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