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The Dynamics of the Theta Method

Stuart, A. M. and Peplow, A. T. (1991) The Dynamics of the Theta Method. SIAM Journal on Scientific and Statistical Computing, 12 (6). pp. 1351-1372. ISSN 0196-5204. https://resolver.caltech.edu/CaltechAUTHORS:20170613-092411747

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Abstract

The dynamics of the theta method for arbitrary systems of nonlinear ordinary differential equations are analysed. Two scalar examples are presented to demonstrate the importance of spurious solutions in determining the dynamics of discretisations. A general system of differential equations is then considered. It is shown that the choice θ = ½ does not generate spurious solutions of period 2 in the timestep n. Using bifurcation theory, it is shown that for θ ≠ ½ the theta method does generate spurious solutions of period 2. The existence and form of spurious solutions are examined in the limit △t ⟶ 0. The existence of spurious steady solutions in a predictor-corrector method is proved to be equivalent to the existence of spurious period 2 solutions in the Euler method. The theory is applied to several examples from nonlinear parabolic equations. Numerical continuation is used to trace out the spurious solutions as Lit is varied. Timestepping experiments are presented to demonstrate the effect of the spurious solutions on the dynamics and some complementary theoretical results are proved. In particular, the linear stability restriction △t/△ x^2 ≤ ½ for the Euler method applied to the heat equation is generalised to cope with a nonlinear problem. This naturally introduces a restriction on △t in terms of the initial data; this restriction is necessary to avoid the effect of spurious periodic solutions.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1137/0912074DOIArticle
http://epubs.siam.org/doi/abs/10.1137/0912074PublisherArticle
Additional Information:© 1991 Society for Industrial and Applied Mathematics. Submitted: 06 December 1989. Accepted: 03 October 1990. The authors thank both referees, whose comments have improved the presentation. A.M. Stuart is grateful to A.R. Mitchell and C.M. Elliott for several helpful discussions.
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ15
Issue or Number:6
Record Number:CaltechAUTHORS:20170613-092411747
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170613-092411747
Official Citation:The Dynamics of the Theta Method A. M. Stuart and A. T. Peplow SIAM Journal on Scientific and Statistical Computing 1991 12:6, 1351-1372
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78155
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:13 Jun 2017 17:01
Last Modified:03 Oct 2019 18:05

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