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

TL;DR

This paper develops a spectral stability theory for one-step methods solving time-dependent linear ODEs, using Lyapunov and Sacker-Sell spectra, and introduces adaptive methods based on stiffness indicators.

Contribution

It introduces a Lyapunov and Sacker-Sell spectral stability framework for one-step methods and develops adaptive explicit-implicit schemes based on time-dependent stability analysis.

Findings

Spectral stability analysis accurately predicts numerical solution behavior.

The proposed stiffness indicator effectively guides method switching.

Global error bounds are established for nonautonomous and nonlinear ODEs.

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, 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.