Underlying one-step methods and nonautonomous stability of general linear methods
Provides a theoretical stability analysis tool for general linear methods solving nonautonomous ODEs, which is incremental for numerical analysis researchers.
The paper extends underlying one-step theory to strictly stable general linear methods for nonautonomous ODEs with global Lipschitz condition, and combines it with Lyapunov/Sacker-Sell spectral theory to analyze stability, yielding a stability diagnostic for such methods.
We generalize the theory of underlying one-step methods to strictly stable general linear methods (GLMs) solving nonautonomous ordinary differential equations (ODEs) that satisfy a global Lipschitz condition. We combine this theory with the Lyapunov and Sacker-Sell spectral stability theory for one-step methods developed in [34,35,36] to analyze the stability of a strictly stable GLM solving a nonautonomous linear ODE. These results are applied to develop a stability diagnostic for the solution of nonautonomous linear ODEs by strictly stable GLMs.