User:Tohline/PGE/Hybrid Scheme - Revision history

Tohline: /* Step 1 */

2019-03-11T21:58:40Z

Step 1

← Older revision		Revision as of 21:58, 11 March 2019
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:NormanWilson78D.png\|700px\|center\|Norman & Wilson (1978)]]		<!-- [[File:NormanWilson78D.png\|700px\|center\|Norman & Wilson (1978)]] -->
	<!-- [[Image:AAAwaiting01.png\|~~400px~~\|center\|Norman & Wilson (1978)]] ~~-->~~		[[Image:AAAwaiting01.png\|600px\|center\|Norman & Wilson (1978)]]
	</td>		</td>
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Tohline: /* Step 1 */

2016-11-19T22:19:02Z

Step 1

← Older revision		Revision as of 22:19, 19 November 2016
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	<tr>		<tr>
	<td align="center">		<td align="center">
	~~<!--~~ [[File:NormanWilson78D.png\|~~650px~~\|center\|Norman & Wilson (1978)]] -->		[[File:NormanWilson78D.png\|700px\|center\|Norman & Wilson (1978)]]
	[[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]]		<!-- [[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]] -->
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	<td align="left">		<td align="left">
	[[File:NewTohline97C.png\|~~650px~~\|center\|New & Tohline (1997)]]		[[File:NewTohline97C.png\|700px\|center\|New & Tohline (1997)]]
	</td>		</td>
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Tohline: /* Angular Momentum Conservation */

2016-11-18T20:16:34Z

Angular Momentum Conservation

← Older revision		Revision as of 20:16, 18 November 2016
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	As foreshadowed above — see the [[##Recognizing_Statements_of_Conservation\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).		As foreshadowed above — see the [[#Recognizing_Statements_of_Conservation\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).

	Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,		Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,

Tohline: /* Angular Momentum Conservation */

2016-11-18T20:15:36Z

Angular Momentum Conservation

← Older revision		Revision as of 20:15, 18 November 2016
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	As foreshadowed above — see the [[#~~Recognizing_Statements_of_Conservatio~~\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).		As foreshadowed above — see the [[##Recognizing_Statements_of_Conservation\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).

	Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,		Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,

Tohline: /* Angular Momentum Conservation */

2016-11-18T20:14:55Z

Angular Momentum Conservation

← Older revision		Revision as of 20:14, 18 November 2016
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	As foreshadowed above — see the [[~~User:Tohline/PGE/Hybrid_Scheme~~#Recognizing_Statements_of_Conservatio\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).		As foreshadowed above — see the [[#Recognizing_Statements_of_Conservatio\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).

	Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,		Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,

Tohline: /* Step 1 */

2016-11-07T02:30:41Z

Step 1

← Older revision		Revision as of 02:30, 7 November 2016
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	</div>		</div>

	<div id="NormanWilson78">		<div align="center" id="NormanWilson78">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">
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	</div>		</div>

	<div id="NewTohline97">		<div align="center" id="NewTohline97">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">

Tohline: /* Step 1 */

2016-11-07T02:29:04Z

Step 1

← Older revision		Revision as of 02:29, 7 November 2016
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	</td></tr>		</td></tr>
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			</div>

	<div id="NormanWilson78">		<div id="NormanWilson78">
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			<div id="NewTohline97">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">

Tohline: /* Step 1 */ add span tag to NormanWilson78 equation image

2016-11-07T02:21:26Z

Step 1: add span tag to NormanWilson78 equation image

← Older revision		Revision as of 02:21, 7 November 2016
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	</table>		</table>

			<div id="NormanWilson78">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">
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	</tr>		</tr>
	</table>		</table>
			</div>

	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">

Tohline: /* Step 1 */

2015-07-12T00:17:07Z

Step 1

← Older revision		Revision as of 00:17, 12 July 2015
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	ApJ, vol. 224, pp. 497-511 © [http://aas.org/ American Astronomical Society]		ApJ, vol. 224, pp. 497-511 © [http://aas.org/ American Astronomical Society]
	</td></tr>		</td></tr>
			<tr>
			<td align="center">
			<!-- [[File:NormanWilson78D.png\|650px\|center\|Norman & Wilson (1978)]] -->
			[[Image:AAAwaiting01.png\|400px\|center\|Norman & Wilson (1978)]]
			</td>
			</tr>
	<tr>		<tr>
	<td align="left">		<td align="left">
	~~[[File:NormanWilson78D.png\|650px\|center\|Norman & Wilson (1978)]]~~		Note: For complete correspondence with equations derived herein, set <math>(\gamma-1)E \rightarrow P</math> in all three component equations.
	~~<!-- [[Image:AAAwaiting01.png\|650px\|center\|Norman & Wilson (1978)]] -->~~
	Note: For complete correspondence, set <math>(\gamma-1)E \rightarrow P</math> in all three component equations.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Step 1 */ Modified display of Norman & Wilson's equations as well as display of New & Tohline's equations

2015-07-11T23:57:50Z

Step 1: Modified display of Norman & Wilson's equations as well as display of New & Tohline's equations

← Older revision		Revision as of 23:57, 11 July 2015
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	</table>		</table>
			</td></tr>
			</table>


			<table border="1" cellpadding="5" width="80%">
			<tr><td align="center">
			Paragraph extracted without modification from p. 499 of [http://adsabs.harvard.edu/abs/1978ApJ...224..497N M. L. Norman & J. R. Wilson (1978)]<p></p>
			"''The Fragmentation of Isothermal Rings and Star Formation''"<p></p>
			ApJ, vol. 224, pp. 497-511 © [http://aas.org/ American Astronomical Society]
	</td></tr>		</td></tr>
	<tr>		<tr>
	<td align="left">		<td align="left">
	~~<!--~~ [[File:~~NormanWilson78C~~.png\|650px\|center\|Norman & Wilson (1978)]] -->		[[File:NormanWilson78D.png\|650px\|center\|Norman & Wilson (1978)]]
	[[Image:AAAwaiting01.png\|650px\|center\|Norman & Wilson (1978)]]		<!-- [[Image:AAAwaiting01.png\|650px\|center\|Norman & Wilson (1978)]] -->
	Note: For complete correspondence, set <math>(\gamma-1)E \rightarrow P</math> in all three component equations.		Note: For complete correspondence, set <math>(\gamma-1)E \rightarrow P</math> in all three component equations.
	</td>		</td>
	</tr>		</tr>
			</table>


			<table border="1" cellpadding="5" width="80%">
			<tr><td align="center">
			Equations extracted<sup>&dagger;</sup> from [http://adsabs.harvard.edu/abs/1997ApJ...490..311N K. C. B. New & J. E. Tohline (1997)]<p></p>
			"''The Relative Stability Against Merger of Close, Compact Binaries''"<p></p>
			ApJ, vol. 490, pp. 311-327 © [http://aas.org/ American Astronomical Society]
			</td></tr>
	<tr>		<tr>
	<td align="left">		<td align="left">
	[[File:~~NewTohline97B~~.png\|650px\|center\|New & Tohline (1997)]]		[[File:NewTohline97C.png\|650px\|center\|New & Tohline (1997)]]
	Note: When comparing this set of equations to the set presented by Norman & Wilson (1978), the definitions of the variables, <math>~\mathcal{S}</math> and <math>~\mathcal{T}</math>, must be swapped.
	</td>		</td>
	</tr>		</tr>
			<tr><td align="left">
			<sup>&dagger;</sup>Mathematical expressions displayed here, as a single digital image, with presentation order & layout modified from the original publication.<p></p>

			Note: When comparing this set of equations to the set presented by Norman & Wilson (1978), the definitions of the variables, <math>~\mathcal{S}</math> and <math>~\mathcal{T}</math>, must be swapped.
			</td></tr>
	</table>		</table>
	</div>		</div>


	[<font color="red">Comment by J. E. Tohline (April 7, 2014)</font>] This is the set of equations that my research group has been using to simulate a wide variety of astrophysical fluid flows over the past twenty years. This is no longer our method of choice, however. A numerical algorithm based on the hybrid scheme, as summarized below, is far preferable to an algorithm that is based on this more familiar, traditional set of equations for several reasons:		[<font color="red">Comment by J. E. Tohline (April 7, 2014)</font>] This is the set of equations that my research group has been using to simulate a wide variety of astrophysical fluid flows over the past twenty years. This is no longer our method of choice, however. A numerical algorithm based on the hybrid scheme, as summarized below, is far preferable to an algorithm that is based on this more familiar, traditional set of equations for several reasons: