Summary
Stability regions of θ-methods for the linear delay differential test equations
where τ is a positive constant, are presented. In the case thatp andq are real constant coefficients, necessary and sufficient conditions on the stepsize for the stability of a θ-method are obtained. Furthermore, whenp andq are complex coefficients, sufficient conditions for the stability of the θ-methods are also given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arndt, H.: Numerical Solution of Retarded Initial Value Problems: Local and Global Error and Stepsize Control. Numer. Math.43, 343–360 (1984)
Barwell, V.K.: Special Stability Problems for Functional Differential Equations. BIT15, 130–135 (1975)
Bellman, R., Cooke, K.L.: Differential-Difference Equations. New York: Academic Press 1963
Bickart, T.A.: P-stable andP[α,β]-stable integration/interpolation methods in the solution of Retarded Differential-Difference Equations. BIT22, 464–476 (1982)
Capdeville, M., Seguier, P.: Stabilité absolue des méthodes RKR. Dept. de Mathématiques, Université de Pau 1983
Cryer, C.W.: Highly Stable Multistep Methods for Retarded Differential Equations. SIAM Numer. Anal.11, 788–797 (1974)
Cushing, J.M.: Integrodifferential equations and delay models in population dynamics. Lect. Notes Biomath., 20th Ed. Berlin: Springer 1977
Driver, R.D.: Ordinary and Delay Differential Equations. Appl. Mathem. Sci., 20th Ed. Berlin Heidelberg New York: Springer 1977
El'sgol'tz, L.E., Norkin, S.B.: Introduction to the theory and application of Differential Equations with Deviating Arguments. Mathematics in Science and Engineering. Vol. 105. New York: Academic Press 1973
Grande, T.: Numerical methods for the integration of delay differential equations Thesis. Dpto. Mathemática Aplicada, Uniz. Zaragoza 1986
Hale, J.K.: Functional Differential Equations. New York: Springer 1971
Jackiewicz, Z.: Asymptotic Stability Analysis of θ-Methods for Functional Differential Equations. Numer. Math.43, 389–396 (1984)
Marden, M.: Geometry of polynomials. American Mathematical Society: Providence, Rhode Island 1966
Miller, J.J.H.: On the location of zeros of certain classes of polynomials with application to numerical analysis. J. Inst. Math. Appl.8, 397–406 (1971)
Neves, K.W.: Control of Interpolatory Error in Retarded Differential Equations. ACM Trans. Math. Software7, 421–444 (1981)
Oberle, H.J., Pesch, H.J.: Numerical Treatment of Delay Differential Equations by Hermite-Interpolation. Numer. Math.37, 235–257 (1981)
Oppelstrup, J.: The RKFHB4 Method for Delay-Differential Equations. Numerical Treatment of Differential Equations. Proceedings Oberwolfach. Lect. Notes N. 631 (1976)
Roth, M.G.: Difference Methods for Stiff Delay Differential Equations. Dpto. of Comp. Science. Univ. of Illinois at Urbana. Champaign Urbana, IL 61801. Thesis (1980)
Tavernini, L.: Linear Multistep Methods for the Numerical Solution of Volterra Functional Differential Equations. Appl. Anal.1, 169–185 (1973)
Van der Houwen, P.J.: Stability in linear multistep methods for pure delay equations. Mathematisch. Centrum, Amsterdam 1983
Watanabe, D.S., Roth, M.G.: The stability of Difference formulas for Delay Differential Equations. SIAM J. Numer. Anal.22, 132–145 (1985)
Wiederholt, L.F.: Stability of Multistep Methods for Delay Differential Equations. Math. Comput.30, 283–290 (1976)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Calvo, M., Grande, T. On the asymptotic stability ofθ-methods for delay differential equations. Numer. Math. 54, 257–269 (1989). https://doi.org/10.1007/BF01396761
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01396761