Editing AxisymmetricConfigurations/PoissonEq (section)

===Cook (1977)===

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<td align="center" bgcolor="red">&nbsp; 3D &nbsp;</td>
<td align="center" bgcolor="lightgreen">&nbsp; Cyl &nbsp;</td>
<td align="center" bgcolor="yellow">&nbsp; Cyl &nbsp;</td>
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Key results from [https://www.osti.gov/servlets/purl/7294335 T. L. Cook's (1977) doctoral dissertation] &#8212; titled, ''Three-Dimensional Dynamics of Protostellar Evolution'' &#8212; were published as [http://adsabs.harvard.edu/abs/1978ApJ...225.1005C T. L. Cook &amp; F. H. Harlow (1978, ApJ, 225, 1005 - 1020)]. Cook's three-dimensional hydrodynamic simulations were conducted on a cylindrical-coordinate mesh.  While few details of the technique used to solve the Poisson equation are provided in the ApJ article, &sect;II.B of [https://www.osti.gov/servlets/purl/7294335 Cook's (1977) dissertation] explains that values of the gravitational potential across the ''interior'' regions of the mesh were obtained by solving the ''differential representation'' of the Poisson equation, subject to the boundary conditions at each boundary point, calculated by "performing a numerical integration over all mass points" using the ''integral representation'' of the Poisson equation, namely,
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<math>~ \Phi_B(\varpi,\phi,z)</math>
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<math>~=</math>
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<math>~ 
-G \int \frac{1}{|\vec{x}^{~'} - \vec{x}|} ~\rho(\vec{x}^{~'}) d^3x^' \, ,
</math>
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with,
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<math>~\frac{1}{|\vec{x}^{~'} - \vec{x}|}</math>
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<math>~=</math>
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<math>~
\biggl[ \varpi^2 + (\varpi^')^2 - 2\varpi \varpi^' \cos|\phi - \phi^'| + (z - z^')^2 \biggr]^{- 1 / 2} \, .
</math>
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[https://www.osti.gov/servlets/purl/7294335 Cook (1977)], p. 15, Eq. (II-16)<br />
See also: [http://adsabs.harvard.edu/abs/2007AmJPh..75..724S Selvaggi, Salon &amp; Chari (2007)] &sect;II, eq. (2)
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(Note that the leading factor of the gravitational constant, <math>~G</math>, does not appear explicitly in Cook's equation II-16, although it should.)  Presumably the coordinate locations of the boundary cells, <math>~(\varpi,\phi,z)</math>, were in no case coincident with the coordinate locations of any of the ''interior'' grid cells, <math>~(\varpi^',\phi^',z^')</math>, so there was no danger that the integration would encounter a singularity as a consequence of the distance, <math>~|\vec{x}^{~'} - \vec{x}|</math>, being zero.