Strathprints Home | Open Access | Browse | Search | User area | Copyright | Help | Library Home | SUPrimo

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.

Item type: Article
ID code: 200
Keywords: global error, interpolation, tolerance proportionality, delay ordinary differential equations, differential equations, Mathematics
Subjects: Science > Mathematics
Department: Faculty of Science > Mathematics and Statistics
Related URLs:
    Depositing user: Ms Sarah Scott
    Date Deposited: 15 Mar 2006
    Last modified: 12 Mar 2012 10:35
    URI: http://strathprints.strath.ac.uk/id/eprint/200

    Actions (login required)

    View Item