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Unified approach to spurious solutions introduced by time discretisation. Part I. Basic theory

Iserles, A. ; Peplow, A. T. LU orcid and Stuart, A. M. (1991) In SIAM Journal on Numerical Analysis 28(6). p.1723-1751
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... (More)

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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publication status
published
subject
in
SIAM Journal on Numerical Analysis
volume
28
issue
6
pages
29 pages
publisher
Society for Industrial and Applied Mathematics
external identifiers
  • scopus:0026402428
ISSN
0036-1429
DOI
10.1137/0728086
language
English
LU publication?
no
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b627994a-5c83-4e71-a250-03c73197ef3d
date added to LUP
2021-02-15 19:58:54
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2021-10-10 03:41:12
@article{b627994a-5c83-4e71-a250-03c73197ef3d,
  abstract     = {{<p>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)<sup>M</sup> 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.</p>}},
  author       = {{Iserles, A. and Peplow, A. T. and Stuart, A. M.}},
  issn         = {{0036-1429}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{1723--1751}},
  publisher    = {{Society for Industrial and Applied Mathematics}},
  series       = {{SIAM Journal on Numerical Analysis}},
  title        = {{Unified approach to spurious solutions introduced by time discretisation. Part I. Basic theory}},
  url          = {{http://dx.doi.org/10.1137/0728086}},
  doi          = {{10.1137/0728086}},
  volume       = {{28}},
  year         = {{1991}},
}