$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Finite difference discretizations
The nonlinear term \( (\dfc(u)u^{\prime})^{\prime} \) behaves just
as a variable coefficient term \( (\dfc(x)u^{\prime})^{\prime} \) wrt
discretization:
$$ [-D_x\dfc D_x u +au = f]_i$$
Written out at internal points:
$$
\begin{align*}
-\frac{1}{\Delta x^2}
\left(\dfc_{i+\half}(u_{i+1}-u_i) -
\dfc_{i-\half}(u_{i}-u_{i-1})\right)
+ au_i &= f(u_i)
\end{align*}
$$
\( \dfc_{i+\half} \): two choices
$$
\begin{align*}
\dfc_{i+\half} &\approx
\dfc(\half(u_i + u_{i+1})) =
[\dfc(\overline{u}^x)]^{i+\half}
\\
\dfc_{i+\half} &\approx
\half(\dfc(u_i) + \dfc(u_{i+1})) = [\overline{\dfc(u)}^x]^{i+\half}
\end{align*}
$$
