Editing PGE/AStarScheme (section)

===Conservation of Angular Momentum===

Now, the first and third of these expressions are indeed the ones we are utilizing in our hydrocode.  but the middle one, expressing the time-rate-of-change of the azimuthal velocity, has been implemented in a different form, namely, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\partial_t(\rho \varpi v_\theta) + \nabla\cdot(\rho\varpi v_\theta \boldsymbol{v} )
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~ - \varpi \nabla_\theta P - \varpi \rho \nabla_\theta \Phi \, .
</math>
  </td>
</tr>
</table>
</div>

The $64,000 question is, "<font color="maroon">Are these equivalent expressions?</font>"  Well, let's play with the left-hand-side of this last expression.
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<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\partial_t(\rho \varpi v_\theta) + \nabla\cdot(\rho\varpi v_\theta \boldsymbol{v} )
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\varpi \partial_t(\rho v_\theta) + (\rho v_\theta) \partial_t(\varpi) + \varpi \nabla\cdot(\rho v_\theta \boldsymbol{v}) + (\rho v_\theta 
\boldsymbol{v}\cdot\nabla)\varpi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp; &nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\varpi \{ \partial_t(\rho v_\theta) +  \nabla\cdot(\rho v_\theta \boldsymbol{v}) \} + (\rho v_\theta v_\varpi)
</math>
  </td>
</tr>
</table>
</div>
It is easy to see, therefore, that the two expressions are equivalent; but the latter one is used in preference to the former because it provides a direct statement of conservation of angular momentum.  Specifically, when the external forces (due to gradients in the gravitational potential and pressure) balance, our selected "conservative" finite-difference scheme will guarantee that the physical quantity <math>~A = \rho\varpi v_\theta</math> is conserved globally to precisely the same degree of accuracy as mass is conserved.