Error control for initial value problems with discontinuities and delays

D.J. Higham

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

When using software for ordinary differential equation (ODE) initial value problems, it is not unreasonable to expect the global error to decrease linearly with the user-supplied error tolerance. For standard ODEs, conditions on an algorithm that guarantee such 'tolerance proportionality' asymptotically (as the error tolerance tends to zero) were derived by Stetter. Here we extend the analysis to cover a certain class of ODEs with low-order derivative discontinuities, and the class of ODEs with constant delays. We show that standard error control techniques will be successful if discontinuities are handled correctly and delay terms are calculated with sufficiently accurate interpolants. It is perhaps surprising that several delay ODE algorithms that have been proposed do not use sufficiently accurate interpolants to guarantee asymptotic proportionality. Our theoretical results are illustrated numerically.
LanguageEnglish
Pages315-330
Number of pages15
JournalApplied Numerical Mathematics
Volume12
Issue number4
DOIs
Publication statusPublished - Jun 1993

Fingerprint

Initial value problems
Error Control
Initial Value Problem
Tolerance
Discontinuity
Interpolants
Ordinary differential equation
Ordinary differential equations
Standard error
Delay Differential Equations
Linearly
Cover
Tend
Derivative
Decrease
Software
Zero
Term
Derivatives
Class

Keywords

  • Delay ordinary differential equations
  • discontinuity
  • global error
  • interpolation
  • local error
  • defect
  • residual
  • tolerance proportionality
  • Runge-Kutta
  • numerical mathematics

Cite this

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title = "Error control for initial value problems with discontinuities and delays",
abstract = "When using software for ordinary differential equation (ODE) initial value problems, it is not unreasonable to expect the global error to decrease linearly with the user-supplied error tolerance. For standard ODEs, conditions on an algorithm that guarantee such 'tolerance proportionality' asymptotically (as the error tolerance tends to zero) were derived by Stetter. Here we extend the analysis to cover a certain class of ODEs with low-order derivative discontinuities, and the class of ODEs with constant delays. We show that standard error control techniques will be successful if discontinuities are handled correctly and delay terms are calculated with sufficiently accurate interpolants. It is perhaps surprising that several delay ODE algorithms that have been proposed do not use sufficiently accurate interpolants to guarantee asymptotic proportionality. Our theoretical results are illustrated numerically.",
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Error control for initial value problems with discontinuities and delays. / Higham, D.J.

In: Applied Numerical Mathematics, Vol. 12, No. 4, 06.1993, p. 315-330.

Research output: Contribution to journalArticle

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