2015/11/28 by Andrew Steyer, Steyer, Andrew J., Erik S. Van Vleck +1
Mathematics · Engineering · Physics and Astronomy · #Numerical methods for differential equations #Stability and Controllability of Differential Equations #Model Reduction and Neural Networks
paper · pdf · doi:10.48550/arxiv.1511.08943
Approximation theory for Lyapunov and Sacker-Sell spectra based upon QR\ntechniques is used to analyze the stability of a one-step method solving a\ntime-dependent, linear, ordinary differential equation (ODE) initial value\nproblem in terms of the local error. Integral separation is used to\ncharacterize the conditioning of stability spectra calculations. In an\napproximate sense the stability of the numerical solution by a one-step method\nof a time-dependent linear ODE using real-valued, scalar, time-dependent,\nlinear test equations is justified. This analysis is used to approximate\nexponential growth/decay rates on finite and infinite time intervals and\nestablish global error bounds for one-step methods approximating uniformly\nstable trajectories of nonautonomous and nonlinear ODEs. A time-dependent\nstiffness indicator and a one-step method that switches between explicit and\nimplicit Runge-Kutta methods based upon time-dependent stiffness are developed\nbased upon the theoretical results.\n