A Lyapunov and Sacker-Sell spectral stability theory for one-step methods
This work provides a theoretical foundation for stability analysis of numerical methods for nonautonomous ODEs, benefiting computational scientists and engineers by enabling adaptive method selection based on time-dependent stiffness.
The paper develops a Lyapunov and Sacker-Sell spectral stability theory for one-step methods solving time-dependent linear ODEs, using QR-based approximation to analyze stability and conditioning. It establishes global error bounds for uniformly stable trajectories and introduces a time-dependent stiffness indicator that enables switching between explicit and implicit Runge-Kutta methods.
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.