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A necessary condition for $ A$-stability of multistep multiderivative methods


Author: Rolf Jeltsch
Journal: Math. Comp. 30 (1976), 739-746
MSC: Primary 65L05
MathSciNet review: 0431690
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Abstract: The region of absolute stability of multistep multiderivative methods is studied in a neighborhood of the origin. This leads to a necessary condition for A-stability. For methods where $ \rho (\zeta )/(\zeta - 1)$ has no roots of modulus 1 this condition can be checked very easily. For Hermite interpolatory and Adams type methods a necessary condition for A-stability is found which depends only on the error order and the number of derivatives used at $ ({x_{n + k}},{y_{n + k}})$.


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

  • [1] R. L. BROWN, "Multi-derivative numerical methods for the solution of stiff ordinary differential equations," Dept. of Comput. Sci., Univ. of Illinois, Report UIUCDCS-R-74-672, 1974.
  • [2] S. D. Conte, Elementary numerical analysis: An algorithmic approach, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1965. MR 0202267
  • [3] B. L. EHLE, On Padé Approximations to the Exponential Function and A-stable methods for the Numerical Solution of Initial Value Problems, Res. Rep. CSRR 2010, Dept. of Appl. Anal. and Comput. Sci., Univ. of Waterloo, Canada, 1969.
  • [4] W. H. Enright, Second derivative multistep methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 11 (1974), 321–331. MR 0351083
  • [5] C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898
  • [6] Eberhard Griepentrog, Mehrschrittverfahren zur numerischen Integration von gewöhnlichen Differentialgleichungssystemen und asymptotische Exaktheit, Wiss. Z. Humboldt-Univ. Berlin Math.-Natur. Reihe 19 (1970), 637–653 (German, with Russian, English and French summaries). MR 0321300
  • [7] Rolf Jeltsch, Stiff stability and its relation to 𝐴₀- and 𝐴(0)-stability, SIAM J. Numer. Anal. 13 (1976), no. 1, 8–17. MR 0411174
  • [8] Rolf Jeltsch, Multistep methods using higher derivatives and damping at infinity, Math. Comp. 31 (1977), no. 137, 124–138. MR 0428716, 10.1090/S0025-5718-1977-0428716-7
  • [9] Rolf Jeltsch, Multistep multiderivative methods and Hermite-Birkhoff interpolation, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) Utilitas Math. Publ., Winnipeg, Man., 1976, pp. 417–428. Congressus Numerantium, No. XVI. MR 0436600

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1976-0431690-X
Keywords: Linear k-step methods using higher derivatives, behavior of the region of absolute stability at the origin, necessary condition for A-stability, Hermite interpolatory multistep multiderivative methods, Adams-type multistep multiderivative methods
Article copyright: © Copyright 1976 American Mathematical Society