Stability

Observations:

• Numerical solution has constant amplitude (desired!), but phase error
• Constant amplitude requires $\sin^{-1}(\omega\Delta t/2)$ to be real-valued $\Rightarrow |\omega\Delta t/2| \leq 1$
• $\sin^{-1}(x)$ is complex if $|x| > 1$, and then $\tilde\omega$ becomes complex

• Set $\tilde\omega = \tilde\omega_r + i\tilde\omega_i$
• Since $\sin^{-1}(x)$ has a *negative* imaginary part for $x>1$, $\exp{(i\omega\tilde t)}=\exp{(-\tilde\omega_i t)}\exp{(i\tilde\omega_r t)}$ leads to exponential growth $e^{-\tilde\omega_it}$ when $-\tilde\omega_i t > 0$
• This is instability because the qualitative behavior is wrong