The tolerance proportionality of adaptive ODE solvers

D.J. Higham

Research output: Contribution to journalArticle

22 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)227-236
Number of pages9
JournalJournal of Computational and Applied Mathematics
Volume45
Issue number1-2
DOIs
Publication statusPublished - 8 Apr 1993

Keywords

  • global error
  • interpolation
  • tolerance proportionality
  • delay ordinary differential equations
  • differential equations

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