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Theta Method Dynamics

Published online by Cambridge University Press:  01 February 2010

Graeme J. Barclay ,
David F. Griffiths  and
Desmond J. Higham
Graeme J. Barclay
Affiliation:
Department of Mathematics, University of StrathclydeGlasgow, G1 1XH, ra.gbar@maths.strath.ac.uk
David F. Griffiths
Affiliation:
Department of Mathematics, University of Dundee, Dundee, DD1 4NH, dfg@mcs.dundee.ac.uk, http://www.mcs.dundee.ac.uk:8080/dfg/homepage.html
Desmond J. Higham
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow, G1 1XH, djh@maths.strath.ac.uk, http://www.maths.strath.ac.uk/aas96106/

Abstract

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Long-term solutions of the theta method applied to scalar nonlinear differential equations are studied in this paper. In the case where the equation has a stable steady state, lower bounds on the basin of non-oscillatory, monotonic attraction for the theta method are derived. Spurious period two solutions are then analysed. Under mild assumptions, precise results are obtained concerning the generic nature and stability of these solutions for small timesteps. Particular problem classes are studied, and direct connections are made between the existence and stability of period two solutions and the dynamics of the theta method. The analysis is extended to a wide class of semi-discretized partial differential equations. Numerical examples are given.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 3 , 2000 , pp. 27 - 43
Copyright
Copyright © London Mathematical Society 2000

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