Study guide: Finite difference methods for vibration problems

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The computational algorithm

\( u^0=I \)

compute \( u^1 \)

for \( n=1,2,\ldots,N_t-1 \):

compute \( u^{n+1} \)

More precisly expressed in Python:

t = linspace(0, T, Nt+1)  # mesh points in time
dt = t[1] - t[0]          # constant time step.
u = zeros(Nt+1)           # solution

u[0] = I
u[1] = u[0] - 0.5*dt**2*w**2*u[0]
for n in range(1, Nt):
    u[n+1] = 2*u[n] - u[n-1] - dt**2*w**2*u[n]

Note: w is consistently used for \( \omega \) in my code.