Picture of virus under microscope

Research under the microscope...

The Strathprints institutional repository is a digital archive of University of Strathclyde research outputs.

Strathprints serves world leading Open Access research by the University of Strathclyde, including research by the Strathclyde Institute of Pharmacy and Biomedical Sciences (SIPBS), where research centres such as the Industrial Biotechnology Innovation Centre (IBioIC), the Cancer Research UK Formulation Unit, SeaBioTech and the Centre for Biophotonics are based.

Explore SIPBS research

The tolerance proportionality of adaptive ODE solvers

Higham, D.J. (1993) The tolerance proportionality of adaptive ODE solvers. Journal of Computational and Applied Mathematics, 45 (1-2). pp. 227-236. ISSN 0377-0427

Full text not available in this repository. (Request a copy from the Strathclyde author)

Abstract

Modern software for solving ordinary differential equation (ODE) initial-value problems requires the user to specify the ODE and choose a value for the error tolerance. The software can be thought of as a black box with a dial - turning the dial changes the accuracy and expense of the integration process. It is therefore of interest to know how the global error varies with the error tolerance. In this work, we look at explicit Runge-Kutta methods and show that with any standard error control method, and ignoring higher-order terms, the global error in the numerical solution behaves like a known rational power of the error tolerance. This generalises earlier work of Stetter, who found sufficient conditions for the global error to be linear in the tolerance. We also display the order of the next-highest term. We then analyse continuous Runge-Kutta schemes, and show what order of interpolation is necessary and sufficient for the continuous approximation to inherit the tolerance proportionality of the discrete formula. Finally we extend the results to the case of ODE systems with constant delays, thereby generalising some previous results of the author.