Editing Appendix/Ramblings/51BiPolytropeStability/BetterInterface (section)

====Third Approximation====

Let's assume that we know the three quantities, <math>x_{J-1}, x_J</math>, and <math>(x_J)^' \equiv (dx/d\tilde{r})_J</math> and want to project forward to determine, <math>x_{J+1}</math>.  We should assume that, locally, the displacement function <math>x</math> is quadratic in <math>\tilde{r}</math>, that is,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a + b\tilde{r} + c\tilde{r}^2 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{dx}{d\tilde{r}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b + 2c\tilde{r} \, ,
</math>
  </td>
</tr>
</table>
where we have three unknowns, <math>a, b, c</math>.  These can be determined by appropriately combining the three relations,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>(x_J)^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b + 2c\tilde{r}_J \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>x_J</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a + b\tilde{r}_J + c\tilde{r}_J^2 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>x_{J-1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a + b(\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 \, .
</math>
  </td>
</tr>
</table>
The difference between the last two expressions gives,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_J - x_{J-1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ b\tilde{r}_J + c\tilde{r}_J^2 \biggr]
-
\biggl[ b(\tilde{r}_{J}-\Delta\tilde{r}) + c(\tilde{r}_{J}-\Delta\tilde{r})^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
b\Delta\tilde{r} 
+ 
c(2\tilde{r}_J \Delta\tilde{r} - \Delta\tilde{r}^2) \, .
</math>
  </td>
</tr>
</table>
Combining this with the first of the three expressions then gives,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\Delta\tilde{r} \biggl[(x_J)^' - 2c\tilde{r}_J  \biggr]
+ 
c(2\tilde{r}_J \Delta\tilde{r} - \Delta\tilde{r}^2) 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - x_{J-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
c(2\tilde{r}_J \Delta\tilde{r} - \Delta\tilde{r}^2) 
-c\biggl[ 2\tilde{r}_J \Delta\tilde{r}  \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
-c \Delta\tilde{r}^2 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
c  
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\Delta\tilde{r}^2}\biggl[
-x_J + x_{J-1} + (x_J)^'\Delta\tilde{r}
\biggr] 
\, .
</math>
  </td>
</tr>
</table>
Hence,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
b
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J)^' - 2c\tilde{r}_J 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(x_J)^' + \frac{2\tilde{r}_J}{\Delta \tilde{r}^2}\biggl[
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
\biggr] 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
a
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - b\tilde{r}_J -c\tilde{r}_J^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J
-\biggl\{ (x_J)^' + \frac{2\tilde{r}_J}{\Delta \tilde{r}^2}\biggl[
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
\biggr] \biggr\}\tilde{r}_J
- \biggl\{
\frac{1}{\Delta\tilde{r}^2}\biggl[
-x_J + x_{J-1} + (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}\tilde{r}_J^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - (x_J)^'\tilde{r}_J
-\biggl\{  \frac{2\tilde{r}_J^2}{\Delta \tilde{r}^2}\biggl[
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
\biggr] \biggr\}
+ \biggl\{
\frac{\tilde{r}_J^2}{\Delta\tilde{r}^2}\biggl[
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - (x_J)^'\tilde{r}_J
-\frac{\tilde{r}_J^2}{\Delta \tilde{r}^2}\biggl[
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
\biggr] 
\, .
</math>
  </td>
</tr>
</table>
As a result,

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
a 
\biggr\}
+ (\tilde{r}_J +\Delta\tilde{r}) \biggl\{
b
\biggr\}
+ (\tilde{r}_J+\Delta\tilde{r})^2 \biggl\{
c
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
x_J - (x_J)^'\tilde{r}_J
-\frac{\tilde{r}_J^2}{\Delta \tilde{r}^2}\biggl[
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}
+ (\tilde{r}_J +\Delta\tilde{r}) \biggl\{
(x_J)^' + \frac{2\tilde{r}_J}{\Delta \tilde{r}^2}\biggl[
x_J - x_{J-1} - (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ (\tilde{r}_J^2 + 2\tilde{r}_J \Delta\tilde{r} +\Delta\tilde{r}^2) \biggl\{
\frac{1}{\Delta\tilde{r}^2}\biggl[
-x_J + x_{J-1} + (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J - (x_J)^'\tilde{r}_J
+ \tilde{r}_J^2\biggl\{\frac{1}{\Delta \tilde{r}^2}\biggl[
-x_J + x_{J-1} + (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}
+ (x_J)^'(\tilde{r}_J +\Delta\tilde{r}) 

- 2\tilde{r}_J(\tilde{r}_J +\Delta\tilde{r}) \biggl\{
\frac{1}{\Delta \tilde{r}^2}\biggl[
-x_J + x_{J-1} + (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ (\tilde{r}_J^2 + 2\tilde{r}_J \Delta\tilde{r} +\Delta\tilde{r}^2) \biggl\{
\frac{1}{\Delta\tilde{r}^2}\biggl[
-x_J + x_{J-1} + (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J + (x_J)^'\Delta\tilde{r} 

+ \biggl[ \tilde{r}_J^2 - 2\tilde{r}_J(\tilde{r}_J +\Delta\tilde{r})+ (\tilde{r}_J^2 + 2\tilde{r}_J \Delta\tilde{r} +\Delta\tilde{r}^2)\biggr]
\biggl\{\frac{1}{\Delta \tilde{r}^2}\biggl[
-x_J + x_{J-1} + (x_J)^'\Delta\tilde{r}
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_J + (x_J)^'\Delta\tilde{r} 
+ 
\biggl[
-x_J + x_{J-1} + (x_J)^'\Delta\tilde{r}
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1} + 2(x_J)^'\Delta\tilde{r} \, .
</math>
  </td>
</tr>
</table>
<font color="red"><b>GOOD!</b></font>  This is the same as our [[#1stapprox|first approximation expression]] stated above.


<table border=1 align="center" cellpadding="10" width="80%"><tr><td bgcolor="lightgreen" align="left">
This is test ...
<table border="1" align="center" cellpadding="5">

  <tr>
<td align="center" bgcolor="white"><math>\Delta\tilde{r}</math></td>
<td align="center" bgcolor="white"><math>x_J</math></td>
<td align="center" bgcolor="white"><math>x_{J-1}</math></td>
<td align="center" bgcolor="white"><math>(x_J)^'</math></td>
<td align="center" bgcolor="white"><math>(x_{J-1})^'</math></td>
  </tr>

  <tr>
<td align="center" bgcolor="white">0.001936393</td>
<td align="center" bgcolor="white">-4.695376</td>
<td align="center" bgcolor="white">-4.547832</td>
<td align="center" bgcolor="white">-116.0119</td>
<td align="center" bgcolor="white">-76.19513</td>
  </tr>
</table>
<tr><td bgcolor="white" align="center">

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_{J+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{J-1} + 2(x_J)^'\Delta\tilde{r}
=
-4.997121
\, .
</math>
  </td>
</tr>
</table>
</td></tr>
</td></tr></table>