Editing AxisymmetricConfigurations/PoissonEq (section)

==Overview==

The set of [[PGE#Principal_Governing_Equations|Principal Governing Equations]] that serves as the foundation of our study of the structure, stability, and dynamical evolution of self-gravitating fluids contains an equation of motion (the ''Euler'' equation) that includes an acceleration due to local gradients in the (Newtonian) gravitational potential, <math>\Phi</math>.  As has been pointed out in an [[PGE/PoissonOrigin#Origin_of_the_Poisson_Equation|accompanying chapter that discusses the origin of the Poisson equation]], the mathematical definition of this acceleration is fundamentally drawn from Isaac Newton's inverse-square law of gravitation, but takes into account that our fluid systems are not ensembles of point-mass sources but, rather, are represented by a continuous ''distribution'' of mass via the function, <math>\rho(\vec{x},t)</math>.  As indicated, in our study, <math>\rho</math> may depend on time as well as space.  The acceleration felt at any point in space may be obtained by integrating over the accelerations exerted by each differential mass element.  Alternatively &#8212; and more commonly &#8212; as has been explicitly demonstrated in, respectively, [[PGE/PoissonOrigin#Step_1|Step 1]] and [[PGE/PoissonOrigin#Step_3|Step 3]] of the same accompanying chapter, at any point in time the spatial variation of the gravitational potential, <math>\Phi(\vec{x})</math>, is determined from <math>~\rho(\vec{x})</math> via either an ''integral'' or a ''differential'' equation as follows:


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<table border="1" cellpadding="8" align="center" width="80%">
<tr><th align="center" colspan="2"><font size="+0">Table 1: &nbsp;Poisson Equation</font></th></tr>
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  <th align="center">Integral Representation</th>
  <th align="center">Differential Representation </th>
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<table border="0" align="center">
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  <td align="right">
<math>\Phi(\vec{x})</math>
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  <td align="center">
<math>=</math>
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  <td align="left">
<math>-G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
  </td>
</tr>
</table>
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  <td align="center">
{{ Math/EQ_Poisson01 }}
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While it is possible in some restricted situations to determine analytic expressions for the matched pair of functions, <math>\Phi </math> and <math>\rho</math>,  that satisfy the Poisson equation, modeling the vast majority of interesting astrophysical problems requires the develop of a numerical scheme to solve the Poisson equation.  In what follows, our aim is twofold:  (a) To recount &#8212; in a reasonable amount of detail &#8212; the steps that we have taken over the past, approximately forty years to develop more and more accurate and efficient ways to solve the Poisson equation in full three-dimensional generality; and (b) to list/summarize alternative techniques that have been successfully employed by other research groups over the years.  

<table border="0" align="center" cellpadding="5" width="100%">
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<td align="center">
  <table border="1"><tr><td align="center" bgcolor="red">&nbsp; &nbsp; &nbsp;</td></tr></table>
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<td align="left" colspan="2">Dimensionality &hellip;</td>
<td align="left">2D &nbsp; or &nbsp; 3D</td>
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<tr>
<td align="center">
  <table border="1"><tr><td align="center" bgcolor="lightgreen">&nbsp; &nbsp; &nbsp;</td></tr></table>
</td>
<td align="left" colspan="2">Computational Mesh Used for ''Differential Representation'' &hellip;</td>
<td align="left">[Car]tesian &nbsp; [Cyl]indrical &nbsp; [Sph]erical &nbsp; Gridless ([Lag]rangian)</td>
</tr>

<tr>
<td align="center">
  <table border="1"><tr><td align="center" bgcolor="yellow">&nbsp; &nbsp; &nbsp;</td></tr></table>
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<td align="left" colspan="2">Green's Function for ''Integral Representation'' &hellip;</td>
<td align="left">[Sph]erical &nbsp; [Tor]oidal &nbsp; Gridless ([Lag]rangian)</td>
</tr>
</table>

For each chosen problem, a research group must decide/specify, at a minimum: &nbsp; (1) The ''dimensionality'' of the problem; that is, whether the study will be restricted to 2D (e.g., axisymmetric) systems or whether the problem will be tackled in its full 3D complexity. (2) Whether the gravitational potential ''inside'' the mass distribution will be determined by solving the ''integral representation'' or the ''differential representation'' of the Poisson equation and, if the latter, in what coordinate frame (e.g., cylindrical or spherical) the differential operator and the computational mesh will be based.  (Note that even when the interior solution is obtained by evaluating the ''differential representation'' of the Poisson equation, the ''integral representation'' will likely be employed to evaluate the potential on a boundary that lies outside the mass distribution.)  (3) Which Green's function representation of the term, <math>~|\vec{x}^{~'} - \vec{x}|^{-1}</math>, will be used if/when the ''integral representation'' is evaluated.  As the numerical techniques employed by each research group are introduced, below, a small red/green/yellow boxed icon has been interlaced with the text in an effort to highlight, up front, which choices the group has made.