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

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.

Authors:
 [1];  [2]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Univ. of Kansas, Lawrence, KS (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
OSTI Identifier:
1441387
Report Number(s):
SAND-2018-3923J
Journal ID: ISSN 0006-3835; 662324
Grant/Contract Number:  
AC04-94AL85000; DMS-1419047
Resource Type:
Accepted Manuscript
Journal Name:
BIT Numerical Mathematics
Additional Journal Information:
Journal Volume: 58; Journal ID: ISSN 0006-3835
Publisher:
Springer Nature
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; One-step methods; Stiffness; Lyapunov exponents; Sacker–Sell spectrum; Nonautonomous differential equations

Citation Formats

Steyer, Andrew J., and Van Vleck, Erik S. A Lyapunov and Sacker–Sell spectral stability theory for one-step methods. United States: N. p., 2018. Web. doi:10.1007/s10543-018-0704-2.
Steyer, Andrew J., & Van Vleck, Erik S. A Lyapunov and Sacker–Sell spectral stability theory for one-step methods. United States. doi:10.1007/s10543-018-0704-2.
Steyer, Andrew J., and Van Vleck, Erik S. Fri . "A Lyapunov and Sacker–Sell spectral stability theory for one-step methods". United States. doi:10.1007/s10543-018-0704-2. https://www.osti.gov/servlets/purl/1441387.
@article{osti_1441387,
title = {A Lyapunov and Sacker–Sell spectral stability theory for one-step methods},
author = {Steyer, Andrew J. and Van Vleck, Erik S.},
abstractNote = {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.},
doi = {10.1007/s10543-018-0704-2},
journal = {BIT Numerical Mathematics},
number = ,
volume = 58,
place = {United States},
year = {2018},
month = {4}
}

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