A Lyapunov and Sacker–Sell spectral stability theory for onestep methods
Abstract
Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a onestep method solving a timedependent (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 onestep method of a nonautonomous linear ODE using realvalued, 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 onestep methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A timedependent stiffness indicator and a onestep method that switches between explicit and implicit Runge–Kutta methods based upon timedependent stiffness are developed based upon the theoretical results.
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Univ. of Kansas, Lawrence, KS (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
 OSTI Identifier:
 1441387
 Report Number(s):
 SAND20183923J
Journal ID: ISSN 00063835; 662324
 Grant/Contract Number:
 AC0494AL85000; DMS1419047
 Resource Type:
 Accepted Manuscript
 Journal Name:
 BIT Numerical Mathematics
 Additional Journal Information:
 Journal Volume: 58; Journal ID: ISSN 00063835
 Publisher:
 Springer Nature
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Onestep 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 onestep methods. United States: N. p., 2018.
Web. doi:10.1007/s1054301807042.
Steyer, Andrew J., & Van Vleck, Erik S. A Lyapunov and Sacker–Sell spectral stability theory for onestep methods. United States. doi:10.1007/s1054301807042.
Steyer, Andrew J., and Van Vleck, Erik S. Fri .
"A Lyapunov and Sacker–Sell spectral stability theory for onestep methods". United States. doi:10.1007/s1054301807042. https://www.osti.gov/servlets/purl/1441387.
@article{osti_1441387,
title = {A Lyapunov and Sacker–Sell spectral stability theory for onestep 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 onestep method solving a timedependent (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 onestep method of a nonautonomous linear ODE using realvalued, 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 onestep methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A timedependent stiffness indicator and a onestep method that switches between explicit and implicit Runge–Kutta methods based upon timedependent stiffness are developed based upon the theoretical results.},
doi = {10.1007/s1054301807042},
journal = {BIT Numerical Mathematics},
number = ,
volume = 58,
place = {United States},
year = {2018},
month = {4}
}
Web of Science
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