$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
Damping of a discontinuity; Backward Euler scheme
Discrete model:
$$ [D_t^- u = \dfc D_xD_x]^n_i $$
results in a (tridiagonal) linear system
$$
- F u^n_{i-1} + \left(1+ 2F \right) u^{n}_i - F u^n_{i+1} =
u_{i-1}^{n-1}
$$
where
$$ F = \dfc\frac{\Delta t}{\Delta x^2} $$
is the mesh Fourier number
