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A special stability problem for linear multistep methods

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Abstract

The trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property. For this method error bounds are derived which are valid under rather general conditions. In order to make sure that the error remains bounded ast → ∞, even though the product of the Lipschitz constant and the step-size is quite large, one needs not to assume much more than that the integral curve is uniformly asymptotically stable in the sense of Liapunov.

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References

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  2. Antosiewicz, H.-A.,A survey of Liapunov's second method, in Contributions to the theory of non-linear oscillation, vol. IV., S. Lefschetz (ed.), Ann. Math. Studies, No. 41, Ch VIII, Princeton 1958.

  3. Dahlquist, G.,Stability and error bounds in the numerical integration of ordinary differential equations, Dissertation, Stockholm 1958. Also in Trans. Roy. Inst. Technol. Stockholm, Nr. 130, (1959).

  4. Dahlquist, G.,Stability questions for some numerical methods for ordinary differential equations, To appear in Proc. Symposia on Applied Mathematics, vol. 15, „Interactions between Mathematical Research and High-Speed Computing, 1962.“

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  7. Robertson, H. H.,Some new formulae for the numerical integration of ordinary differential equations, Information Processing, UNESCO, Paris, pp. 106–108.

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Authors and Affiliations

  1. Royal Institute of Technology, Stockholm, Sweden

    Germund G. Dahlquist

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  1. Germund G. Dahlquist
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The preparation of this paper was partly sponsored by the Office of Naval Research and the US Army Research Office (Durham). Reproduction in whole or in part is permitted for any purpose of the US Government.

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Dahlquist, G.G. A special stability problem for linear multistep methods. BIT 3, 27–43 (1963). https://doi.org/10.1007/BF01963532

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  • DOI: https://doi.org/10.1007/BF01963532

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