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November 8, 2013, 12:05 
Dissipation in upwind scheme remedies!!

#1 
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Tarun Malik
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Hi,
I am currently working on numerically solving partial differential equation (basically NavierStokes equations ) for a particular problem. I wrote a Matlab code. For higher Reynolds number, I am using first order upwind scheme. But I am getting quite a dissipative solution. I searched on the internet and came across the use of deferred correction for convective terms. But now, the iterations are diverging. Can anyone suggest me what other schemes can i use? or what changes can i consider in the upwind scheme ? Thanks in advance. 

November 8, 2013, 12:43 

#2  
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Filippo Maria Denaro
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November 8, 2013, 12:51 

#3 
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Tarun Malik
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Thanks for your response.
I would like to add that it is assured that the flow is laminar. I tried using central difference scheme but the solution is diverging for high reynolds number. I am getting solution for high reynolds number for 1st order upwind scheme but as i mentioned they are dissipative. I was thinking about high resolution schemes > http://www.cfdonline.com/Wiki/Appro...emes_.28HRS.29 But i dont know which one to choose. Thanks. 

November 8, 2013, 13:11 

#4  
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Filippo Maria Denaro
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November 10, 2013, 12:43 

#5 
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Tarun Malik
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hello again,
As suggested, i tried to decrease the grid size such that the cell Pe <2 . I ran into two problems. 1. The solution converged only for low reynolds number, not for medium to high value of Re. 2. I ended up with large number of linear equations are to be solved. As such large system necessarily require the iteration solvers, I am using Gauss Siedel with relaxation. Is there any other faster iteration solvers? what about Multigrid method? Thanks. 

November 10, 2013, 13:23 

#6  
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Filippo Maria Denaro
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1. if the cell Re number is less than 1, you are solving all the scales of the flow, provided that the dt is within the stability constraint your numerical solution must be stable. 2. the SOR method is quite useful also for large system. Often, more sophisticated methods such as Multigrid, BCGSTAB, GMRES, etc, are efficient in terms of reducted number of iterations but the computational cost of the single workunit can be much more expensive than tenth of SOR iteration. 

November 10, 2013, 13:35 

#7 
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Tarun Malik
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Thank you for your response.
However, the problem is steady. I am basically solving stream vorticity equations(so an outer iterative procedure is also required). Is there any other condition for steady problem ? On the other hand, I also wrote a code for 2nd order upwind scheme. There the solution is converging for the required Reynolds number. But the solution is highly dampened (or dissipative). Any insights? 

November 10, 2013, 14:48 

#8 
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Tarun Malik
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Thank you for your response.
However, the problem is steady. I am basically solving stream vorticity equations(so an outer iterative procedure is also required). Is there any other condition for steady problem ? On the other hand, I also wrote a code for 2nd order upwind scheme. There the solution is converging for the required Reynolds number. But the solution is highly dampened (or dissipative). Any insights? 

November 10, 2013, 15:03 

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Hassan Nemati
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November 10, 2013, 15:20 

#10  
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Filippo Maria Denaro
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November 16, 2013, 09:22 

#11 
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Tarun Malik
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yes, i have tried to reduce the relaxation factor.
This problem on which i am working on is two dimensional. I have read about effectiveness of upwind scheme in onedimensional from Patankar. While using an iterative procedure to solve a set of PDEs, I am getting negative coefficients of the variable (as in ApTp=AeTe+AwTw; Ae <0) for some of the grid points. I don't know if this happens because i am using upwind in 2D problem. How do i tackle this "negative coefficient" problem? Thanks in advance. 

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