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Shock in the Inviscid Burgers Equation

0

$begingroup$

I'm trying to implement the Finite-Volume method using local Lax-Friedrichs flux function. I found an example to the Riemann-problem, here is the solution plot:

So from characteristics $$u=begincases
1, x<frac12t \
0, x>frac12t
endcases$$
How can we see from the plot where the shock is?

pde numerical-methods characteristics

share|cite|improve this question

asked Mar 27 at 17:20

dxdydzdxdydz

49110

$endgroup$

add a comment | 

0

$begingroup$

I'm trying to implement the Finite-Volume method using local Lax-Friedrichs flux function. I found an example to the Riemann-problem, here is the solution plot:

So from characteristics $$u=begincases
1, x<frac12t \
0, x>frac12t
endcases$$
How can we see from the plot where the shock is?

pde numerical-methods characteristics

share|cite|improve this question

asked Mar 27 at 17:20

dxdydzdxdydz

49110

$endgroup$

add a comment | 

$begingroup$

I'm trying to implement the Finite-Volume method using local Lax-Friedrichs flux function. I found an example to the Riemann-problem, here is the solution plot:

So from characteristics $$u=begincases
1, x<frac12t \
0, x>frac12t
endcases$$
How can we see from the plot where the shock is?

pde numerical-methods characteristics

share|cite|improve this question

asked Mar 27 at 17:20

dxdydzdxdydz

49110

$endgroup$

share|cite|improve this question

asked Mar 27 at 17:20

dxdydzdxdydz

49110

share|cite|improve this question

asked Mar 27 at 17:20

dxdydzdxdydz

49110

asked Mar 27 at 17:20

dxdydzdxdydz

49110

asked Mar 27 at 17:20

dxdydzdxdydz

49110

1 Answer
1

active

oldest

votes

0

$begingroup$

If you want a numerical way of finding the shock, you could take your array $u[i,j]$ and consider the partial derivative in the x-direction:
dudx = diff(u) / dx

And then if you contour plot dudx, you should get an image where the shock shows up as yellow. This should be along the line $t = 2x$, which you can see in your plot above.

share|cite|improve this answer

answered Mar 27 at 18:50

gdepaulgdepaul

3759

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I tried to follow your advice, and I got smth like this: imgur.com/FLzCyMo, seems a little bit awkward, I picked the number of nodes $x$ to be $100$ on $[-3,3], dt = 0.01$ Can you show what I should get?
  $endgroup$
  – dxdydz
  Mar 27 at 22:40
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That’s more or less what you should get, with the exception of the loop that juts our from the origin to the left.
  $endgroup$
  – gdepaul
  Mar 27 at 23:10
  

  
  
  
  

add a comment | 

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0

$begingroup$

If you want a numerical way of finding the shock, you could take your array $u[i,j]$ and consider the partial derivative in the x-direction:
dudx = diff(u) / dx

And then if you contour plot dudx, you should get an image where the shock shows up as yellow. This should be along the line $t = 2x$, which you can see in your plot above.

share|cite|improve this answer

answered Mar 27 at 18:50

gdepaulgdepaul

3759

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I tried to follow your advice, and I got smth like this: imgur.com/FLzCyMo, seems a little bit awkward, I picked the number of nodes $x$ to be $100$ on $[-3,3], dt = 0.01$ Can you show what I should get?
  $endgroup$
  – dxdydz
  Mar 27 at 22:40
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That’s more or less what you should get, with the exception of the loop that juts our from the origin to the left.
  $endgroup$
  – gdepaul
  Mar 27 at 23:10
  

  
  
  
  

add a comment | 

0

$begingroup$

If you want a numerical way of finding the shock, you could take your array $u[i,j]$ and consider the partial derivative in the x-direction:
dudx = diff(u) / dx

And then if you contour plot dudx, you should get an image where the shock shows up as yellow. This should be along the line $t = 2x$, which you can see in your plot above.

share|cite|improve this answer

answered Mar 27 at 18:50

gdepaulgdepaul

3759

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I tried to follow your advice, and I got smth like this: imgur.com/FLzCyMo, seems a little bit awkward, I picked the number of nodes $x$ to be $100$ on $[-3,3], dt = 0.01$ Can you show what I should get?
  $endgroup$
  – dxdydz
  Mar 27 at 22:40
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That’s more or less what you should get, with the exception of the loop that juts our from the origin to the left.
  $endgroup$
  – gdepaul
  Mar 27 at 23:10
  

  
  
  
  

add a comment | 

$begingroup$

If you want a numerical way of finding the shock, you could take your array $u[i,j]$ and consider the partial derivative in the x-direction:
dudx = diff(u) / dx

And then if you contour plot dudx, you should get an image where the shock shows up as yellow. This should be along the line $t = 2x$, which you can see in your plot above.

share|cite|improve this answer

answered Mar 27 at 18:50

gdepaulgdepaul

3759

$endgroup$

share|cite|improve this answer

answered Mar 27 at 18:50

gdepaulgdepaul

3759

answered Mar 27 at 18:50

gdepaulgdepaul

3759

answered Mar 27 at 18:50

gdepaulgdepaul

3759