BIT Numerical Mathematics

, Volume 38, Issue 1, pp 44–57

Analysis of the dynamics of local error control via a piecewise continuous residual

  • D. J. Higham
  • A. M. Stuart
Article

Abstract

Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under theassumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for thematlab ode23 algorithm [10] when applied to a variety of problems.

Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation—dissipative, contractive and gradient systems are analysed in this way.

AMS subject classification

34C35 34D05 65L07 65L20 65L50 

Key words

Error control continuous interpolants dissipativity contractivity gradient systems 

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Copyright information

© Swets & Zeitlinger 1998

Authors and Affiliations

  • D. J. Higham
    • 1
  • A. M. Stuart
    • 2
  1. 1.Department of MathematicsUniversity of StrathclydeGlasgowScotland
  2. 2.Scientific Computing and Computational Mathematics Program Division of Mechanics and ComputationStanford UniversityStanfordUSA

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