$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
A centered finite difference scheme; step 1 and 2
- Strategy: follow the four steps of the finite difference method.
- Step 1: Introduce a time mesh, here uniform on \( [0,T] \): \( t_n=n\Delta t \)
- Step 2: Let the ODE be satisfied at each mesh point:
$$
u^{\prime\prime}(t_n) + \omega^2u(t_n) = 0,\quad n=1,\ldots,N_t
$$
