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

摘要

Approximation theory for Lyapunov and Sacker-Sell spectra based upon QRtechniques is used to analyze the stability of a one-step method solving atime-dependent, linear, ordinary differential equation (ODE) initial valueproblem in terms of the local error. Integral separation is used tocharacterize the conditioning of stability spectra calculations. In anapproximate sense the stability of the numerical solution by a one-step methodof a time-dependent linear ODE using real-valued, scalar, time-dependent,linear test equations is justified. This analysis is used to approximateexponential growth/decay rates on finite and infinite time intervals andestablish global error bounds for one-step methods approximating uniformlystable trajectories of nonautonomous and nonlinear ODEs. A time-dependentstiffness indicator and a one-step method that switches between explicit andimplicit Runge-Kutta methods based upon time-dependent stiffness are developedbased upon the theoretical results.