Theta method dynamics
Graeme J. Barclay, David F. Griffiths and Desmond J. Higham
Abstract: 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.
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