Like in this question Finite difference derivatives for angle variables with jumps, I am trying to calculate numerically the derivative of an angle $\alpha \in [-\pi, \pi]$ :
$$ \omega = \frac{d\alpha}{dt}$$
I have been able to use the forward difference :
$$ \omega_t \simeq \frac{\Delta \alpha_{t+1, t} }{\Delta t}$$
With:
$$\Delta \alpha_{t+1, t} = \alpha_{t+1} - \alpha_{t} \mod 2\pi $$
Then, in order to express $\Delta \alpha_{t+1, t} \in [-\pi, \pi]$, if $\Delta \alpha_{t+1, t} > \pi$, I take $\Delta \alpha_{t+1, t} = \Delta \alpha_{t+1, t} - 2 \pi$.
Question: How can I calculate finite differences of higher orders?
For example, in the central difference you need to divide by 2: $$ x'(t) \simeq \frac{x_{t+1} - x_{t-1}}{2\Delta t}$$ How should I do this division when using an angle?
If I apply it before taking the modulus, two similar angles in different turns should give $0$ but for example $\alpha_{t+1} = \pi$ and $\alpha_{t-1} = -\pi$ would give $\frac{\alpha_{t+1} - \alpha_{t-1}}{2} = \pi$.
If I apply it after, I would be reducing the interval to $\Delta \alpha_{t+1, t} \in [0, \pi]$!
In particular, I would be interested in realizing the Richardson 4th point extrapolation: $$ x'(t) \simeq \frac{-x_{t+2} + 8 x_{t+1} - 8 x_{t-1} + x_{t-2}}{12\Delta t}$$