Order of $\theta$-methods for IVP

The truncation error of the method is $$ \epsilon = \Delta t\left(\frac{1}{2} - \theta\right) y''(t_{n+1/2}) -\frac{\Delta t^2}{12} y'''(t_{n+1/2}) + O(\Delta t^3) $$ When $\theta$ approaches $\frac{1}{2}$ the linear term of the error tends to zero and finally is zero when $\theta = \frac{1}{2}$.

Let $\Delta t_0 = \left|(6-12\theta) \frac{y''(t_{n+1/2})}{y'''(t_{n+1/2})}\right|$. For $\Delta t \gg \Delta t_0$ the linear term is small and the method effectively behaves like a second order one, but when $\Delta t \ll \Delta t_0$ the linear term dominates and the method behaves like a first order one. Note the closer $\theta$ is to $1/2$ the smaller is $\Delta t_0$. Finally for $\theta = 1/2$, $\Delta t_0 = 0$.