$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Boundary conditions
- At \( i=N_x \): \( u_i=0 \).
- At \( i=0 \): \( \dfc(u)u^{\prime}=C \)
$$ [\dfc(u)D_{2x}u = C]_0$$
$$
\dfc(u_0)\frac{u_{1} - u_{-1}}{2\Delta x} = C
$$
The fictitious value \( u_{-1} \) can, as usual, be eliminated with the aid
of the scheme at \( i=0 \)
