Some recent advances in validated methods for IVPs for ODEs

Authors:
Kenneth R. Jackson;Nedialko S. Nedialkov
Affiliations:
Department of Computer Science, University of Toronto, Toronto, ON, M5S 3G4, Canada;Department of Computing and Software, McMaster University, Hamilton, ON, L8S 4L7, Canada
Venue:
Applied Numerical Mathematics
Year:
2002

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Abstract

Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced.We present a brief overview of interval Taylor series (ITS) methods for IVPs for ODEs and discuss some recent advances in the theory of validated methods for IVPs for ODEs. In particular, we discuss an interval Hermite-Obreschkoff (IHO) scheme for computing rigorous bounds on the solution of an IVP for an ODE, the stability of ITS and IHO methods, and a new perspective on the wrapping effect, where we interpret the problem of reducing the wrapping effect as one of finding a more stable scheme for advancing the solution.