Computation of stability in this problem

$A < 0$ if $$\frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t} < 0$$ To avoid oscillatory solutions we must have $A> 0$ and $$\begin{equation} \Delta t < \frac{1}{(1-\theta)a}\ \end{equation}$$

• Always fulfilled for Backward Euler
• $\Delta t \leq 1/a$ for Forward Euler
• $\Delta t \leq 2/a$ for Crank-Nicolson