$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Step 1: Discretizing the domain
Mesh in time:
$$
\begin{equation}
0 = t_0 < t_1 < t_2 < \cdots < t_{N_t-1} < t_{N_t} = T \end{equation}
$$
Mesh in space:
$$
\begin{equation}
0 = x_0 < x_1 < x_2 < \cdots < x_{N_x-1} < x_{N_x} = L \end{equation}
$$
Uniform mesh with constant mesh spacings \( \Delta t \) and \( \Delta x \):
$$
\begin{equation}
x_i = i\Delta x,\ i=0,\ldots,N_x,\quad
t_i = n\Delta t,\ n=0,\ldots,N_t
\end{equation}
$$
