Operator notation; ODE

With $[D_tD_t u]^n$ as the finite difference approximation to $u^{\prime\prime}(t_n)$ we can write $$[D_tD_t u + \omega^2 u = 0]^n$$

$[D_tD_t u]^n$ means applying a central difference with step $\Delta t/2$ twice: $$[D_t(D_t u)]^n = \frac{[D_t u]^{n+\half} - [D_t u]^{n-\half}}{\Delta t}$$ which is written out as $$\frac{1}{\Delta t}\left(\frac{u^{n+1}-u^n}{\Delta t} - \frac{u^{n}-u^{n-1}}{\Delta t}\right) = \frac{u^{n+1}-2u^n + u^{n-1}}{\Delta t^2} \tp$$