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. doi:10.1007/BF02510916. 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.
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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. | ||||||||||||
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Subject Keywords: | Error control, continuous interpolants, dissipativity, contractivity, gradient systems. | ||||||||||||
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Issue or Number: | 1 | ||||||||||||
Classification Code: | AMS subject classification: 34C35, 34D05, 65L07, 65L20, 65L50 | ||||||||||||
DOI: | 10.1007/BF02510916 | ||||||||||||
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: | 15 Nov 2021 17:37 |
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