Global error versus tolerance for explicit Runge-Kutta methods

Desmond J. Higham

Research output: Contribution to journalArticlepeer-review

30 Citations (Scopus)


Initial value solvers typically input a problem specification and an error tolerance, and output an approximate solution. Faced with this situation many users assume, or hope for, a linear relationship between the global error and the tolerance. In this paper we examine the potential for such 'tolerance proportionality' in existing explicit Runge-Kutta algorithms. We take account of recent developments in the derivation of high-order formulae, defect control strategies, and interpolants for continuous solution and first derivative approximations. Numerical examples are used to verify the theoretical predictions. The analysis draws on the work of Stetter, and the numerical testing makes use of the nonstiff DETEST package of Enright and Pryce.
Original languageEnglish
Pages (from-to)457-480
Number of pages23
JournalIMA Journal of Numerical Analysis
Issue number4
Publication statusPublished - Oct 1991


  • Runge-Kutta Methods
  • numerical mathematics
  • high-order formulae
  • DETEST package


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