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BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 3/2018

13.04.2018

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

verfasst von: Andrew J. Steyer, Erik S. Van Vleck

Erschienen in: BIT Numerical Mathematics | Ausgabe 3/2018

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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.
Vorheriger Artikel Efficient preconditioner of one-sided space fractional diffusion equation
Nächster Artikel An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg–Landau equation

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Metadaten
Titel
A Lyapunov and Sacker–Sell spectral stability theory for one-step methods
verfasst von
Andrew J. Steyer
Erik S. Van Vleck
Publikationsdatum
13.04.2018
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 3/2018
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-018-0704-2

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