The Dynamics of the Theta Method

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 $\theta = \frac{1}{2}$ does not generate spurious solutions of period 2 in the timestep n. Using bifurcation theory, it is shown that for $\theta \ne \frac{1}{2}$ the theta method does generate spurious solutions of period 2. The existence and form of spurious solutions are examined in the limit $\Delta t \to 0$. 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 $\Delta t/\Delta x^2 \leq \frac{1}{2}$ for the Euler method applied to the heat equation is generalised to cope with a nonlinear problem. This naturally introduces a restriction on $\Delta t$ in terms of the initial data; this restriction is necessary to avoid the effect of spurious periodic solutions.