A Lyapunov and Sacker-Sell spectral stability theory for one-step methods (Q1783382)

The authors study the stability of numerical solutions of linear and nonlinear initial value problems for ordinary time-dependent differential equations. For the linear time-dependent systems they apply an orthogonal transformation, which puts the matrix of the system in upper triangular form, that is suitable to investigate the spectral properties; the eigenvalues for systems with constant coefficients generalize the Lyapunov and Sacker-Sell spectrum. The integral separation of the time-dependent upper-diagonal matrix is used to characterize the conditioning of the spectra calculations. It is shown that for properly integral separated continuous systems the spectral properties are well approximated by the discrete system. If the spectra of such systems are in the left half plane, the zero solution is an exponentially stable equilibrium. The authors prove that stability and convergence properties are preserved also for nonlinear systems for properly chosen stepsizes. They suggest an indicator for the stiffness of the system based on the distance between the right-most and left-most spectrum interval and construct a one-step method, which switches between an explicit and implicit Runge-Kutta method.