BIT Numerical Mathematics

, Volume 58, Issue 3, pp 749–781 | Cite as

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

  • Andrew J. Steyer
  • Erik S. Van Vleck


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.


One-step methods Stiffness Lyapunov exponents Sacker–Sell spectrum Nonautonomous differential equations 

Mathematics Subject Classification

65L04 65L05 65P40 34D08 34D09 


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Authors and Affiliations

  1. 1.Sandia National LaboratoriesAlbuquerqueUSA
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA

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