A centered finite difference scheme; step 3

Step 3: Approximate derivative(s) by finite difference approximation(s). Very common (standard!) formula for $u^{\prime\prime}$: $$u^{\prime\prime}(t_n) \approx \frac{u^{n+1}-2u^n + u^{n-1}}{\Delta t^2}$$

Use this discrete initial condition together with the ODE at $t=0$ to eliminate $u^{-1}$: $$\frac{u^{n+1}-2u^n + u^{n-1}}{\Delta t^2} = -\omega^2 u^n$$