Probabilistic and deterministic convergence proofs for software for initial value problems

Andrew M. Stuart

Abstract: The numerical solution of initial value problems for ordinary differential equations is frequently performed by means of adaptive algorithms with user-input tolerance τ. The time-step is then chosen according to an estimate, based on small time-step heuristics, designed to try and ensure that an approximation to the local error commited is bounded by τ. A question of natural interest is to determine how the global error behaves with respect to the tolerance τ. This has obvious practical interest and also leads to an interesting problem in mathematical analysis. The primary difficulties arising in the analysis are that: (i) the time-step selection mechanisms used in practice are discontinuous as functions of the specified data; (ii) the small time-step heuristics underlying the control of the local error can break down in some cases. In this paper an analysis is presented which incorporates these two difficulties.

External IDs:dblp:journals/na/Stuart97