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Summary of the analysis
We can draw three important conclusions:
- The key parameter in the formulas is \( p=\omega\Delta t \).
- 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