User:Tohline/PGE/AStarScheme - Revision history

Tohline: /* ASIDE: Relevant to hydrocode development */

2015-07-13T20:11:22Z

ASIDE: Relevant to hydrocode development

← Older revision		Revision as of 20:11, 13 July 2015
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	==ASIDE: Relevant to hydrocode development==		==ASIDE: Relevant to hydrocode development==
	===Curvature Terms===		===Curvature Terms===
	~~Coverting [''sic'']~~ from a Lagrangian to an Eulerian time-derivative, Equation [I.A.5] becomes,		Converting from a Lagrangian to an Eulerian time-derivative, Equation [I.A.5] becomes,
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	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">
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	</table>		</table>
	</div>		</div>
	Now, if you're not working in Cartesian coordinates, care must be taken when dealing with the second term on the left-hand-side of this ~~equaiton~~ because when the gradient operator acts on a vector quantity (in this case, <math>~\rho\boldsymbol{v}</math>), various curvature terms will arise reflecting the fact that, in general, the unit vectors of your curvilinear coordinate system point in different directions as the fluid moves to different locations in space. Quite generally, though, for the <math>~j^\mathrm{th}</math> component of the equation of motion we may isolate these curvature terms as follows:		Now, if you're not working in Cartesian coordinates, care must be taken when dealing with the second term on the left-hand-side of this equation because when the gradient operator acts on a vector quantity (in this case, <math>~\rho\boldsymbol{v}</math>), various curvature terms will arise reflecting the fact that, in general, the unit vectors of your curvilinear coordinate system point in different directions as the fluid moves to different locations in space. Quite generally, though, for the <math>~j^\mathrm{th}</math> component of the equation of motion we may isolate these curvature terms as follows:
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	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">
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	</div>		</div>


	===Conservation of Angular Momentum===		===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,		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,
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	</math>		</math>
	</td>		</td>
			</tr>

	<tr>		<tr>
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	</math>		</math>
	</td>		</td>
	~~</tr>~~
	</tr>		</tr>
	</table>		</table>

Tohline: /* Hybrid Advection Scheme (Background) */ Add final comment in preface

2014-03-03T20:05:44Z

Hybrid Advection Scheme (Background): Add final comment in preface

← Older revision		Revision as of 20:05, 3 March 2014
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	''March 1, 2014'' by Joel E. Tohline		''March 1, 2014'' by Joel E. Tohline

	As is mentioned in the preface to our [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme\|primary discussion of the Hybrid Advection Scheme]], my early notes on ~~the~~ topic ~~have been preserved, as they~~ were included in my earliest version of this web-based H_Book. The relevant page can be accessed [http://www.phys.lsu.edu/astro/H_Book.current/Context/PGE/angmom.conserve.html via this link]; it is an html file whose linux time stamp is <font color="red">August 27, 2000</font>. Here we re-present the content of ~~those~~ "year 2000" notes because the "symbol" fonts utilized throughout the original html page seem now not to be properly translated by some web browsers.		As is mentioned in the preface to our [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme\|primary discussion of the Hybrid Advection Scheme]], my early notes on this topic were included in my earliest version of this web-based H_Book. The relevant page can be accessed [http://www.phys.lsu.edu/astro/H_Book.current/Context/PGE/angmom.conserve.html via this link]; it is an html file whose linux time stamp is <font color="red">August 27, 2000</font>. Here we re-present the content of these "year 2000" notes because the symbol fonts utilized throughout the original html page seem now not to be properly translated by some web browsers.

			As my group discussed this proposed new algorithmic approach to advecting angular momentum over the first decade of the new millennium, it was often referred to as the "A* scheme," where, <math>~A^* \equiv (A_\mathrm{rot} + \rho\varpi^2\Omega_f)</math> was the inertial-frame angular momentum density as given by the sum of variables whose values were tracked in the rotating frame.

	{{LSU_HBook_header}}		{{LSU_HBook_header}}

Tohline: /* The So-called A* Scheme */ Finished typing in original "year 2000" notes

2014-03-03T19:51:42Z

The So-called A* Scheme: Finished typing in original "year 2000" notes

Show changes

Tohline: /* Hybrid Advection Scheme (Background) */

2014-03-03T00:38:57Z

Hybrid Advection Scheme (Background)

← Older revision		Revision as of 00:38, 3 March 2014
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	''March 1, 2014'' by Joel E. Tohline		''March 1, 2014'' by Joel E. Tohline

	As is mentioned in the preface to our ~~primary discussion of the~~ [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme\|Hybrid Advection Scheme]], my early notes on the topic have been preserved, as they were included in my earliest version of this web-based H_Book. The relevant page can be accessed [http://www.phys.lsu.edu/astro/H_Book.current/Context/PGE/angmom.conserve.html via this link]; it is an html file whose linux time stamp is <font color="red">August 27, 2000</font>. Here we re-present the content of those "year 2000" notes because the "symbol" fonts utilized throughout the original html page seem now not to be properly translated by some web browsers..		As is mentioned in the preface to our [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme\|primary discussion of the Hybrid Advection Scheme]], my early notes on the topic have been preserved, as they were included in my earliest version of this web-based H_Book. The relevant page can be accessed [http://www.phys.lsu.edu/astro/H_Book.current/Context/PGE/angmom.conserve.html via this link]; it is an html file whose linux time stamp is <font color="red">August 27, 2000</font>. Here we re-present the content of those "year 2000" notes because the "symbol" fonts utilized throughout the original html page seem now not to be properly translated by some web browsers.

	{{LSU_HBook_header}}		{{LSU_HBook_header}}

Tohline: /* ASIDE: Relevant to hydrocode development */ More paragraphs from old "year 2000" notes

2014-03-03T00:37:00Z

ASIDE: Relevant to hydrocode development: More paragraphs from old "year 2000" notes

← Older revision		Revision as of 00:37, 3 March 2014
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	==ASIDE: Relevant to hydrocode development==		==ASIDE: Relevant to hydrocode development==
			===Curvature Terms===
	Coverting [''sic''] from a Lagrangian to an Eulerian time-derivative, Equation [I.A.5] becomes,		Coverting [''sic''] from a Lagrangian to an Eulerian time-derivative, Equation [I.A.5] becomes,
	<div align="center">		<div align="center">
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	</table>		</table>
	</div>		</div>

			===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.
			<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 \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>
			<td align="right">

			</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>
			</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.

			===Notation Adjustment===
			In what follows, the governing PDEs will be expressed in terms of velocities that are measured in the context of a rotating reference frame. In our [http://www.phys.lsu.edu/astro/H_Book.current/Context/PGE/angmom.conserve.html original html-formatted notes], these variables are differentiated from inertial-frame variables by color; specifically, rotating-frame variables are colored orange. Here we will abandon the color labeling and, instead, use <math>~\boldsymbol{u}</math> to represent the velocity as measured in a frame rotating about the <math>~z</math>-axis with angular frequency, <math>~\Omega_f</math>. That is, <math>~\boldsymbol{u}</math> is related to the inertial-frame velocity <math>~\boldsymbol{v}</math> through the expression,
			<div align="center">
			<table border="0" cellpadding="3">
			<tr>
			<td align="right">
			<math>
			~\boldsymbol{u}
			</math>
			</td>
			<td align="center">
			<math>~=~</math>
			</td>
			<td align="left">
			<math>
			~\boldsymbol{v} - \boldsymbol{\hat{e}}_\theta \varpi\Omega_f \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			===The So-called A* Scheme===
			Now, in a rotating frame of reference, this preferred form of the azimuthal component of the equation of motion takes the form,
			<div align="center">
			<table border="0" cellpadding="3">
			<tr>
			<td align="right">
			<math>
			\partial_t(\rho \varpi u_\theta) + \nabla\cdot(\rho\varpi u_\theta \boldsymbol{u} )
			</math>
			</td>
			<td align="center">
			<math>~=~</math>
			</td>
			<td align="left">
			<math>
			~ - \varpi \nabla_\theta P - \varpi \rho \nabla_\theta \Phi -2\rho\varpi \Omega_f u_\varpi \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Notice that a new term has appeared due to the coriolis force. Traditionally, in numerical hydrodynamics, this new term has been treated explicitly as a source term. hence, this component of the equation of motion no longer takes on a strictly conservative form, and the adopted "conservative" finite-difference is no longer a particularly useful tool to guarantee that the angular momentum is globally conserved even when the external forces due to pressure and gravity balance one another.

			To derive a form of this equation that is a lot more suited to a "conservative" finite-difference implementation, note that,

	=Related Discussions=		=Related Discussions=

Tohline: Add couple of lines at beginning to force table of contents

2014-03-02T23:50:21Z

Add couple of lines at beginning to force table of contents

← Older revision		Revision as of 23:50, 2 March 2014
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			<!-- __NOTOC__ -->
	=Hybrid Advection Scheme (Background)=		=Hybrid Advection Scheme (Background)=
	''March 1, 2014'' by Joel E. Tohline		''March 1, 2014'' by Joel E. Tohline

Tohline: /* Hybrid Advection Scheme (Background) */ Type first 1/3 of content

2014-03-02T22:35:47Z

Hybrid Advection Scheme (Background): Type first 1/3 of content

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Tohline: Begin page that re-presents my notes from August of 2000.

2014-03-02T22:05:09Z

Begin page that re-presents my notes from August of 2000.

New page

=Hybrid Advection Scheme (Background)=
==Preface==

''March 1, 2014'' by Joel E. Tohline

As is mentioned in the preface to our primary discussion of the [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]], my early notes on the topic have been preserved, as they were included in my earliest version of this web-based H_Book. The relevant page can be accessed [http://www.phys.lsu.edu/astro/H_Book.current/Context/PGE/angmom.conserve.html via this link]; it is an html file whose linux time stamp is <font color="red">August 27, 2000</font>. Here we re-present the content of those "year 2000" notes because the "symbol" fonts utilized throughout the original html page seem now not to be properly translated by some web browsers..

{{LSU_HBook_header}}

==Setting the Stage==

=Related Discussions=
* Euler equation viewed from a [http://www.vistrails.org/index.php?title=User:Tohline/PGE/RotatingFrame rotating frame of reference] or [[Main Page]].
* An [http://www.vistrails.org/index.php/User:Tohline/PGE/ConservingMomentum earlier draft of this "Euler equation" presentation].

{{LSU_HBook_footer}}