$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
The stencil for the first time level
- Problem: the stencil for \( n=1 \) involves \( u^{-1}_i \), but time
\( t=-\Delta t \) is outside the mesh
- Remedy: use the initial condition \( u_t=0 \) together with the
stencil to eliminate \( u^{-1}_i \)
Initial condition:
$$ [D_{2t}u=0]^0_i\quad\Rightarrow\quad u^{-1}_i=u^1_i$$
Insert in stencil \( [D_tD_tu = c^2D_xD_x]^0_i \) to get
$$
\begin{equation}
u_i^1 = u^0_i - \half
C^2\left(u^{n}_{i+1}-2u^{n}_{i} + u^{n}_{i-1}\right)
\tag{10}
\end{equation}
$$
