$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Summary of the analysis
We can draw three important conclusions:
- The key parameter in the formulas is \( p=\omega\Delta t \) (dimensionless)
- Period of oscillations: \( P=2\pi/\omega \)
- Number of time steps per period: \( N_P=P/\Delta t \)
- \( \Rightarrow\ p=\omega\Delta t = 2\pi/ N_P \sim 1/N_P \)
- The smallest possible \( N_P \) is 2 \( \Rightarrow \) $p\in (0,\pi]$
- For \( p\leq 2 \) the amplitude of \( u^n \) is constant (stable solution)
- \( u^n \) has a relative phase error
\( \tilde\omega/\omega \approx 1 + \frac{1}{24}p^2 \), making numerical
peaks occur too early
