https://www.vistrails.org//index.php?title=User:Tohline/PGE/Hybrid_Scheme_Preface&feed=atom&action=historyUser:Tohline/PGE/Hybrid Scheme Preface - Revision history2024-03-28T21:19:55ZRevision history for this page on the wikiMediaWiki 1.36.2https://www.vistrails.org//index.php?title=User:Tohline/PGE/Hybrid_Scheme_Preface&diff=10152&oldid=prevTohline at 23:06, 11 July 20152015-07-11T23:06:46Z<p></p>
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<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Hybrid Advection Scheme (Preface)=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Hybrid Advection Scheme (Preface)=</div></td></tr>
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<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{LSU_HBook_header}}</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We delayed implementing this A* advection scheme in our production code for a number of years, primarily because it was unclear to me how to derive &#8212; and, therefore, fully justify &#8212; this hybrid inertial/rotating-frame advection scheme in full three-dimensional generality. How was the Coriolis term in the radial component of the equation of motion to be concurrently handled, for example? [http://etd.lsu.edu/docs/available/etd-07052010-224427/ Jay Call's dissertation research] focused precisely on this question (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner 2010]). He derived a complete description of the hybrid advection scheme in a fully relativistic and generalized coordinate framework. Jay showed that it is indeed valid to advect inertial-frame quantities across a rotating grid, in the manner suggested by my simpler A* scheme derivation. In addition &#8212; and more importantly &#8212; he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We delayed implementing this A* advection scheme in our production code for a number of years, primarily because it was unclear to me how to derive &#8212; and, therefore, fully justify &#8212; this hybrid inertial/rotating-frame advection scheme in full three-dimensional generality. How was the Coriolis term in the radial component of the equation of motion to be concurrently handled, for example? [http://etd.lsu.edu/docs/available/etd-07052010-224427/ Jay Call's dissertation research] focused precisely on this question (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner 2010]). He derived a complete description of the hybrid advection scheme in a fully relativistic and generalized coordinate framework. Jay showed that it is indeed valid to advect inertial-frame quantities across a rotating grid, in the manner suggested by my simpler A* scheme derivation. In addition &#8212; and more importantly &#8212; he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Go to: [[User:</del>Tohline<del style="font-weight: bold; text-decoration: none;">/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">''July 11, 2015'' by Joel E. </ins>Tohline</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Primarily in the context of [http://etd.lsu.edu/docs/available/etd-04022014-210318/ Zach Byerly's doctoral dissertation research], my group has since demonstrated that this hybrid advection scheme works extremely well. The key refereed publication resulting from this work is [http://adsabs.harvard.edu/abs/2014ApJS..212...23B Byerly, Adelstein-Lelbach, Tohline, &amp; Marcello (2014)].</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">''July 11, 2015'' by Joel E. </del>Tohline</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Go to: [[User:</ins>Tohline<ins style="font-weight: bold; text-decoration: none;">/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Primarily in the context of [http://etd.lsu.edu/docs/available/etd-04022014-210318/ Zach Byerly's doctoral dissertation research], my group has since demonstrated that this hybrid advection scheme works extremely well. The key refereed publication resulting from this work is [http://adsabs.harvard.edu/abs/2014ApJS..212...23B Byerly, Adelstein-Lelbach, Tohline, &amp; Marcello (2014)].</del></div></td><td colspan="2"></td></tr>
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</table>Tohlinehttps://www.vistrails.org//index.php?title=User:Tohline/PGE/Hybrid_Scheme_Preface&diff=10148&oldid=prevTohline: /* Hybrid Advection Scheme (Preface) */ Insert references to electronic dissertations of Jay Call and Zach Byerly2015-07-11T19:35:13Z<p><span dir="auto"><span class="autocomment">Hybrid Advection Scheme (Preface): </span> Insert references to electronic dissertations of Jay Call and Zach Byerly</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:35, 11 July 2015</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>One day I noticed that, while imposing some fairly reasonable constraints, the Coriolis term could be removed from the "source" term and folded into the divergence term on the left-hand side of the angular momentum conservation equation. This manipulation of terms seemed to be saying that the undesirable Coriolis term would disappear while employing a rotating coordinate mesh if the variable that was advected through the grid was the ''inertial-frame'' angular momentum density, rather than the ''rotating-frame'' angular momentum density. This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion. My early notes on this 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 here, which is an html file whose linux time stamp is August 27, 2000]. The symbol fonts utilized throughout this old html page seem now to be readable only through Microsoft's Internet Explorer web browser. Hence, for posterity sake, I have retyped this "year 2000" set of notes into an [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|accompanying page of this wiki]]. As the notes indicate, my group began referring to this as the [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|"A* scheme"]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>One day I noticed that, while imposing some fairly reasonable constraints, the Coriolis term could be removed from the "source" term and folded into the divergence term on the left-hand side of the angular momentum conservation equation. This manipulation of terms seemed to be saying that the undesirable Coriolis term would disappear while employing a rotating coordinate mesh if the variable that was advected through the grid was the ''inertial-frame'' angular momentum density, rather than the ''rotating-frame'' angular momentum density. This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion. My early notes on this 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 here, which is an html file whose linux time stamp is August 27, 2000]. The symbol fonts utilized throughout this old html page seem now to be readable only through Microsoft's Internet Explorer web browser. Hence, for posterity sake, I have retyped this "year 2000" set of notes into an [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|accompanying page of this wiki]]. As the notes indicate, my group began referring to this as the [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|"A* scheme"]].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>We delayed implementing this A* advection scheme in our production code for a number of years, primarily because it was unclear to me how to derive &#8212; and, therefore, fully justify &#8212; this hybrid inertial/rotating-frame advection scheme in full three-dimensional generality. How was the Coriolis term in the radial component of the equation of motion to be concurrently handled, for example? Jay Call's dissertation research focused precisely on this question (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner 2010]). He derived a complete description of the hybrid advection scheme in a fully relativistic and generalized coordinate framework. Jay showed that it is indeed valid to advect inertial-frame quantities across a rotating grid, in the manner suggested by my simpler A* scheme derivation. In addition &#8212; and more importantly &#8212; he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We delayed implementing this A* advection scheme in our production code for a number of years, primarily because it was unclear to me how to derive &#8212; and, therefore, fully justify &#8212; this hybrid inertial/rotating-frame advection scheme in full three-dimensional generality. How was the Coriolis term in the radial component of the equation of motion to be concurrently handled, for example? <ins style="font-weight: bold; text-decoration: none;">[http://etd.lsu.edu/docs/available/etd-07052010-224427/ </ins>Jay Call's dissertation research<ins style="font-weight: bold; text-decoration: none;">] </ins>focused precisely on this question (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner 2010]). He derived a complete description of the hybrid advection scheme in a fully relativistic and generalized coordinate framework. Jay showed that it is indeed valid to advect inertial-frame quantities across a rotating grid, in the manner suggested by my simpler A* scheme derivation. In addition &#8212; and more importantly &#8212; he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
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<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">''July 11, 2015'' by Joel E. Tohline</ins></div></td></tr>
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<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Primarily in the context of [http://etd.lsu.edu/docs/available/etd-04022014-210318/ Zach Byerly's doctoral dissertation research], my group has since demonstrated that this hybrid advection scheme works extremely well. The key refereed publication resulting from this work is [http://adsabs.harvard.edu/abs/2014ApJS..212...23B Byerly, Adelstein-Lelbach, Tohline, &amp; Marcello (2014)].</ins></div></td></tr>
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</table>Tohlinehttps://www.vistrails.org//index.php?title=User:Tohline/PGE/Hybrid_Scheme_Preface&diff=7050&oldid=prevTohline at 23:16, 4 March 20142014-03-04T23:16:03Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:16, 4 March 2014</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10">Line 10:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>During the mid-90s, when Kimberly Barker New was conducting her dissertation research ([http://adsabs.harvard.edu/abs/1997ApJ...490..311N see New &amp; Tohline 1997]), we started using a ''rotating'' cylindrical coordinate mesh. On a grid that was spinning at a suitably chosen frequency, advection of fluid ''through'' the grid could be minimized and this, in turn, reduced the undesirable effects of numerical diffusion. We adopted a fairly standard algorithmic approach very similar to the one used by [http://adsabs.harvard.edu/abs/1978ApJ...224..497N Norman &amp; Wilson (1978)] and acknowledged that we were making a tradeoff: While the shift to a rotating cylindrical coordinate mesh reduced the effects of numerical diffusion, the shift introduced a rather ugly Coriolis "source" term into two components of the equation of motion. This made it more difficult to ensure conservation of angular momentum.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>During the mid-90s, when Kimberly Barker New was conducting her dissertation research ([http://adsabs.harvard.edu/abs/1997ApJ...490..311N see New &amp; Tohline 1997]), we started using a ''rotating'' cylindrical coordinate mesh. On a grid that was spinning at a suitably chosen frequency, advection of fluid ''through'' the grid could be minimized and this, in turn, reduced the undesirable effects of numerical diffusion. We adopted a fairly standard algorithmic approach very similar to the one used by [http://adsabs.harvard.edu/abs/1978ApJ...224..497N Norman &amp; Wilson (1978)] and acknowledged that we were making a tradeoff: While the shift to a rotating cylindrical coordinate mesh reduced the effects of numerical diffusion, the shift introduced a rather ugly Coriolis "source" term into two components of the equation of motion. This made it more difficult to ensure conservation of angular momentum.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>One day I noticed that, while imposing some fairly reasonable constraints, the Coriolis term could be removed from the "source" term and folded into the divergence term on the left-hand side of the angular momentum conservation equation. This manipulation of terms seemed to be saying that the undesirable Coriolis term would disappear while employing a rotating coordinate mesh if the variable that was advected through the grid was the ''inertial-frame'' angular momentum density, rather than the ''rotating-frame'' angular momentum density. This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion. My notes on this 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 here, which is an html file whose linux time stamp is August 27, 2000]. The symbol fonts utilized throughout this old html page seem now to be readable only through Microsoft's Internet Explorer web browser. Hence, for posterity sake, I have retyped this "year 2000" set of notes into an [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|accompanying page of this wiki]]. As the notes indicate, my group began referring to this as the [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|"A* scheme"]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>One day I noticed that, while imposing some fairly reasonable constraints, the Coriolis term could be removed from the "source" term and folded into the divergence term on the left-hand side of the angular momentum conservation equation. This manipulation of terms seemed to be saying that the undesirable Coriolis term would disappear while employing a rotating coordinate mesh if the variable that was advected through the grid was the ''inertial-frame'' angular momentum density, rather than the ''rotating-frame'' angular momentum density. This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion. My <ins style="font-weight: bold; text-decoration: none;">early </ins>notes on this 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 here, which is an html file whose linux time stamp is August 27, 2000]. The symbol fonts utilized throughout this old html page seem now to be readable only through Microsoft's Internet Explorer web browser. Hence, for posterity sake, I have retyped this "year 2000" set of notes into an [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|accompanying page of this wiki]]. As the notes indicate, my group began referring to this as the [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|"A* scheme"]].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We delayed implementing this A* advection scheme in our production code for a number of years, primarily because it was unclear to me how to derive &#8212; and, therefore, fully justify &#8212; this hybrid inertial/rotating-frame advection scheme in full three-dimensional generality. How was the Coriolis term in the radial component of the equation of motion to be concurrently handled, for example? Jay Call's dissertation research focused precisely on this question (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner 2010]). He derived a complete description of the hybrid advection scheme in a fully relativistic and generalized coordinate framework. Jay showed that it is indeed valid to advect inertial-frame quantities across a rotating grid, in the manner suggested by my simpler A* scheme derivation. In addition &#8212; and more importantly &#8212; he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We delayed implementing this A* advection scheme in our production code for a number of years, primarily because it was unclear to me how to derive &#8212; and, therefore, fully justify &#8212; this hybrid inertial/rotating-frame advection scheme in full three-dimensional generality. How was the Coriolis term in the radial component of the equation of motion to be concurrently handled, for example? Jay Call's dissertation research focused precisely on this question (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner 2010]). He derived a complete description of the hybrid advection scheme in a fully relativistic and generalized coordinate framework. Jay showed that it is indeed valid to advect inertial-frame quantities across a rotating grid, in the manner suggested by my simpler A* scheme derivation. In addition &#8212; and more importantly &#8212; he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">This component set has been spelled out in, for example, equations (5) - (7) of and equations (11), (12), &amp; (3) of .</del></div></td><td colspan="2"></td></tr>
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</table>Tohlinehttps://www.vistrails.org//index.php?title=User:Tohline/PGE/Hybrid_Scheme_Preface&diff=7049&oldid=prevTohline: /* Hybrid Advection Scheme (Preface) */ Power outage in the middle of editing2014-03-04T23:11:54Z<p><span dir="auto"><span class="autocomment">Hybrid Advection Scheme (Preface): </span> Power outage in the middle of editing</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:11, 4 March 2014</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Throughout my research career, I have sought new or modified techniques and algorithms that would allow my group to perform more accurate numerical simulations of astrophysical fluid flows. At the beginning &#8212; following the advice of my dissertation co-advisors, Peter Bodenheimer and David Black &#8212; I adopted a cylindrical, rather than Cartesian, computational grid. When a cylindrical coordinate system is used, one of the components of the equation of motion can be written in a form that fairly naturally conserves angular momentum, and this was quite a desirable feature, given that our investigations were focusing on analyzing the stability of rotating configurations. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Throughout my research career, I have sought new or modified techniques and algorithms that would allow my group to perform more accurate numerical simulations of astrophysical fluid flows. At the beginning &#8212; following the advice of my dissertation co-advisors, Peter Bodenheimer and David Black &#8212; I adopted a cylindrical, rather than Cartesian, computational grid. When a cylindrical coordinate system is used, one of the components of the equation of motion can be written in a form that fairly naturally conserves angular momentum, and this was quite a desirable feature, given that our investigations were focusing on analyzing the stability of rotating configurations. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>During the mid-90s, when Kimberly Barker New was conducting her dissertation research, we started using a ''rotating'' cylindrical coordinate mesh. On a grid that was spinning at a suitably chosen frequency, advection of fluid ''through'' the grid could be minimized and this, in turn, reduced the undesirable effects of numerical diffusion. We adopted a fairly standard algorithmic approach very similar to the one used by Norman &amp; Wilson (<del style="font-weight: bold; text-decoration: none;">1980</del>) and acknowledged that we were making a tradeoff: While the shift to a rotating cylindrical coordinate mesh reduced the effects of numerical diffusion, the shift introduced a rather ugly Coriolis "source" term into two components of the equation of motion. This made it more difficult to ensure conservation of angular momentum.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>During the mid-90s, when Kimberly Barker New was conducting her dissertation research <ins style="font-weight: bold; text-decoration: none;">([http://adsabs.harvard.edu/abs/1997ApJ...490..311N see New &amp; Tohline 1997])</ins>, we started using a ''rotating'' cylindrical coordinate mesh. On a grid that was spinning at a suitably chosen frequency, advection of fluid ''through'' the grid could be minimized and this, in turn, reduced the undesirable effects of numerical diffusion. We adopted a fairly standard algorithmic approach very similar to the one used by <ins style="font-weight: bold; text-decoration: none;">[http://adsabs.harvard.edu/abs/1978ApJ...224..497N </ins>Norman &amp; Wilson (<ins style="font-weight: bold; text-decoration: none;">1978</ins>)<ins style="font-weight: bold; text-decoration: none;">] </ins>and acknowledged that we were making a tradeoff: While the shift to a rotating cylindrical coordinate mesh reduced the effects of numerical diffusion, the shift introduced a rather ugly Coriolis "source" term into two components of the equation of motion. This made it more difficult to ensure conservation of angular momentum.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">During my continuing efforts to develop an improved computational fluid dynamics algorithm, one </del>day I noticed that, while imposing some fairly reasonable constraints, the Coriolis term could be removed from the "source" term and folded into the divergence term on the left-hand side of the angular momentum conservation equation. This manipulation of terms seemed to be saying that the undesirable Coriolis term would disappear while employing a rotating coordinate mesh if the variable that was advected through the grid was the ''inertial-frame'' angular momentum density, rather than the ''rotating-frame'' angular momentum density. This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion. My notes on this 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 here, which is an html file whose linux time stamp is August 27, 2000]. The <del style="font-weight: bold; text-decoration: none;">"</del>symbol<del style="font-weight: bold; text-decoration: none;">" </del>fonts utilized throughout this old html page seem now to be readable only through Microsoft's Internet Explorer web browser. Hence, for posterity sake, I have retyped this "year 2000" set of notes into an [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|accompanying page of this wiki]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">One </ins>day I noticed that, while imposing some fairly reasonable constraints, the Coriolis term could be removed from the "source" term and folded into the divergence term on the left-hand side of the angular momentum conservation equation. This manipulation of terms seemed to be saying that the undesirable Coriolis term would disappear while employing a rotating coordinate mesh if the variable that was advected through the grid was the ''inertial-frame'' angular momentum density, rather than the ''rotating-frame'' angular momentum density. This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion. My notes on this 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 here, which is an html file whose linux time stamp is August 27, 2000]. The symbol fonts utilized throughout this old html page seem now to be readable only through Microsoft's Internet Explorer web browser. Hence, for posterity sake, I have retyped this "year 2000" set of notes into an [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|accompanying page of this wiki]]. <ins style="font-weight: bold; text-decoration: none;"> As the notes indicate, my group began referring to this as the [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|"A* scheme"]].</ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">We delayed implementing this A* advection scheme in our production code for a number of years, primarily because it was unclear to me how to derive &#8212; and, therefore, fully justify &#8212; this hybrid inertial/rotating-frame advection scheme in full three-dimensional generality. How was the Coriolis term in the radial component of the equation of motion to be concurrently handled, for example? Jay Call's dissertation research focused precisely on this question (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner 2010]). He derived a complete description of the hybrid advection scheme in a fully relativistic and generalized coordinate framework. Jay showed that it is indeed valid to advect inertial-frame quantities across a rotating grid, in the manner suggested by my simpler A* scheme derivation. In addition &#8212; and more importantly &#8212; he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. </ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]</div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">This component set has been spelled out in, for example, equations (5) - (7) of and equations (11), (12), &amp; (3) of .</ins></div></td></tr>
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</table>Tohlinehttps://www.vistrails.org//index.php?title=User:Tohline/PGE/Hybrid_Scheme_Preface&diff=7044&oldid=prevTohline: Initial content for full preface2014-03-03T20:40:38Z<p>Initial content for full preface</p>
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=Hybrid Advection Scheme (Preface)=<br />
Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]<br />
<br />
''March 1, 2014'' by Joel E. Tohline<br />
<br />
Throughout my research career, I have sought new or modified techniques and algorithms that would allow my group to perform more accurate numerical simulations of astrophysical fluid flows. At the beginning &#8212; following the advice of my dissertation co-advisors, Peter Bodenheimer and David Black &#8212; I adopted a cylindrical, rather than Cartesian, computational grid. When a cylindrical coordinate system is used, one of the components of the equation of motion can be written in a form that fairly naturally conserves angular momentum, and this was quite a desirable feature, given that our investigations were focusing on analyzing the stability of rotating configurations. <br />
<br />
During the mid-90s, when Kimberly Barker New was conducting her dissertation research, we started using a ''rotating'' cylindrical coordinate mesh. On a grid that was spinning at a suitably chosen frequency, advection of fluid ''through'' the grid could be minimized and this, in turn, reduced the undesirable effects of numerical diffusion. We adopted a fairly standard algorithmic approach very similar to the one used by Norman &amp; Wilson (1980) and acknowledged that we were making a tradeoff: While the shift to a rotating cylindrical coordinate mesh reduced the effects of numerical diffusion, the shift introduced a rather ugly Coriolis "source" term into two components of the equation of motion. This made it more difficult to ensure conservation of angular momentum.<br />
<br />
During my continuing efforts to develop an improved computational fluid dynamics algorithm, one day I noticed that, while imposing some fairly reasonable constraints, the Coriolis term could be removed from the "source" term and folded into the divergence term on the left-hand side of the angular momentum conservation equation. This manipulation of terms seemed to be saying that the undesirable Coriolis term would disappear while employing a rotating coordinate mesh if the variable that was advected through the grid was the ''inertial-frame'' angular momentum density, rather than the ''rotating-frame'' angular momentum density. This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion. My notes on this 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 here, which is an html file whose linux time stamp is August 27, 2000]. The "symbol" fonts utilized throughout this old html page seem now to be readable only through Microsoft's Internet Explorer web browser. Hence, for posterity sake, I have retyped this "year 2000" set of notes into an [[User:Tohline/PGE/AStarScheme#Hybrid_Advection_Scheme_.28Background.29|accompanying page of this wiki]].<br />
<br />
Go to: [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Hybrid Advection Scheme]]<br />
<br />
{{LSU_HBook_footer}}</div>Tohline