$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\dfc}{\alpha} % diffusion coefficient $$
Forward Euler applied to an initial plug profile
\( N_x=50 \). The method results in a growing, unstable solution if \( F>0.5 \).
Choosing \( F=0.5 \) gives a strange saw tooth-like curve.
Lowering \( F \) to 0.25 gives a smooth (expected) solution.
