Theta method dynamics

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.