Numerische Mathematik

, Volume 42, Issue 3, pp 359–377 | Cite as

Stability of multistep-methods on variable grids

  • Rolf Dieter Grigorieff


This paper is concerned with the stability of multistep methods for ordinary initial-value problems on grids with variable mesh-sizes. A necessary and sufficient condition for stability is given from which generalizations of recent results by Gear et al. and by Zlatev can be obtained as special cases. As an application the stability of the variable BDF-formulas is treated.

Subject Classifications

AMS 65L05 65L07 CR: 5.17 


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  1. 1.
    Albrecht, P.: Die numerische Behandlung gewöhnlicher Differentialgleichungen. München: Hanser 1979Google Scholar
  2. 2.
    Ansorge, R.: Differenzenapproximationen partieller Anfangswertaufgaben. Stuttgart: Teubner 1978Google Scholar
  3. 3.
    Dahlquist, G., Liniger, W., Nevanlinna, O.: Stability of Two-Step Methods for Variable Integration Steps. IBM Res. Rpt. Nr. 36967, 27 S., New York 1980Google Scholar
  4. 4.
    Gear, C.W., Tu, K.W.: The Effect of Variable Mesh Size on the Stability of Multistep Methods. SIAM J. Numer. Anal.11, 1025–1043 (1974)Google Scholar
  5. 5.
    Gear, C.W., Watanabe, D.S.: Stability and Convergence of Variable Order Multistep Methods. SIAM J. Numer. Anal.11, 1044–1058 (1974)Google Scholar
  6. 6.
    Grigorieff, R.D.: Numerik gewöhnlicher Differentialgleichungen. Bd. 2. Stuttgart: Teubner 1977Google Scholar
  7. 7.
    Lapidus, L., Schiesser, W.E. (eds.): Numerical Methods for Differential Systems. New York: Academic Press 1976Google Scholar
  8. 8.
    März, R.: Variable Multistep Methods. Preprint Nr. 7 der Humboldt-Universität, 31 S., Berlin 1981Google Scholar
  9. 9.
    März, R.: Zur Stabilität und Konsistenz variabler Verfahren. In: Numer. Bhdlg. von Dgln. Wiss. Beiträge der Martin-Luther-Universität Halle-Wittenberg. Strehmel, K. (ed.). S. 87–91, 1981Google Scholar
  10. 10.
    Piotrowski, P.: Stability, Consistency and Convergence of Variablek-Step Methods for Numerical Integration of Large Systems of Ordinary Differential Equations. Lect. Notes in Math.109, 221–227 (1969)Google Scholar
  11. 11.
    Spijker, M.N.: Convergence and Stability of Step-by-Step Methods for the Numerical Solution of Initial-Value Problems. Numer. Math.8, 161–177 (1966)Google Scholar
  12. 12.
    Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Berlin, Heidelberg, New York: Springer 1973Google Scholar
  13. 13.
    Stummel, F.: Biconvergence, bistability and consistency of one step methods for the numerical solution of initial value problems. In: Proc. Conf. Numer. Anal. Dublin 1974, 197–211. London: Academic Press 1975Google Scholar
  14. 14.
    Zlatev, Z.: Stability Properties of Variable Stepsize Variable Formula Methods. Numer. Math.31, 175–182 (1978)Google Scholar
  15. 15.
    Zlatev, Z.: Zero-Stability Properties of the Three-Ordinate Variable Stepsize Variable Formula Methods. Numer. Math.37, 157–166 (1981)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Rolf Dieter Grigorieff
    • 1
  1. 1.Fachbereich 3 MathematikTechnische Universität BerlinBerlin 12

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