Difference between revisions of "Comparison of solitons and waves"

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[[ru:Сравнение солитона с волной]]
 
<!--[[ТМ|Кафедра ТМ]] > [[Проект "Термокристалл"]] > [[Compare soliton with wave]] <HR>
 
<!--[[ТМ|Кафедра ТМ]] > [[Проект "Термокристалл"]] > [[Compare soliton with wave]] <HR>
 
[[Виртуальная лаборатория]] > [[Compare soliton with wave]] <HR>-->
 
[[Виртуальная лаборатория]] > [[Compare soliton with wave]] <HR>-->
  
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To simulate the soliton at this booth is used the numerical solution of the Korteweg - de Vries . It has the form:
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To demonstrate a soliton propagation, we numerically solve the Korteweg–de Vries equation. It has the form:
 
::<math>\dot U + UU' + CU''' = 0</math>.
 
::<math>\dot U + UU' + CU''' = 0</math>.
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For the numerical solution gradually find <math>U'</math>, <math>U''</math> и <math>U'''</math>, using the central difference method:
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To obtain a numerical solution we successively calculate derivatives <math>U'</math>, <math>U''</math> and <math>U'''</math>, using the central difference method:
 
::<math>U' = \frac{U_{n+1} - U_{n-1}}{2\Delta x}</math>,
 
::<math>U' = \frac{U_{n+1} - U_{n-1}}{2\Delta x}</math>,
 
::<math>U'' = \frac{U_{n+2} - 2U_{n} + U_{n-2}}{4\Delta x} \approx \frac{U_{n+1} - 2U_{n} + U_{n-1}}{2\Delta x}</math>,
 
::<math>U'' = \frac{U_{n+2} - 2U_{n} + U_{n-2}}{4\Delta x} \approx \frac{U_{n+1} - 2U_{n} + U_{n-1}}{2\Delta x}</math>,
 
::<math>U''' = \frac{U_{n+2} - 2U_{n+1} + 2U_{n-1} - U_{n-2}}{4\Delta x}</math>.
 
::<math>U''' = \frac{U_{n+2} - 2U_{n+1} + 2U_{n-1} - U_{n-2}}{4\Delta x}</math>.
  
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At the stand of the initial moment of movement indicated by the <span style="color:#808">purple outline</span> of the soliton, <span style="background-color:#0bb">turquoise gradient</span> - moving waves.
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The soliton is indicated by the <span style="color:#808">purple outline</span>, the moving wave is shown by a <span style="background-color:#0bb">turquoise gradient</span>.
  
 
{{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Tcvetkov/Equations/Soliton_Wave_compare_v1/Equations.html |width=830 |height=640 |border=0 }}
 
{{#widget:Iframe |url=http://tm.spbstu.ru/htmlets/Tcvetkov/Equations/Soliton_Wave_compare_v1/Equations.html |width=830 |height=640 |border=0 }}
  
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Developer [[Tsvetkov Denis]], when writing program code used by Alexander Yavorsky ([https://github.com/yavalvas/kdf-equation/blob/master/kdf_equation.cpp ссылка]).
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Coded by [[Tsvetkov Denis]], the code written by Alexander Yavorsky ([https://github.com/yavalvas/kdf-equation/blob/master/kdf_equation.cpp link]) was used to accomplish the work.
  
 
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<!--[[Category: Виртуальная лаборатория]]
 
[[Category: Проект "Термокристалл"]]-->
 
[[Category: Проект "Термокристалл"]]-->

Latest revision as of 18:53, 18 January 2017


To demonstrate a soliton propagation, we numerically solve the Korteweg–de Vries equation. It has the form:

[math]\dot U + UU' + CU''' = 0[/math].

To obtain a numerical solution we successively calculate derivatives [math]U'[/math], [math]U''[/math] and [math]U'''[/math], using the central difference method:

[math]U' = \frac{U_{n+1} - U_{n-1}}{2\Delta x}[/math],
[math]U'' = \frac{U_{n+2} - 2U_{n} + U_{n-2}}{4\Delta x} \approx \frac{U_{n+1} - 2U_{n} + U_{n-1}}{2\Delta x}[/math],
[math]U''' = \frac{U_{n+2} - 2U_{n+1} + 2U_{n-1} - U_{n-2}}{4\Delta x}[/math].

The soliton is indicated by the purple outline, the moving wave is shown by a turquoise gradient.

Coded by Tsvetkov Denis, the code written by Alexander Yavorsky (link) was used to accomplish the work.


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