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I am studying this paper on black holes quasi-normal modes, in particular I am reading section III-E (page 10). It deals with the numerical integration of the equation:

$$4\dfrac{\partial^2}{\partial u\partial v} \Phi(u,v) +V(u,v) \Phi(u,v)=0$$

it proposes to use the following formula:

$$\Phi(N)=\Phi(W)+\Phi(E)-\Phi(S)-\dfrac{h^2}{8} V(S)\left[ \Phi(W)+\Phi(E) \right]$$

to calulate the function $\Phi$, where the points W, E, N and S are given by: $S=(u.v)$, $W=(u+h,v)$, $E=(u,v+h)$ and $N=(u+h,v+h)$.

I do not understand this method: it seems to me that I need to know the form of the function $\Phi$ in advance in order to calculate it in the various points, but as far as I understand the function $\Phi$ should be the result of the calculations.

Where am I wrong?

mattiav27
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  • You start with some initial conditions (say $\Phi(0,v) = f(v)$ and $\Phi(u,0) = g(u)$). From there it works like a proof by induction. I already know $\Phi(0,0)$ and $\Phi(h,0)$ and $\Phi(0,h)$, so I can calculate $\Phi(h,h)$ according to the equation. Now I know $\Phi(h,h)$ so I can calculate $\Phi(2h, h)$ using what I know, etc. – Charles Hudgins Jul 08 '20 at 15:09
  • @CharlesHudgins How do I chose the initial function? Does it influence the result of the integration? – mattiav27 Jul 08 '20 at 16:50
  • Usually you don't choose the initial function. It's given to you. The initial function absolutely will influence the results of integration. – Charles Hudgins Jul 08 '20 at 17:03

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