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This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


[[Image:SodShockTubeTest Regions.png|thumb|300px|right|Density Plot after time evolution of t=0.2[-]]]
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
The '''Sod shock tube''' problem, named after Gary A. Sod, is a common test for the accuracy of [[Computational fluid dynamics|computational fluid codes]], like [[Riemann solver]]s,  and was heavily investigated by Sod in 1978.
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


The test consists of a one-dimensional [[Riemann problem]] with the following parameters, for left and right states of an [[ideal gas]].
Registered users will be able to choose between the following three rendering modes:


<center>
'''MathML'''
<math>
:<math forcemathmode="mathml">E=mc^2</math>
\left( \begin{array}{c}\rho_L\\P_L\\v_L\end{array}\right)
=
\left( \begin{array}{c}1.0\\1.0\\0.0\end{array} \right)
</math>
,
<math>
\left( \begin{array}{c}\rho_R\\P_R\\v_R\end{array}\right)
=
\left( \begin{array}{c}0.125\\0.1\\0.0\end{array}\right)
</math>
</center>
where
::*<math>\rho</math> is the density
::*P  is the pressure
::*v is the velocity


The time evolution of this problem can be described by solving the [[Euler equations]].
<!--'''PNG'''  (currently default in production)
Which leads to three characteristics, describing the propagation speed of the
:<math forcemathmode="png">E=mc^2</math>
various regions of the system. Namely the rarefaction wave, the contact discontinuity and
the shock discontinuity.
If this is solved numerically, one can test against the analytical solution,
and get information how well a code captures and resolves shocks and contact discontinuities
and reproduce the correct density profile of the rarefaction wave.


==Analytic derivation==
'''source'''
The different states of the solution are separated by the time evolution of the
:<math forcemathmode="source">E=mc^2</math> -->
three [[Method of characteristics|characteristics]] of the system, which is due to the finite speed
of information propagation. Two of them are equal to the speed
of sound of the both states
::<math>cs_1 = \sqrt{\gamma \frac{P_L}{\rho_L}}</math>
::<math>cs_5 = \sqrt{\gamma \frac{P_R}{\rho_R}}</math>
The first one is the position of the beginning of the rarefaction wave while
the other is the velocity of the propagation of the shock.


Defining:
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
::<math>\Gamma = \frac{\gamma - 1}{\gamma + 1}</math>, <math>\beta = \frac{\gamma - 1}{2 \gamma}</math>
The states after the shock are connected by the [[Rankine-Hugoniot equation|Rankine Hugoniot]]
shock jump conditions.
::<math>\rho_4 = \rho_5 \frac{P_4 + \Gamma P_5}{P_5 + \Gamma P_4}</math>
But to calculate the density in Region 4 we need to know the pressure in that region.
This is related by the contact discontinuity with the pressure in region 3 by
::<math>P_4 = P_3</math>
Unfortunately the pressure in region 3 can only be calculated iteratively, the right
solution is found when <math>u_2</math> equals <math>u_4</math>
::<math>u_4 = \left(P_3' - P_5\right)\sqrt{\frac{1-\Gamma}{\rho_R(P_3'+\Gamma P_5)}}</math>
::<math>u_2 =\left(P_1^\beta-P_3'^\beta\right) \sqrt{\frac{(1-\Gamma^2)P_1^{1/\gamma}}{\Gamma^2 \rho_L}}</math>
::<math>u_2 - u_4 = 0</math>
This function can be evaluated to an arbitrary precision thus giving the pressure in the
region 3
::<math>P_3 = \operatorname{calculate}(P_3,s,s,,)</math>
finally we can calculate
::<math>u_3 = u_5 + \frac{(P_3 - P_5)}{\sqrt{\frac{\rho_5}{2}((\gamma+1)P_3 +(\gamma-1)P_5)}}</math>
::<math>u_4 = u_3</math>
and <math>\rho_3</math> follows from the adiabatic gas law
::<math>\rho_3 = \rho_1 \left(\frac{P_3}{P_1}\right)^{1/\gamma}</math>


==References==
==Demos==
*{{cite journal | title=A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation  Laws | first=G. A. | last=Sod | year=1978 | journal=[[Journal of Computational Physics|J. Comput. Phys.]] | volume=27| pages=1–31 | url=http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=6812922 | doi=10.1016/0021-9991(78)90023-2 |bibcode = 1978JCoPh..27....1S }}
*{{cite book | first=Eleuterio F.| last=Toro| year=1999 | title=Riemann Solvers and Numerical Methods for Fluid Dynamics| publisher=Springer Verlag|location=Berlin | id=ISBN 3-540-65966-8}}


==See also==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
*[[Shock tube]]
*[[Computational fluid dynamics]]


[[Category:Fluid dynamics]]
 
[[Category:Computational fluid dynamics]]
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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