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On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods

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Summary

We study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy/dt=λy and constant stepsizeh. LetS be the region of the absolute stability of the method, and letD be a closed subset ofS (on the Riemann sphere\(\mathbb{C}\)).

It is shown that the solutions of these difference equations are bounded forn≧0, uniformly for λh∈D.S is itself closed in\(\mathbb{C}\) iff ∂S is free of cusps. The question is studed by means of contractivity analysis and a matrix theorem, derived from the matrix theorem of Kreiss.

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Authors and Affiliations

  1. The Royal Institute of Technology, S-10044, Stockholm 70, Sweden

    Germund Dahlquist

  2. Jilin University, People's Republic of China

    Huang Mingyou

  3. Stanford University, USA

    Randall LeVeque

Authors
  1. Germund Dahlquist
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  2. Huang Mingyou
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  3. Randall LeVeque
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Dahlquist, G., Mingyou, H. & LeVeque, R. On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods. Numer. Math. 42, 1–13 (1983). https://doi.org/10.1007/BF01400914

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  • DOI: https://doi.org/10.1007/BF01400914

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