Study guide: Finite difference methods for vibration problems

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Polynomial approximation of the phase error

Taylor series expansion for small $ \Delta t $ gives a formula that is easier to understand:

>>> from sympy import *
>>> dt, w = symbols('dt w')
>>> w_tilde = asin(w*dt/2).series(dt, 0, 4)*2/dt
>>> print w_tilde
(dt*w + dt**3*w**3/24 + O(dt**4))/dt  # note the final "/dt"

$$ \tilde\omega = \omega\left( 1 + \frac{1}{24}\omega^2\Delta t^2\right) + {\cal O}(\Delta t^3) $$ The numerical frequency is too large (to fast oscillations).