  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, 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. In an
approximate sense the stability of the numerical solution by a one-step method
of a time-dependent linear ODE using real-valued, scalar, time-dependent,
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
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.
