$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Modified equation for the first time level
Centered difference for \( u_t(x,0) = V(x) \):
$$ [D_{2t}u = V]^0_i\quad\Rightarrow\quad u^{-1}_i = u^{1}_i - 2\Delta t V_i,$$
Inserting this in the stencil (17) for \( n=0 \) leads to
$$
\begin{equation}
u^{1}_i = u^0_i - \Delta t V_i + {\half}
C^2
\left(u^{n}_{i+1}-2u^{n}_{i} + u^{n}_{i-1}\right) + \half\Delta t^2 f^n_i
\tag{18}
\end{equation}
$$
