$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Discretizing the variable coefficient (1)
The principal idea is to first discretize the outer derivative.
Define
$$ \phi = q(x)
\frac{\partial u}{\partial x}
$$
Then use a centered derivative around \( x=x_i \) for the derivative of \( \phi \):
$$
\left[\frac{\partial\phi}{\partial x}\right]^n_i
\approx \frac{\phi_{i+\half} - \phi_{i-\half}}{\Delta x}
= [D_x\phi]^n_i
$$
