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

Steyer, Andrew J.; Van Vleck, Erik S.

doi:10.1007/s10543-018-0704-2

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

Journal Article · 13 April 2018 · BIT Numerical Mathematics

DOI:https://doi.org/10.1007/s10543-018-0704-2· OSTI ID:1441387

Steyer, Andrew J. ^[1]; Van Vleck, Erik S. ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Univ. of Kansas, Lawrence, KS (United States)

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.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)

Grant/Contract Number:: AC04-94AL85000; DMS-1419047

OSTI ID:: 1441387

Report Number(s):: SAND-2018-3923J; 662324

Journal Information:: BIT Numerical Mathematics, Vol. 58; ISSN 0006-3835

Publisher:: Springer NatureCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 2 works

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Related Subjects

97 MATHEMATICS AND COMPUTING
One-step methods
Stiffness
Lyapunov exponents
Sacker–Sell spectrum
Nonautonomous differential equations

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

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