$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Euler-Cromer is equivalent to the scheme for \( u^{\prime\prime}+\omega^2u=0 \)
- Forward Euler and Backward Euler have error \( \Oof{\Delta t} \)
- What about the overall scheme? Expect \( \Oof{\Delta t} \)...
We can eliminate \( v^n \) and \( v^{n+1} \), resulting in
$$
u^{n+1} = 2u^n - u^{n-1} - \Delta t^2 \omega^2u^{n}
$$
which is the centered finite differrence scheme for \( u^{\prime\prime}+\omega^2u=0 \)!
