Iserles, A. and Peplow, A. T. and Stuart, A. M. (1991) A Unified Approach to Spurious Solutions Introduced by Time Discretisation. Part I: Basic Theory. SIAM Journal on Numerical Analysis, 28 (6). pp. 1723-1751. ISSN 0036-1429. doi:10.1137/0728086. https://resolver.caltech.edu/CaltechAUTHORS:20170612-164247464
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Abstract
The asymptotic states of numerical methods for initial value problems are examined. In particular, spurious steady solutions, solutions with period 2 in the timestep, and spurious invariant curves are studied. A numerical method is considered as a dynamical system parameterised by the timestep h. It is shown that the three kinds of spurious solutions can bifurcate from genuine steady solutions of the numerical method (which are inherited from the differential equation) as h is varied. Conditions under which these bifurcations occur are derived for Runge–Kutta schemes, linear multistep methods, and a class of predictor-corrector methods in a PE(CE)^M implementation. The results are used to provide a unifying framework to various scattered results on spurious solutions which already exist in the literature. Furthermore, the implications for choice of numerical scheme are studied. In numerical simulation it is desirable to minimise the effect of spurious solutions. Classes of methods with desirable dynamical properties are described and evaluated.
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| Additional Information: | © 1991 SIAM. Received by the editors March 19, 1990; accepted for publication (in revised form) November 19, 1990. The research of this author was supported by the United Kingdom Science and Engineering Research Council. We thank J.M. Sanz-Serna for pointing out an error in an early version of Theorem 5.4 and for suggesting its resolution, Theorem 5.6. We also thank T. Eirola for helpful discussions and an anonymous referee for his careful reading of the manuscript. | |||||||||
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| Subject Keywords: | spurious solutions, bifurcation, timestepping, regularity | |||||||||
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| Issue or Number: | 6 | |||||||||
| Classification Code: | AMS(MOS) subject classifications. 65L05, 65L20 | |||||||||
| DOI: | 10.1137/0728086 | |||||||||
| Record Number: | CaltechAUTHORS:20170612-164247464 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20170612-164247464 | |||||||||
| Official Citation: | A Unified Approach to Spurious Solutions Introduced by Time Discretisation. Part I: Basic Theory A. Iserles, A. T. Peplow, and A. M. Stuart SIAM Journal on Numerical Analysis 1991 28:6, 1723-1751 | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 78133 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | George Porter | |||||||||
| Deposited On: | 13 Jun 2017 14:41 | |||||||||
| Last Modified: | 15 Nov 2021 17:37 |
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