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

Higham, D. J. and Stuart, A. M. (1998) Analysis of the dynamics of local error control via a piecewise continuous residual. BIT Numerical Mathematics, 38 (1). pp. 44-57. ISSN 0006-3835. https://resolver.caltech.edu/CaltechAUTHORS:20170613-110046810

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/BF02510916DOIArticle
https://link.springer.com/article/10.1007%2FBF02510916?LI=truePublisherArticle
http://rdcu.be/tryVPublisherFree ReadCube access
Additional Information:© 1998 Swets & Zeitlinger . Received July 1996. Revised July 1997. Communicated by Gustaf Söderlind. Supported by the Engineering and Physical Sciences Research Council under grants GR/H94634 and GR/K80228. Supported by the Office of Naval Research under grant N00014-92-J-1876 and by the National Science Foundation under grant DMS-9201727.
Funders:
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)GR/H94634
Engineering and Physical Sciences Research Council (EPSRC)GR/K80228
Office of Naval Research (ONR)N00014-92-J-1876
NSFDMS-9201727
Subject Keywords:Error control, continuous interpolants, dissipativity, contractivity, gradient systems.
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ37
Issue or Number:1
Classification Code:AMS subject classification: 34C35, 34D05, 65L07, 65L20, 65L50
Record Number:CaltechAUTHORS:20170613-110046810
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170613-110046810
Official Citation:Higham, D.J. & Stuart, A.M. Bit Numer Math (1998) 38: 44. doi:10.1007/BF02510916
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78160
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:13 Jun 2017 18:16
Last Modified:03 Oct 2019 18:06

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