$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
Analysis of the Backward Euler scheme
$$
\begin{equation*} [D_t^- u = \dfc D_xD_x u]^n_q\end{equation*}
$$
$$ u^n_q = A^n e^{ikq\Delta x}$$
$$ A = (1 + 4F\sin^2p)^{-1} $$
$$ u^n_q = (1 + 4F\sin^2p)^{-n}e^{ikq\Delta x} $$
Stability:
We see that \( |A| < 1 \) for all \( \Delta t>0 \)
and that \( A>0 \) (no oscillations)
