Global Bounds on Numerical Error for Ordinary Differential Equations

Classical upper bounds on global numerical errors are much too large for most ordinary differential systems of practical interest. The explanation is that perturbation bounds are generally poorly represented in estimates based on the Lipschitz constant. Consequently, complexity studies that are based on classical Lipschitz estimates are either exceedingly pessimistic or need be restricted to a small subset of problems of marginal relevance. In this paper we review the shortcomings of the classical bounds, explaining the reasons for their inadequacy. We present two modern alternative techniques, that produce much more realistic error bounds: the Alekseev-Gröbner lemma and Lipschitz algebra bounds using the Dahlquist functional. The exposition of both techniques is illustrated with examples of their implementation, and compared with the classical approach. In the concluding section we review a new and speculative approach to approximate ordinary differential systems. This method abandons all standard concepts of time-stepping, matching of terms in a Taylor expansion, etc. Instead, the solution is derived by truncating a Dirichlet expansion. It is accompanied by a useful global error bound and yields itself to easy complexity analysis.