I Schwarzschild geodesics

Summary
Hi, I started reading this ![paper](https://github.com/Python-simulation/Black-hole-simulation-using-python/blob/master/Black%20hole%20simulation.pdf) dealing with the simulation of a Schwarzschild black hole in python, what I didn't understand is how did the author of the paper derive ##v_1 (0) = \frac{1}{D \cdot \tan \alpha}## from ##v_0 (0) = \frac{1}{D}##?
Can anyone here kindly provide an answer.
(Below are screenshots from the paper)
[Mentors' note: edited to fix the Latex]
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what I didn't understand is how did the author of the paper derive v1(0)=1D⋅tanα from v0(0)=1D?
I don't know, and I'm not even sure why those equations make sense. It looks like those equations are taken from reference [5], not derived in the paper itself.

What I don't understand is where the distance ##D## is coming from, since the radial coordinate ##r## in Schwarzschild coordinates does not equal radial distance, and since there is no spatial "center" to the black hole anyway (##r = 0## is not a point at the center, it's a spacelike like to the future of every event inside the horizon). So I'm not sure what the author of the paper thinks he is doing.
 
I watched the video he referred to, the video is in french by the way and doesn't provide any explanation to how the second equation has been derived, except mentioning that ##u' = \frac{1}{D \cdot tan \alpha}## is the derivative of ##u = \frac{1}{D}## when ##\phi = 0##, where ##D## is the distance between the black hole and the observer and ##\alpha## defining the initial direction of the photon.
I've been trying the whole day but with no success, now I'm confused.
 
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xcept mentioning that ##u' = \frac{1}{D \cdot \alpha}## is the derivative of ##u = \frac{1}{D}## when ##\phi = 0##, where ##D## is the distance between the black hole and the observer and ##\alpha## defining the initial direction of the photon.
Hm, ok, if he's using the standard method of rewriting the orbital equations, where ##u = 1 / r##, then I can see why he would say that the value of ##u## at time ##0## (which is what I think he means by ##v_0##, although his notation is very confusing) would be ##1 / D##--but that's still only correct in the approximation where we are very far from the hole, so "distance from the hole" can be taken as a good approximation of the radial coordinate ##r##. But if he means to include trajectories of photons close to the hole's horizon, then this approximation no longer works.

In general, I don't think this is a good source from which to try to learn how to do numerical simulations in GR.
 
If ##u = \frac{1}{D}## is correct as an approximation (the source he stated uses the Euler method to solve the light path around a the Schwarzschild spacetime) then how ##u' = \frac{1}{D \cdot tan \alpha}## (which is confusingly denoted as ##v_1(0)## in the paper) was derived from ##u = \frac{1}{D}## when ##\phi = 0##, do you have any idea please? and what books, articles or any other source would you suggest for me to start with numerical relativity if you don't mind me asking?
 
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If ##u = \frac{1}{D}## is correct as an approximation (the source he stated uses the Euler method to solve the light path around a the Schwarzschild spacetime) then how ##u' = \frac{1}{D \cdot tan \alpha}## (which is confusingly denoted as ##v_1(0)## in the paper) was derived from ##u = \frac{1}{D}## when ##\phi = 0##,
The ##\phi = 0## part is just because he has to choose some particular value of ##\phi## as the "initial" value (the word "initial" is a bit misleading since the trajectory can be extended into both the future and the past from this point).

I'm not sure how he's deriving ##u' = \frac{1}{D \cdot tan \alpha}## since the initial value of ##u## should be a position, not a velocity. Again, I don't think this is a good source to learn from.

what books, articles or any other source would you suggest for me to start with numerical relativity
Unfortunately I don't have any good references to suggest, since I have not studied numerical relativity. Possibly some other posters here might.
 
Thank you @PeterDonis for the valuable time you spent discussing the thread, really appreciate it.
 

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