Study guide: Finite difference methods for vibration problems

$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)} $$

Exact discrete solution

$$ u^n = I\cos\left(\tilde\omega n\Delta t\right),\quad \tilde\omega = \frac{2}{\Delta t}\sin^{-1}\left(\frac{\omega\Delta t}{2}\right) $$

The error mesh function, $$ e^n = \uex(t_n) - u^n = I\cos\left(\omega n\Delta t\right) - I\cos\left(\tilde\omega n\Delta t\right) $$ is ideal for verification and further analysis! $$ e^n = I\cos\left(\omega n\Delta t\right) - I\cos\left(\tilde\omega n\Delta t\right) = -2I\sin\left(t\half\left( \omega - \tilde\omega\right)\right) \sin\left(t\half\left( \omega + \tilde\omega\right)\right) $$