$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Observations of RK and CN methods
- 4th-order Runge-Kutta is very accurate, also for large \( \Delta t \).
- 2th-order Runge-Kutta is almost as bad as Forward and Backward
Euler.
- Crank-Nicolson is accurate, but the amplitude is not as accurate
as the difference scheme for \( u^{\prime\prime}+\omega^2u=0 \).
