$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Experimental investigation of oscillatory solutions
The solution is oscillatory if
$$ u^{n} > u^{n-1}$$
Seems that \( a\Delta t < 1 \) for FE and 2 for CN.