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A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

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Abstract

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.

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Authors and Affiliations

  1. Sandia National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM, 87185-1320, USA

    Andrew J. Steyer

  2. Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS, 66045, USA

    Erik S. Van Vleck

Authors
  1. Andrew J. Steyer
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  2. Erik S. Van Vleck
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Correspondence to Andrew J. Steyer.

Additional information

Communicated by Antonella Zanna Munthe-Kaas.

This research was supported in part by NSF Grant DMS-1419047.

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Steyer, A.J., Van Vleck, E.S. A Lyapunov and Sacker–Sell spectral stability theory for one-step methods. Bit Numer Math 58, 749–781 (2018). https://doi.org/10.1007/s10543-018-0704-2

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  • DOI: https://doi.org/10.1007/s10543-018-0704-2

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