Study guide: Finite difference methods for vibration problems

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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:

$u^{\prime\prime}(t_n) + \omega^2u(t_n) = 0,\quad n=1,\ldots,N_t$