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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, [email protected]
David F. Griffiths
Affiliation:
Department of Mathematics, University of Dundee, Dundee, DD1 4NH, [email protected], http://www.mcs.dundee.ac.uk:8080/dfg/homepage.html
Desmond J. Higham
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow, G1 1XH, [email protected], 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

References

1. Aves, M.A., Griffiths, D.F. and Higham, D.J., ‘Runge–Kutta solutions of a hyperbolic conservation law with source term’, SIAM J. Sci. Stat. Comput. To appear.Google Scholar
2. Griffiths, D.F., Sweby, P.K. and Yee, H.C., ‘On spurious asymptotic numerical solutions of explicit Runge–Kutta methods’, IMA J. Numer. Anal. 12 (1992) 319–338.CrossRefGoogle Scholar
3. Hairer, E. and Wanner, G., Solving ordinary differential equations II; stiff and differential-algebraic problems, 2nd edn (Springer Verlag, 1996).Google Scholar
4. Humphries, A.R., ‘Spurious solutions of numerical methods for initial value problems’, IMA J. Numer. Anal.. 13 (1993) 263–290.Google Scholar
5. Iserles, A., ‘Stability and dynamics of numerical methods for nonlinear ordinary differential equations’, IMA J. Numer. Anal. 10(1990) 1–30.CrossRefGoogle Scholar
6. Lambert, J.D., Numerical methods for ordinary differential systems (John Wiley and Sons, Chichester, 1991).Google Scholar
7. Liniger, W, ‘Global accuracy and A-stability of one- and two-step integration formulae for stiff ordinary differential equations’, Proceedings, Conference on the Numerical Solution of Differential Equations, Lecture Notes in Mathematics 109 (Morris, J.LI., Springer-Verlag, 1969) 188–193.Google Scholar
8. Sanz-Serna, J.M. and Verwer, J.G., ‘A study of the recursion yn+1=yn+[yn]m'’, J. Math. Anal. Appl. 116 (1986) 456–464.CrossRefGoogle Scholar
9. Stuart, A.M. and Humphries, A.R., Dynamical systems and numerical analysis (Cambridge University Press, Cambridge, 1996).Google Scholar
10. Stuart, A.M. and Peplow, A.T., ‘The dynamics of the theta method’, SIAM J. Sci. Stat. Comput. 12 (1991) 1351–1372.CrossRefGoogle Scholar