Linear multistep methods for functionaldifferential equations
Author:
Maarten de Gee
Journal:
Math. Comp. 48 (1987), 633649
MSC:
Primary 65Q05
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.
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 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. 343360. MR 738381 (85h:65142)
 [3]
 H. G. Bock & J. Schlöder, "Numerical solution of retarded differential equations with statedependent timelags," Z. Angew. Math. Mech., v. 61, 1981, pp. 269271.
 [4]
 R. Bulirsch & J. Stoer, "Numerical treatment of ordinary differential equations by extrapolation methods," Numer. Math., v. 8, 1966, pp. 113. MR 0191095 (32:8504)
 [5]
 E. Fehlberg, "Klassische RungeKuttaFormeln fünfter und siebenter Ordnung mit SchrittweitenKontrolle," Computing, v. 4, 1969, pp. 93106. MR 0260179 (41:4807)
 [6]
 A. Feldstein & K. W. Neves, "High order methods for statedependent delay differential equations with nonsmooth solutions," SIAM J. Numer. Anal., v. 21, 1984, pp. 844863. MR 760621 (86c:34127)
 [7]
 M. de Gee, "Smoothness of solutions of functional differential equations," J. Math. Anal. Appl., v. 107, 1985, pp. 103121. 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. 12541272. MR 865955 (88c:65106)
 [9]
 M. de Gee, "The GraggBulirschStoer 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. 355360. 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. 235255. MR 623043 (83a:65077)
 [13]
 J. Oppelstrup, The RKFHB4 Method for Delay Differential Equations, Lecture Notes in Math., vol. 631, SpringerVerlag, 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. 169185. MR 0398131 (53:1986)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708786961
PII:
S 00255718(1987)08786961
Keywords:
Functional differential equations,
delay differential equations,
linear multistep methods,
predictorcorrector methods,
Milne's device
Article copyright:
© Copyright 1987
American Mathematical Society
