$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
Damping of a discontinuity; Forward Euler scheme
Discrete model:
$$ [D_t^+ u = \dfc D_xD_x]^n_i $$
results in the explicit updating formula
$$ u^{n+1}_i = u^n_i + F\left(
u^{n}_{i+1} - 2u^n_i + u^n_{i-1}\right)
$$
