$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Observations from the figures
- Forward Euler has growing amplitude and outward \( (u,v) \) spiral - pumps
energy into the system.
- Backward Euler is opposite: decreasing amplitude, inward sprial,
extracts energy.
- Forward and Backward Euler are useless for vibrations.
- Crank-Nicolson (MidpointImplicit) looks much better.
