Underlying one-step methods and nonautonomous stability of general linear methods
Andrew J. Steyer, Erik S. Van Vleck
TL;DR
This paper extends the theory of underlying one-step methods to general linear methods for nonautonomous ODEs, integrating spectral stability analysis to develop a stability diagnostic tool.
Contribution
It generalizes the stability theory of one-step methods to strictly stable GLMs for nonautonomous ODEs, combining Lyapunov and spectral analysis.
Findings
Developed a stability diagnostic for nonautonomous linear ODEs using GLMs.
Extended the theory of underlying one-step methods to general linear methods.
Applied spectral stability theory to analyze GLM stability in nonautonomous settings.
Abstract
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.
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Taxonomy
TopicsNonlinear Dynamics and Pattern Formation · Fractional Differential Equations Solutions · Numerical methods for differential equations