Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I need to solve the Poisson equation in gravitational case (for galaxy dynamics) with Green's function by applying Fast Fourier Transform.

I don't understand the method used for an isolated system from (Hockney & Eastwood 1981); it says :

The convolution method described will solve a periodic system of sources with an arbitrary form of interaction. No conductors or boundaries are permitted in the system and, if these are present, the capacity matrix method must be used.

By an appropriate choice of Green's function the convolution method enables the potential to be found for an isolated source distribution.By isolated system, like a galaxy of stars, in which the only boundary condition is that the potential decays to zero correctly at infinity (i.e Φ∝r^{-1}).

The method, which we describe in the two dimensions, may be adapted to isolated systems if one is prepared to use only one-quarter of the available mesh points for the source distribution - say, the bottom left-hand corner, defined by 0≤p,q≤n/2. The source distributionover the remaining three-quarters of the system is made identically zero. Taking the interaction of point charges as an example, an interaction potential is constructed as follows :

[tex]G_{p,q}=(p^2+q^2)^{-1/2}\\ G_{n-p,q}}=G_{p,n-q}=G{n-p,n-q}=G_{p,q}\\ G_{0,0}=1\\ for 0≤p,q≤n/2 and p+q≠0[/tex]

When this potential is repeated periodically, one sees that the correct r. At the boundary of this region there is a cusp, and outside the region the potential is incorrect. However,^{-1}potential for a point charge at the origin is obtained within the region -n/2≤p,q≤n/2if we use only the bottom left-hand corner for the charge distributionand use only the potential in this region,the correct potential for an isolated system is obtained. The potential outside the bottom left-hand corner is incorrect, containing as it does not matter, since this potential is never used."I have 2 questions: "The Fourier analysis being more straightforwardly performed in a periodic system, we use the "zero padding trick" for a N x N grid :

Let's consider the 2D case :

1. I compute the Green's function in real space, in the N x N grid

2. I duplicate and mirror the Green's function in the 3 "ghost" quadrants, so that the Green's function is now periodic in the Ng = 2N x 2N grid (see below figure)

3. Set the density field to zero in the 3 ghosts quadrants. (see below figure)

4. Perform a forward FFT on the 2N x 2N grid : multiply the Fourier transform of Green's function by the Fourier transform of density field ( product wich is equal to the Fourier transform of the gravitational potential)

5. Perform a backward FFT of the Fourier potential to get the potential in space.

* Why we duplicate the Green function ? It seems like one wants to make it periodic but why ?

* Why do we "zero padding" on distribution charges ? Is it because the Green function is duplicated and stored on 2N points ?

Thanks in advance

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Poisson equation/zero padding and duplicating Green function

**Physics Forums | Science Articles, Homework Help, Discussion**