Computation of stability in this problem

$|A|\leq 1$ means $-1\leq A\leq 1$ $$\begin{equation} -1\leq\frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t} \leq 1 \tag{30} \end{equation}$$ $-1$ is the critical limit: $$\begin{align*} \Delta t &\leq \frac{2}{(1-2\theta)a},\quad \theta < \half\\ \Delta t &\geq \frac{2}{(1-2\theta)a},\quad \theta > {\half} \end{align*}$$

• Always fulfilled for Backward Euler and Crank-Nicolson
• $\Delta t \leq 2/a$ for Forward Euler