$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Discretization of derivatives at the boundary (2)
$$
\frac{u_{-1}^n - u_1^n}{2\Delta x} = 0
$$
- Problem: \( u_{-1}^n \) is outside the mesh (fictitious value)
- Remedy: use the stencil at the boundary to eliminate \( u_{-1}^n \);
just replace \( u_{-1}^n \) by \( u_{1}^n \)
$$
\begin{equation}
u^{n+1}_i = -u^{n-1}_i + 2u^n_i + 2C^2
\left(u^{n}_{i+1}-u^{n}_{i}\right),\quad i=0 \end{equation}
$$
