$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Computing the first step
- The formula breaks down for \( u^1 \) because \( u^{-1} \) is unknown and
outside the mesh!
- And: we have not used the initial condition \( u^{\prime}(0)=0 \).
Discretize \( u^{\prime}(0)=0 \) by a centered difference
$$
\frac{u^1-u^{-1}}{2\Delta t} = 0\quad\Rightarrow\quad u^{-1} = u^1
$$
Inserted in the scheme for \( n=0 \) gives
$$
u^1 = u^0 - \half \Delta t^2 \omega^2 u^0
$$
