Appendix/Ramblings/51BiPolytropeStability/BetterInterface - Revision history

Jet53man: /* At the Surface */

2023-09-27T20:04:27Z

At the Surface

← Older revision		Revision as of 16:04, 27 September 2023
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	</tr>		</tr>
	</table>		</table>

			This result should be compared with our [[SSC/Stability/BiPolytropes#Eigenfunction_Details\|separate discussion of ''eigenfunction details'']].

	==Discretize for Numerical Integration==		==Discretize for Numerical Integration==

Jet53man: /* Part 2 */

2023-09-12T12:31:48Z

Part 2

← Older revision		Revision as of 08:31, 12 September 2023
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	=Part 2=		=Part 2=

	For a continuation (part 2) of this discussion, go [[Ramblings/51BiPolytropeStability/BetterInterfacePt2\|here]].		For a continuation (part 2) of this discussion, go [[Appendix/Ramblings/51BiPolytropeStability/BetterInterfacePt2\|here]].

	=See Also=		=See Also=


	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man at 12:31, 12 September 2023

2023-09-12T12:31:01Z

Show changes

Jet53man: /* Fourth Approximation */

2023-09-12T12:06:28Z

Fourth Approximation

@@ Line 5,315: / Line 5,315: @@
 </td></tr>
 </td></tr></table>
 ===Interface===

Jet53man: /* Fourth Approximation */

2023-09-11T22:44:37Z

Fourth Approximation

← Older revision		Revision as of 18:44, 11 September 2023
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	</tr>		</tr>
	</table>		</table>
	Higher precision value (from Excel) is <math>x_J = -4.695376 \, ,</math> which precisely matches the input value.		Higher precision value (from Excel) is <math>x_J = -4.695376 \, ,</math> which precisely matches the input value. Also from Excel, <math>x_{J-1} = -4.547832</math> and <math>x_{J+1} = -3.803455 \, .</math>
	</td></tr></table>		</td></tr></table>

Jet53man: /* Fourth Approximation */

2023-09-11T22:31:01Z

Fourth Approximation

← Older revision		Revision as of 18:31, 11 September 2023
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	</tr>		</tr>
	</table>		</table>
	<tr><td bgcolor="white" align="~~center~~">		</td></tr>
			<tr><td bgcolor="white" align="left">

	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">
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	<td align="left">		<td align="left">
	<math>		<math>
	-3.36955 -2.76645 - e\Delta\tilde{r}^3(608.9698) = -53.~~08776~~		-3.36955 -2.76645 - e\Delta\tilde{r}^3(608.9698) = -232.7874
	\, ,		\, ,
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>		<math>
	0.5067329 + e\Delta\tilde{r}^3 (215.7856) = ~~-16~~.~~13040~~		0.5067329 + e\Delta\tilde{r}^3 (215.7856) = +80.819698
	\, ,		\, ,
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>		<math>
	-0.0385505 + e\Delta\tilde{r}^3 ( -25.45794 ) = 1.~~924265~~		-0.0385505 + e\Delta\tilde{r}^3 ( -25.45794 ) = -9.51370
	\, ,		\, ,
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>		<math>
	-0.~~0771003~~		0.3721883
	\, .		\, .
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>		<math>
	-53.~~08776~~ - ~~144~~.~~9475~~ + ~~155~~.~~38026~~ - 55.~~943877 = -98~~.~~5988~~		-232.7874 +726.2442 -768.2108 +270.0593 = -4.68369 \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			Higher precision value (from Excel) is <math>x_J = -4.695376 \, ,</math> which precisely matches the input value.
	<~~/td~~></tr>
	</td></tr></table>		</td></tr></table>

Jet53man: /* Fourth Approximation */

2023-09-11T21:24:22Z

Fourth Approximation

@@ Line 4,778: / Line 4,778: @@
    <td align="left">
 <math>
-x_J  -\frac{\tilde{r}_J}{\Delta\tilde{r}} \biggl[x_J - x_{J-1}\biggr]
+-3.36955 -2.76645 - e\Delta\tilde{r}^3(608.9698) = -53.08776
 \, ,
 </math>
    </td>
@@ Line 4,826: / Line 4,818: @@
 </tr>
 </table>
 </td></tr>
 </td></tr></table>

Jet53man: /* Fourth Approximation */

2023-09-11T21:02:07Z

Fourth Approximation

← Older revision		Revision as of 17:02, 11 September 2023
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	</table>		</table>

			</td></tr></table>

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

			<tr>
			<td align="center" bgcolor="white"><math>\tilde{r}_J = \tilde{r}_i + \Delta\tilde{r}</math></td>
			<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">0.01740039</td>
			<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>a</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			x_J -\frac{\tilde{r}_J}{\Delta\tilde{r}} \biggl[x_J - x_{J-1}\biggr]
			-
			\frac{1}{2\Delta\tilde{r}}\biggl[ (x_J)^' - (x_{J-1})^'\biggr]\biggl[ \tilde{r}_J \Delta\tilde{r} - \tilde{r}_J^2\biggr]

			- e\Delta\tilde{r}^3\biggl\{
			\frac{1}{2\Delta\tilde{r}^2} \biggl[
			\tilde{r}_J\Delta\tilde{r}
			-3\tilde{r}_J^2 \biggr]
			+ \biggl(\frac{\tilde{r}_J}{\Delta\tilde{r}}\biggr)^3
			\biggr\} \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>b \Delta\tilde{r}</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			0.5067329 + e\Delta\tilde{r}^3 (215.7856) = -16.13040
			\, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>c\Delta\tilde{r}^2</math>
			</td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			-0.0385505 + e\Delta\tilde{r}^3 ( -25.45794 ) = 1.924265
			\, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>e \Delta\tilde{r}^3 </math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			-0.0771003
			\, .
			</math>
			</td>
			</tr>
			</table>
			</td></tr>
	</td></tr></table>		</td></tr></table>

Jet53man: /* Third Approximation */

2023-09-11T18:18:42Z

Third Approximation

@@ Line 4,005: / Line 4,005: @@
 </table>
 <font color="red"><b>GOOD!</b></font>  This is the same as our [[#1stapprox|first approximation expression]] stated above.
 ====Fourth Approximation====

Jet53man: /* Fourth Approximation */

2023-09-11T18:15:25Z

Fourth Approximation

← Older revision		Revision as of 14:15, 11 September 2023
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	<tr>		<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</math></td>
	<td align="center" bgcolor="white"><math>x_{J-1}</math></td>		<td align="center" bgcolor="white"><math>x_{J-1}</math></td>
Line 5,142:		Line 5,143:

	<tr>		<tr>
	<td align="center" bgcolor="white"><~~math~~>~~x_J~~<~~/math~~></td>		<td align="center" bgcolor="white">0.001936393</td>
	<td align="center" bgcolor="white">~~<math>x_{J~~-~~1}</math>~~</td>		<td align="center" bgcolor="white">-4.695376</td>
	<td align="center" bgcolor="white">~~<math>(x_J)^'</math>~~</td>		<td align="center" bgcolor="white">-4.547832</td>
	<td align="center" bgcolor="white">~~<math>(x_{J~~-~~1})^'</math>~~</td>		<td align="center" bgcolor="white">-116.0119</td>
			<td align="center" bgcolor="white">-76.19513</td>
	</tr>		</tr>
	</table>		</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>
			\biggl[5x_{J-1} - 4x_J \biggr]
			+
			\biggl[ 4(x_J)^' + 2 (x_{J-1})^'\biggr]\Delta\tilde{r}
			=
			-5.15132
			\, .
			</math>
			</td>
			</tr>
			</table>
			</td></tr>
	</td></tr></table>		</td></tr></table>