Linear multistep methods for functional-differential equations

Author:
Maarten de Gee

Journal:
Math. Comp. **48** (1987), 633-649

MSC:
Primary 65Q05

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878696-1

MathSciNet review:
878696

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Abstract: A new way to define linear multistep methods for functional differential equations is presented, and some of their properties are analyzed. The asymptotic behavior of the global discretization error is investigated. Finally, Milne's device is generalized to functional differential equations. The effect of the nonsmoothness of the exact solution is taken into account.

**[1]**H. Arndt, "The influence of interpolation on the global discretization error in retarded differential equations," in*Differential Difference Equations*(L. Collatz, G. Meinardus and W. Wetterling, eds.), ISNM 62, Birkhäuser Verlag, Basel, 1983.**[2]**H. Arndt, "Numerical solution of retarded initial value problems: Local and global error and stepsize control,"*Numer. Math.*, v. 43, 1984, pp. 343-360. MR**738381 (85h:65142)****[3]**H. G. Bock & J. Schlöder, "Numerical solution of retarded differential equations with state-dependent time-lags,"*Z. Angew. Math. Mech.*, v. 61, 1981, pp. 269-271.**[4]**R. Bulirsch & J. Stoer, "Numerical treatment of ordinary differential equations by extrapolation methods,"*Numer. Math.*, v. 8, 1966, pp. 1-13. MR**0191095 (32:8504)****[5]**E. Fehlberg, "Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle,"*Computing*, v. 4, 1969, pp. 93-106. MR**0260179 (41:4807)****[6]**A. Feldstein & K. W. Neves, "High order methods for state-dependent delay differential equations with nonsmooth solutions,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 844-863. MR**760621 (86c:34127)****[7]**M. de Gee, "Smoothness of solutions of functional differential equations,"*J. Math. Anal. Appl.*, v. 107, 1985, pp. 103-121. MR**786015 (86f:34144)****[8]**M. de Gee, "Asymptotic expansions for the midpoint rule applied to delay differential equations,"*SIAM J. Numer. Anal.*, v. 23, 1986, pp. 1254-1272. MR**865955 (88c:65106)****[9]**M. de Gee, "The Gragg-Bulirsch-Stoer algorithm for delay differential equations." (To appear.)**[10]**B. A. Gottwald & G. Wanner, "A reliable Rosenbrock integrator for stiff differential equations,"*Computing*, v. 26, 1981, pp. 355-360. MR**620404 (83d:65206)****[11]**J. D. Lambert,*Computational Methods in Ordinary Differential Equations*, Wiley, New York, 1973. MR**0423815 (54:11789)****[12]**H. J. Oberle & H. J. Pesch, "Numerical treatment of delay differential equations by Hermite interpolation,"*Numer. Math.*, v. 37, 1981, pp. 235-255. MR**623043 (83a:65077)****[13]**J. Oppelstrup,*The RKFHB4 Method for Delay Differential Equations*, Lecture Notes in Math., vol. 631, Springer-Verlag, Berlin and New York, 1978. MR**0494955 (58:13730)****[14]**L. Tavernini, "Linear multistep methods for the numerical solution of Volterra functional differential equations,"*Applicable Anal.*, v. 1, 1973, pp. 169-185. MR**0398131 (53:1986)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878696-1

Keywords:
Functional differential equations,
delay differential equations,
linear multistep methods,
predictor-corrector methods,
Milne's device

Article copyright:
© Copyright 1987
American Mathematical Society