$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
The linear system for a general \( N_x \)
$$
\begin{equation}
- F_o u^n_{i-1} + \left(1+ 2F_o \right) u^{n}_i - F_o u^n_{i+1} =
u_{i-1}^{n-1}
\tag{11}
\end{equation}
$$
for \( i=1,\ldots,Nx-1 \).
What are the unknowns in the linear system?
- either \( u^n_i \) for \( i=1,\ldots,N_x-1 \) (all internal spatial mesh points)
- or \( u^n_i \), \( i=0,\ldots,N_x \) (all spatial points)
The linear system in matrix notation:
$$
\begin{equation*} AU = b,\quad U=(u^n_0,\ldots,u^n_{N_x})
\end{equation*}
$$
