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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
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] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729 (24 #B1772)
  • [3] Rolf Jeltsch, Multistep multiderivative methods for the numerical solution of initial value problems of ordinary differential equations, Department of Mathematics, University of Kentucky, Lexington, Ky., 1976. Seminar Notes 1975–76. MR 0461915 (57 #1897)
  • [4] J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976), no. 2, 189–202. MR 0431691 (55 #4686)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1978-0501999-1
PII: S 0025-5718(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