On the Numerical Approximability of Stable Dynamical Systems

Abstract

This paper discusses the numerical approximation of stable dynamical systems of ordinary differential equations by general time-stepping methods. The traditional error analysis for this approximation yields estimates for the global discretization error in terms of the local truncations errors with constants generally growing exponentially in time due to the use of a discrete Gronwall inequality. In special cases of monotone systems, global error estimates can be shown to hold uniformly in time provided that also the time stepping scheme possesses certain monotonicity properties. The standard example is the backward Euler method. However, for more general time-stepping schemes, this is not clear even in the monotone case. It is shown here how in general situations certain qualitative stability properties (exponential and quasi-exponential stability) of the solution to be approximated can be used for extending local error estimates to global ones holding uniformly in time. Further, the approximate solutions are likewise stable and, in the autonomous case, tend to equilibrium points.