$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The Euler-Cromer method; idea
2x2 system for \( u^{\prime\prime}+\omega^2u=0 \):
$$
\begin{align*}
v^{\prime} &= -\omega^2u\\
u^{\prime} &= v
\end{align*}
$$
Forward-backward discretization:
- Update \( v \) with Forward Euler
- Update \( u \) with Backward Euler, using latest \( v \)
$$
\begin{align}
[D_t^+v &= -\omega^2u]^n\\
[D_t^-u &= v]^{n+1}
\end{align}
$$
