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:
-
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- 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. https://doi.org/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. https://doi.org/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 = {Fri Apr 13 00:00:00 EDT 2018},
month = {Fri Apr 13 00:00:00 EDT 2018}
}
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Works referenced in this record:
Stiffness 1952–2012: Sixty years in search of a definition
journal, June 2014
- Söderlind, Gustaf; Jay, Laurent; Calvo, Manuel
- BIT Numerical Mathematics, Vol. 55, Issue 2
On the Error in the Product QR Decomposition
journal, January 2010
- Van Vleck, Erik S.
- SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 4
Fine Structure of the Dichotomy Spectrum
journal, March 2012
- Pötzsche, Christian
- Integral Equations and Operator Theory, Vol. 73, Issue 1
Lyapunov Spectral Intervals: Theory and Computation
journal, January 2002
- Dieci, Luca; Van Vleck, Erik S.
- SIAM Journal on Numerical Analysis, Vol. 40, Issue 2
Impulses and Physiological States in Theoretical Models of Nerve Membrane
journal, July 1961
- FitzHugh, Richard
- Biophysical Journal, Vol. 1, Issue 6
Lyapunov and Sacker–Sell Spectral Intervals
journal, July 2006
- Dieci, Luca; Van Vleck, Erik S.
- Journal of Dynamics and Differential Equations, Vol. 19, Issue 2
Stable Attracting Sets in Dynamical Systems and in Their One-Step Discretizations
journal, October 1986
- Kloeden, P. E.; Lorenz, J.
- SIAM Journal on Numerical Analysis, Vol. 23, Issue 5
On the error in computing Lyapunov exponents by QR Methods
journal, September 2005
- Dieci, Luca; Vleck, Erik S. Van
- Numerische Mathematik, Vol. 101, Issue 4
Convergence and stability in the numerical integration of ordinary differential equations
journal, December 1956
- Dahlquist, Germund
- MATHEMATICA SCANDINAVICA, Vol. 4
A 3(2) pair of Runge - Kutta formulas
journal, January 1989
- Bogacki, P.; Shampine, L. F.
- Applied Mathematics Letters, Vol. 2, Issue 4
Additive Runge–Kutta schemes for convection–diffusion–reaction equations
journal, January 2003
- Kennedy, Christopher A.; Carpenter, Mark H.
- Applied Numerical Mathematics, Vol. 44, Issue 1-2
Time-Varying Linearization and the Perron Effects
journal, April 2007
- Leonov, G. A.; Kuznetsov, N. V.
- International Journal of Bifurcation and Chaos, Vol. 17, Issue 04
Die Stabilit�tsfrage bei Differentialgleichungen
journal, December 1930
- Perron, Oskar
- Mathematische Zeitschrift, Vol. 32, Issue 1
Unitary Integrators and Applications to Continuous Orthonormalization Techniques
journal, February 1994
- Dieci, Luca; Russell, Robert D.; Van Vleck, Erik S.
- SIAM Journal on Numerical Analysis, Vol. 31, Issue 1
A special stability problem for linear multistep methods
journal, March 1963
- Dahlquist, Germund G.
- BIT, Vol. 3, Issue 1
Difference Methods for Stiff Ordinary Differential Equations
journal, February 1978
- Kreiss, Heinz-Otto
- SIAM Journal on Numerical Analysis, Vol. 15, Issue 1
The general problem of the stability of motion
journal, March 1992
- Lyapunov, A. M.
- International Journal of Control, Vol. 55, Issue 3
Detecting exponential dichotomy on the real line: SVD and QR algorithms
journal, January 2011
- Dieci, Luca; Elia, Cinzia; Van Vleck, Erik
- BIT Numerical Mathematics, Vol. 51, Issue 3
The equivalence of algebraic stability andAN-stability
journal, December 1987
- Butcher, J. C.
- BIT, Vol. 27, Issue 4
On the Compuation of Lyapunov Exponents for Continuous Dynamical Systems
journal, February 1997
- Dieci, Luca; Russell, Robert D.; Van Vleck, Erik S.
- SIAM Journal on Numerical Analysis, Vol. 34, Issue 1
On invariant closed curves for one-step methods
journal, January 1987
- Beyn, Wolf-J�rgen
- Numerische Mathematik, Vol. 51, Issue 1
Computation of orthonormal factors for fundamental solution matrices
journal, October 1999
- Dieci, Luca; Van Vleck, Erik S.
- Numerische Mathematik, Vol. 83, Issue 4
A stability property of implicit Runge-Kutta methods
journal, December 1975
- Butcher, J. C.
- BIT, Vol. 15, Issue 4
Embeddedsdirk-methods of basic order three
journal, December 1984
- Nørsett, Syvert P.; Thomsen, Per G.
- BIT, Vol. 24, Issue 4
A spectral theory for linear differential systems
journal, March 1978
- Sacker, Robert J.; Sell, George R.
- Journal of Differential Equations, Vol. 27, Issue 3
Approximating Lyapunov exponents and Sacker–Sell spectrum for retarded functional differential equations
journal, May 2013
- Breda, Dimitri; Van Vleck, Erik
- Numerische Mathematik, Vol. 126, Issue 2
Perturbation theory for the approximation of stability spectra by QR methods for sequences of linear operators on a Hilbert space
journal, July 2012
- Badawy, M.; Van Vleck, E.
- Linear Algebra and its Applications, Vol. 437, Issue 1
Underlying one-step methods and nonautonomous stability of general linear methods
journal, January 2018
- J. Steyer, Andrew; S. Van Vleck, Erik
- Discrete & Continuous Dynamical Systems - B, Vol. 23, Issue 7
Quasi stage order conditions for SDIRK methods
journal, August 2002
- Cameron, Frank; Palmroth, Mikko; Piché, Robert
- Applied Numerical Mathematics, Vol. 42, Issue 1-3
Fast and Slow Waves in the FitzHugh–Nagumo Equation
journal, January 1997
- Krupa, Martin; Sandstede, Björn; Szmolyan, Peter
- Journal of Differential Equations, Vol. 133, Issue 1
The structurally stable linear systems on the half-line are those with exponential dichotomies
journal, July 1979
- Palmer, Kenneth J.
- Journal of Differential Equations, Vol. 33, Issue 1
Stability Criteria for Implicit Runge–Kutta Methods
journal, February 1979
- Burrage, Kevin; Butcher, J. C.
- SIAM Journal on Numerical Analysis, Vol. 16, Issue 1
Computation of a few Lyapunov exponents for continuous and discrete dynamical systems
journal, July 1995
- Dieci, Luca; Van Vleck, Erik S.
- Applied Numerical Mathematics, Vol. 17, Issue 3
Dichotomy spectra of triangular equations
journal, June 2015
- Pötzsche, Christian
- Discrete and Continuous Dynamical Systems, Vol. 36, Issue 1