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 $\omega$?