$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
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
What is the consequence of complex \( \tilde\omega \)?
- 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
