Editing Appendix/Ramblings/HybridSchemeOld (section)

===Recognizing Statements of Conservation===
When dealing with the compressible fluid equations, we will often encounter hyperbolic PDEs of the following form:
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, ,
</math>
  </td>
</tr>
</table>
</div>
where we are using <math>~\vec{v}</math> to represent the velocity field of the fluid as viewed from an ''inertial frame of reference'', and the total (as opposed to partial) time derivative indicates the time-rate of change of <math>~\psi</math> is being measured in a so-called ''Lagrangian'' fashion, that is, at the location of some fluid element and ''moving along with'' that fluid element.  

When we encounter a situation in which the "source" term, <math>~S</math>, on the right-hand side is zero, we will be able to identify the scalar variable, <math>~\psi</math>, as the volume density of some conserved quantity.  For example, the continuity equation &#8212; which is a mathematical statement of mass conservation &#8212; has the form,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
{{ Math/EQ_Continuity01}}
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; or, equivalently, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">
<math>
\frac{d\ln\rho}{dt} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~- \nabla\cdot \vec{v} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~\rho</math> is the mass per unit volume or, simply, the mass density of the fluid element.  Clearly, when the mass of a Lagrangian fluid element is conserved, the fluid element's mass density changes only in accordance with the divergence of the local velocity field. 

Similarly, if we are following the evolution of a fluid that expands and contracts adiabatically, we should expect to encounter an equation of the form, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{ds}{dt} + s\nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
0 \, ,
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; or, equivalently, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">
<math>
\frac{d\ln s}{dt} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~- \nabla\cdot \vec{v} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~s</math> is the entropy density of a Lagrangian fluid element.  Or, if an axisymmetric distribution of fluid is moving in an axisymmetric potential, we should expect the azimuthal component of the fluid's angular momentum to be conserved, in which case we should expect to encounter a dynamical equation of the form,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d(\rho \varpi v_\phi)}{dt} + (\rho \varpi v_\phi) \nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
0 \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~\varpi</math> is the Lagrangian fluid element's (cylindrical radial) distance measured from the symmetry axis of the underlying potential and <math>~v_\phi = \varpi\dot\phi</math> is the azimuthal component of the inertial velocity field, <math>~\vec{v}</math>, at the location of the fluid element.