$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Discretizing the variable coefficient (2)
Then discretize the inner operators:
$$
\phi_{i+\half} = q_{i+\half}
\left[\frac{\partial u}{\partial x}\right]^n_{i+\half}
\approx q_{i+\half} \frac{u^n_{i+1} - u^n_{i}}{\Delta x}
= [q D_x u]_{i+\half}^n
$$
Similarly,
$$
\phi_{i-\half} = q_{i-\half}
\left[\frac{\partial u}{\partial x}\right]^n_{i-\half}
\approx q_{i-\half} \frac{u^n_{i} - u^n_{i-1}}{\Delta x}
= [q D_x u]_{i-\half}^n
$$
