$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
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
