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Complete characterization of multistep methods with an interval of periodicity for solving $ y\sp{\prime\prime}=f(x,\,y)$


Author: Rolf Jeltsch
Journal: Math. Comp. 32 (1978), 1108-1114
MSC: Primary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1978-0501999-1
MathSciNet review: 501999
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Abstract: Linear multistep methods for the second order differential equation $ y'' = - {\lambda ^2}y$, $ \lambda $ real, are said to have an interval of periodicity if for a fixed $ \lambda $ and a stepsize sufficiently small the numerical solution neither explodes nor decays. We give a very simple necessary and sufficient condition under which a linear multistep method has an interval of periodicity. This condition is then applied to multistep methods with an optimal error order.


References [Enhancements On Off] (What's this?)

  • [1] L. K. AHLFORS, Complex Analysis, McGraw-Hill, New York, 1953.
  • [2] P. HENRICI, Discrete Variable Methods for Ordinary Differential Equations, Wiley, New York, 1962. MR 0135729 (24:B1772)
  • [3] R. JELTSCH, Multistep Multiderivative Methods for the Numerical Solution of Initial Value Problems of Ordinary Differential Equations, Seminar Notes 1975-76, Dept. of Mathematics, University of Kentucky, 1976. MR 0461915 (57:1897)
  • [4] J. D. LAMBERT & I. A. WATSON, "Symmetric multistep methods for periodic initial value problems," J. Inst. Math. Appl., v. 18, 1976, pp. 189-202. MR 0431691 (55:4686)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0501999-1
Keywords: Linear multistep methods, second order differential equations, orbital stability, interval of periodicity, optimal methods, growth parameters
Article copyright: © Copyright 1978 American Mathematical Society

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