Andrew J. Steyer

Yu-Min Chung, Andrew J. Steyer, Erik S. Van Vleck

Techniques are developed for decoupling dissipative differential equations. The approach considered is based upon obtaining a sufficient gap in the time dependent linear portion of the equation that corresponds to the linear variational equation. This is done using an orthogonal change of variables that has proven useful in the computation of Lyapunov to decompose the differential equation in terms of slow and fast variables. Numerically this is accomplished in our implementation using smooth, time dependent Householder reflectors. The the nonlinear decoupling transformation or inertial manifold is obtained by solving a boundary value problem (BVP) which allows for a Newton iteration as opposed to the traditional Lyapunov-Perron approach via a fixed point iteration. Finally, the efficacy of the technique is shown using some challenging examples.

Andrew J. Steyer, Erik S. Van Vleck

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.

Andrew J. Steyer, Erik S. Van Vleck

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.