Summary
Retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme. Hence one can control the global error of the modified problem but not directly the actual global error of the original problem. In this paper we give an estimate for the actual global error in terms of controllable quantities. Further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems. Along with the new definition we are led to developing a reliable basis for a step selection scheme.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arndt, H.: Zur Schrittweitensteuerung bei retardierten Anfangswertproblemen, Z. Angew. Math. Mech.63, 337–338 (1983)
Arndt, H.: The influence of interpolation on the global error in retarded differential equations. In: Differential-difference equations (Collatz, Meinardus, Wetterling, eds.), ISNM Vol. 62, pp. 9–17, Boston: Birkhäuser 1983
Cryer, C.W.: Numerical methods for functional differential equations. In: Schmitt, K. (ed.). Delay and functional differential equations and their applications. New York: Academic Press 1972
Dekker, K., Verwer, J.G.: Estimating the global error of Runge-Kutta approximations. In: Differential-difference equations (Collatz Meinardus, Wetterling, eds.), ISNM Vol. 62, pp. 55–71, Boston: Birkhäuser 1983
Driver, R.D.: Existence theory for a delay-differential system, contributions to differential equations1, 317–336 (1963)
Driver, R.D.: Ordinary and delay differential equations. Berlin-Heidelberg-New York: Springer 1977
Fox, L., Mayers, D.F., Ockendon, J.R., Tayler, A.B.: On a functional differential equation. J. Inst. Math. Appl.8, 271–307 (1971)
Hale, J.: Theory of functional differential equations. Berlin-Heidelberg-New York: Springer 1977
Haverkamp, R.: Ein numerisches Verfahren zur Lösung retardierter Differentialgleichungen. Z. Angew. Math. Mech.63, 351–352 (1983)
van der Houwen, P.J.: Construction of integration formulas for initial value problems. Amsterdam: North-Holland Publishing Company 1977
van der Houwen, P.J., Sommeijer, B.P.: Improved absolute stability of predictor-corrector methods for retarded differential equations. In: Differential-difference, equations (Collatz, Meinardus, Wetterling, eds.) ISNM Vol. 62, pp. 137–148, Boston: Birkhäuser 1983
Neves, K.W.: Automatic integration of functional differential equations: An approach. ACM Trans. Math. Software1 357–368 (1975)
Neves, K.W.: Algorithm 497 - automatic integration of functional differential equations. Collected Algorithms from ACM (1975)
Neves, K.W., Feldstein, A.: Characterization of jump discontinuities for state dependent delay differential equations. J. Math. Anal. Appl.56, 689–707 (1976)
Oberle, H.J., Pesch, H.J.: Numerical treatment of delay differential equations by Hermiteinterpolation. TUM-M8001, Techn. Univ. München 1980
Oberle, H.J., Pesch, H.J.: Numerical treatment of delay differential equations by Hermiteinterpolation. Numer. Math.37, 235–255 (1981)
Oppelstrup, J.: The RKFHB4 method for delay-differential equations. Lecture Notes in Mathematics631, 133–146. Berlin-Heidelberg-New York: Springer 1976
Oppelstrup, J.: One-step methods with interpolation for Volterra functional differential equations. TRITA-NA-7623 Roy. Inst. Tech. Stockholm 1976
Shampine, L.F., Watts, H.A.: Global error estimation for ordinary differential equations. ACM Trans. Math. Software2, 172–186 (1976)
Stetter, H.J.: Numerische Lösung von Differentialgleichungen mit nacheilendem Argument. Z. Angew. Math. Mech.45, T79–80 (1965)
Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Berlin-Heidelberg-New York: Springer 1973
Stetter, H.J.: Global error estimation in Adams PC-Codes. ACM Trans. Math. Software5, 415–430 (1979)
Stetter, H.J.: Interpolation and error estimation in Adams PC-Codes. SIAM J. Numer. Anal.16, 311–323 (1979)
Tavernini, L.: The approximate solution of Volterra differential systems with state-dependent time lags. SIAM J. Numer. Anal.15, 1039–1052 (1978)
Waltman, P.: Deterministic threshold models in the theory of epidemics. Lecture Notes in Biomathematics 1. Berlin-Heidelberg-New York: Springer 1974
Werner, H.: Ein einfacher Existenzbeweis für das deterministische Epidemienmodell von Waltman and Hoppensteadt. Schriftenreihe des Rechenzentrums der Universität Münster 45, 1980
Werner, H., Schaback, R.: Praktische mathematik II, 2. Auflage. Berlin-Heidelberg-New York: Springer 1979
Bock, H.G., Schlöder, J.: Numerical solution of retarded differential equations with statedependent time lags. Z. Angew. Math. Mech.61, T269–271 (1981)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arndt, H. Numerical solution of retarded initial value problems: Local and global error and stepsize control. Numer. Math. 43, 343–360 (1984). https://doi.org/10.1007/BF01390178
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01390178