Summary
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.
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References
Albrecht, P.: Die numerische Behandlung gewöhnlicher Differentialgleichungen. München: Hanser 1979
Ansorge, R.: Differenzenapproximationen partieller Anfangswertaufgaben. Stuttgart: Teubner 1978
Dahlquist, G., Liniger, W., Nevanlinna, O.: Stability of Two-Step Methods for Variable Integration Steps. IBM Res. Rpt. Nr. 36967, 27 S., New York 1980
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)
Gear, C.W., Watanabe, D.S.: Stability and Convergence of Variable Order Multistep Methods. SIAM J. Numer. Anal.11, 1044–1058 (1974)
Grigorieff, R.D.: Numerik gewöhnlicher Differentialgleichungen. Bd. 2. Stuttgart: Teubner 1977
Lapidus, L., Schiesser, W.E. (eds.): Numerical Methods for Differential Systems. New York: Academic Press 1976
März, R.: Variable Multistep Methods. Preprint Nr. 7 der Humboldt-Universität, 31 S., Berlin 1981
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, 1981
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)
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)
Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Berlin, Heidelberg, New York: Springer 1973
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 1975
Zlatev, Z.: Stability Properties of Variable Stepsize Variable Formula Methods. Numer. Math.31, 175–182 (1978)
Zlatev, Z.: Zero-Stability Properties of the Three-Ordinate Variable Stepsize Variable Formula Methods. Numer. Math.37, 157–166 (1981)