Netherlands, 1969.**[10]**L. F. WIEDERHOLT,*Numerical Integration of Delay Differential Equations*, Ph. D. Thesis, University of Wisconsin, 1970.**[11]**P. HENRICI,*Discrete Variable Methods in Ordinary Differential Equations*, Wiley, New York, 1962, p. 218. MR**24**#B1772. MR**0135729 (24:B1772)****[12]**R. L. CRANE & R. J. LAMBERT, "Stability of a generalized corrector formula,"*J. Assoc. Comput. Mach.*, v. 9, 1962, pp. 104-117. MR**24**#B1283. MR**0135233 (24:B1283)****[13]**A. HALANAY,*Differential Equations*:*Stability, Oscillations, Time Lags*, Academic Press, New York, 1966, p. 349. MR**35**#6938. MR**0216103 (35:6938)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65Q05,
65L05

Retrieve articles in all journals with MSC: 65Q05, 65L05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0398132-4

Keywords:
Delay (or retarded) differential equations,
stability of integration methods

Article copyright:
© Copyright 1976
American Mathematical Society