$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
\( A \) is very sparse: a tridiagonal matrix
$$
\begin{equation}
A =
\left(
\begin{array}{cccccccccc}
A_{0,0} & A_{0,1} & 0
&\cdots &
\cdots & \cdots & \cdots &
\cdots & 0 \\
A_{1,0} & A_{1,1} & 0 & \ddots & & & & & \vdots \\
0 & A_{2,1} & A_{2,2} & A_{2,3} &
\ddots & & & & \vdots \\
\vdots & \ddots & & \ddots & \ddots & 0 & & & \vdots \\
\vdots & & \ddots & \ddots & \ddots & \ddots & \ddots & & \vdots \\
\vdots & & & 0 & A_{i,i-1} & A_{i,i} & A_{i,i+1} & \ddots & \vdots \\
\vdots & & & & \ddots & \ddots & \ddots &\ddots & 0 \\
\vdots & & & & &\ddots & \ddots &\ddots & A_{N_x-1,N_x} \\
0 &\cdots & \cdots &\cdots & \cdots & \cdots & 0 & A_{N_x,N_x-1} & A_{N_x,N_x}
\end{array}
\right)
\tag{12}
\end{equation}
$$
