# Problem concerning spatial descretization order

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 December 18, 2017, 00:00 Problem concerning spatial descretization order #1 New Member   A-400M Join Date: Apr 2012 Posts: 12 Rep Power: 14 Hello. I have been making 2d euler code for airfoil geometry. I used Roe scheme for flux treatment. When i used first order scheme for spatial descretization, calculation was well converged. But i used second order scheme, error showed up. Specific symptom is below. --------------------------------------------------------------------------------- Starting calculation, residual decreased well up to around 0.005 order during several itreations. But after this iteration, residual increased up to around 0.037 and in the result, error showed up. I recognized that when residual increase, density on the leading edge become lower than order of density value which is calculated by compressible flow theory. And then density on the leading edge continue to decrease(=residual continue to increase) and become lower than it of freestream. In the result, error show up. (Freestream Mach number is 0.7) -------------------------------------------------------------------------------- When i used first order scheme, this error phenomenon didnt take place. If you had experience like this or you could guess cause of this error, please give me an advice about what should i check? When invicid wall boundary treatment is wrong in second order scheme, this phenomenon can happen? Thanks. Last edited by waven; December 18, 2017 at 01:20.

 December 18, 2017, 07:54 #2 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,794 Rep Power: 71 Are you using second order central or upwind discretization? In the former case have you considered that central discretizations for inviscid flow drive to numerical oscillations and finally can produce the solution to blow-up? You should provide details..

 December 18, 2017, 08:23 #3 New Member   A-400M Join Date: Apr 2012 Posts: 12 Rep Power: 14 I used the second order upwind as shown in below description. q(j,k)_left=q(j-1,k)+(q(j,k)-q(j-1,k))/2 q(j,k)_right=q(j+1,k)-(q(j+2,k)-q(j+1,k))/2

December 18, 2017, 08:35
#4
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Filippo Maria Denaro
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Quote:
 Originally Posted by waven I used the second order upwind as shown in below description. q(j,k)_left=q(j-1,k)+(q(j,k)-q(j-1,k))/2 q(j,k)_right=q(j+1,k)-(q(j+2,k)-q(j+1,k))/2
And how do you ensure the upwind criterion? I see only fixed stencils

 December 19, 2017, 05:29 #5 New Member   A-400M Join Date: Apr 2012 Posts: 12 Rep Power: 14 My descriptin cant be considered as second order upwind? If that, could you let me know the exact description and texts or paper containing the exact description?

December 19, 2017, 05:38
#6
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Filippo Maria Denaro
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Quote:
 Originally Posted by waven My descriptin cant be considered as second order upwind? If that, could you let me know the exact description and texts or paper containing the exact description?
upwind means that you adapt the stencil to the velocity direction

 December 19, 2017, 06:19 #7 New Member   A-400M Join Date: Apr 2012 Posts: 12 Rep Power: 14 I used cell centered scheme. so i think that my description means the second order upwind at the east face of cell. if you think i was wrong, could you correct my wrong description? you can correct it easily with copy && paste.

December 19, 2017, 11:38
#8
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Filippo Maria Denaro
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Quote:
 Originally Posted by waven I used cell centered scheme. so i think that my description means the second order upwind at the east face of cell. if you think i was wrong, could you correct my wrong description? you can correct it easily with copy && paste.
if you consider an FD discretization at the node i, a second order upwind stencil uses i,i-1,i-2 for u>0 and i,i+1,i+2 for u<0.

In a FV discretization, the flux at the face left is discretized over i-2,i-1,i for u>0 and i-1,i,i+1 for u<0, this is at third order of accuracy

 December 19, 2017, 20:46 #9 New Member   A-400M Join Date: Apr 2012 Posts: 12 Rep Power: 14 you suggested third order scheme for FVM, How is second order scheme described? and what does the 'u' mean? is it normal velocity at the face? could you let me know the text related to your description? it is hard to totally understand your description without front and back context.

December 20, 2017, 03:28
#10
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Filippo Maria Denaro
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Quote:
 Originally Posted by waven you suggested third order scheme for FVM, How is second order scheme described? and what does the 'u' mean? is it normal velocity at the face? could you let me know the text related to your description? it is hard to totally understand your description without front and back context.
u is the convective velocity, have a look to any CFD textbook for the upwind criterion and to the Leonard papers for the third order upwind scheme (QUICK/QUICKEST)

December 20, 2017, 04:38
#11
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A. S.
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Quote:
 Originally Posted by waven I used the second order upwind as shown in below description. q(j,k)_left=q(j-1,k)+(q(j,k)-q(j-1,k))/2 q(j,k)_right=q(j+1,k)-(q(j+2,k)-q(j+1,k))/2
You will need limiter to limit your reconstruction. In case of shock q(j+1)-q(j) would be discontinuous and lead to error. Refer to Blazek book for further help.

 December 20, 2017, 05:21 #12 Senior Member   Join Date: Oct 2011 Posts: 242 Rep Power: 16 I agree with apoorv, you may read the book of Blazek: Computational Fluid Dynamics: Principles and Applications Especially Chapter 4 where you will find many things you need for structured finite volume scheme of Euler equations.

 December 20, 2017, 08:40 #13 New Member   A-400M Join Date: Apr 2012 Posts: 12 Rep Power: 14 Thanks a lot to apoorv and naffrancois. I refered to Blazek book and studied MUSCL limiter. But as far as i know, limiter is not needed in subsonic condition like my case such as Mach number 0.4~0.7. Because shock is not generated. Limiter can solve this problem?

 December 21, 2017, 07:58 #14 Member   A. S. Join Date: Apr 2009 Location: Raipur (INDIA) Posts: 54 Rep Power: 17 Yes limiting should help, as it will avoid anyover-shot and undershoot, so as to reduce CFL a bit. Also can you give me detail of solver explicit (how many rk loop) or implicit; what cfl you are running

 December 27, 2017, 02:10 #15 New Member   A-400M Join Date: Apr 2012 Posts: 12 Rep Power: 14 Thanks a lot apoorv. CFL is 0.15, time integration method is first order explicit. When i used limiter, flux from south at the cell above first layer cell on the wall had error. wall boundary cell had no problem but right above cell had error due to error at the south-dir flux. apoorv, could you give me an advice on what should i check?

 Tags inviscid, spatial, wall treatment