CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

Formula for 2nd order upwind scheme for non-uniform grids?

Register Blogs Community New Posts Updated Threads Search

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   August 30, 2012, 04:57
Default Formula for 2nd order upwind scheme for non-uniform grids?
  #1
Senior Member
 
TWB
Join Date: Mar 2009
Posts: 414
Rep Power: 19
quarkz is on a distinguished road
Hi,

I've seen the formula for 2nd order upwind scheme for uniform grids as:

phi_e = 1.5*phi_P - 0.5*phi_W for u_e > 0

phi_e = 1.5*phi_E - 0.5*phi_EE for u_e < 0

Is there a formula meant for non-uniform grids as well?

Thanks
quarkz is offline   Reply With Quote

Old   August 30, 2012, 05:57
Default
  #2
Senior Member
 
Join Date: Dec 2011
Location: Madrid, Spain
Posts: 134
Rep Power: 15
michujo is on a distinguished road
Hi, you can try to derive it. Just take the Taylor expansions around the nodes E and EE, taking into account the different grid spacing \Delta x_i for node i.

The formulas you showed is a particular case where the grid spacing is constant. Under this assumption, the grid spacings appearing in numerator and denominator of your expressions when you solve for phi_e cancel out and \Delta x disappears from the equation. However, the different \Delta x_i will appear in the expression of phi_e if you consider grid non-uniformity.

I hope it helps.

Cheers,
Michujo.
michujo is offline   Reply With Quote

Old   August 30, 2012, 18:25
Default
  #3
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,882
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
Quote:
Originally Posted by quarkz View Post
Hi,

I've seen the formula for 2nd order upwind scheme for uniform grids as:

phi_e = 1.5*phi_P - 0.5*phi_W for u_e > 0

phi_e = 1.5*phi_E - 0.5*phi_EE for u_e < 0

Is there a formula meant for non-uniform grids as well?

Thanks
Be careful that upwind schemes are usually first, third, fifth ... order accurate... the reason is to have a local truncation error that has a dissipative behaviour ... the above formulas are based on a linear extrapolation of the value, that is highly unstable in general.

I suggest to construct a second degree polynomial on non-uniform stencil in such a way that if u_e>0 then Phi_W,Phi_P, Phi_E are involved (the counterpart for u_e<0 involves Phi_P,Phi_E, Phi_EE).
This way the use of Phi_P ensures better stability properties
FMDenaro is offline   Reply With Quote

Reply


Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
second order schemes marine OpenFOAM 67 April 11, 2022 19:19
Free jet simulation msarkar OpenFOAM Running, Solving & CFD 39 May 11, 2021 14:21
[swak4Foam] Air Conditioned room groovyBC Sebaj OpenFOAM Community Contributions 7 October 31, 2012 15:16
singularity? mihaipruna OpenFOAM Running, Solving & CFD 5 April 24, 2012 18:18
second order FD upwind scheme Heinz Wilkening Main CFD Forum 2 November 3, 1998 15:33


All times are GMT -4. The time now is 21:50.