Study guide: Finite difference methods for vibration problems

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Solving for the numerical frequency

The scheme with $ u^n=I\exp{(i\omega\tilde\Delta t\, n)} $ inserted gives $$ -I\exp{(i\tilde\omega t)}\frac{4}{\Delta t^2}\sin^2(\frac{\tilde\omega\Delta t}{2}) + \omega^2 I\exp{(i\tilde\omega t)} = 0 $$ which after dividing by $ I\exp{(i\tilde\omega t)} $ results in $$ \frac{4}{\Delta t^2}\sin^2(\frac{\tilde\omega\Delta t}{2}) = \omega^2 $$ Solve for $ \tilde\omega $: $$ \tilde\omega = \pm \frac{2}{\Delta t}\sin^{-1}\left(\frac{\omega\Delta t}{2}\right) $$

Phase error because $ \tilde\omega \neq \omega $.

Note: dimensionless number $ p=\omega\Delta t $ is the key parameter
(i.e., no of time intervals per period is important, not $ \Delta t $ itself)

But how good is the approximation $ \tilde\omega $ to $ \omega $?