# A few simple questions about linearUpwind and limitedLinear

 Register Blogs Members List Search Today's Posts Mark Forums Read

 September 28, 2011, 13:58 A few simple questions about linearUpwind and limitedLinear #1 Senior Member     Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 553 Rep Power: 18 I am a writing about the schemes that I used in my simulations in my thesis and could use some clarification on the linearUpwind and limitedLinear schemes. I have read second order schemes and Alberto did a good explanation, but I need a little more clarification. linearUpwind: I know linearUpwind is second order and bounded by a limiter (e.g. cellMDLimited from cellMDLimited vs. cellLimited). This is different than the limiters used in limitedLinear. In Fluent documentation, the definition of second order upwind is given as: where and are the face value of phi and the cell centered values of phi in the upstream cells respectively. I am reading through the code of linearUpwind and I see: Code:  GeometricField& sfCorr = tsfCorr(); const surfaceScalarField& faceFlux = this->faceFlux_; const labelList& owner = mesh.owner(); const labelList& neighbour = mesh.neighbour(); const volVectorField& C = mesh.C(); const surfaceVectorField& Cf = mesh.Cf(); GeometricField ::type, fvPatchField, volMesh> gradVf = gradScheme_().grad(vf); forAll(faceFlux, facei) { if (faceFlux[facei] > 0) { label own = owner[facei]; sfCorr[facei] = (Cf[facei] - C[own]) & gradVf[own]; } else { label nei = neighbour[facei]; sfCorr[facei] = (Cf[facei] - C[nei]) & gradVf[nei]; } } I see the mesh.C and mesh.Cf and I am led to believe that through quickly reading the code, these are the cell centroid and face centroid, thus producing an r vector that is dotted with the gradient between the owner and neighbor cells (&gradVf) controlled by gradSchemes. I then see some similar code to take care of the boundaries. When is the sfCorr used to correct the value of the face? I guess I am just interested in generating a discussion and confirming or clarifying my thoughts. limitedLinear This again is second order and is bounded using a sweby limiter (second order schemes post 16). I am just a little fuzzy on what the final simplified form of the equation is for limitedLinear (I admit I haven't looked at it as much as linearUpwind). I am out of my office and will look through Richard Pletcher, John Tannehill and Dale Anderson when I get a chance. Laslty, correct me if I'm wrong but the parameter following a definition of a limitedLinear declaration is passed to the limiter, where 1 uses the limiter and 0 does not (with anything in between being a blend). Sorry for the massive post, and any clarification would be fantastic and if I find an answer, I will post the result here. And for those interested on how I did the math in the forum, look at (Guide: Writing Equations in LaTeX on the CFD Online Forums) Dan mm.abdollahzadeh, zhulianhua, skeptik and 3 others like this. Last edited by chegdan; September 29, 2011 at 13:24.

 September 28, 2011, 19:05 #2 Senior Member     Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 553 Rep Power: 18 on the limitedLinear, it seems that my knowledge of limiters is lacking and it may turn out to be a simple answer. Looking in Dr. Jasak's thesis (pg 98), I found that: where is the value of phi at the face for a higher order scheme. For limitedLinear, it is just the central difference scheme as the "HO" and the limiter () is a Sweby limiter (http://www.cfd-online.com/Forums/ope...ar-scheme.html). if there is more to add here or there are corrections then please do. Dan rajibroy likes this.

 September 29, 2011, 04:48 #3 New Member   Chris Join Date: Jun 2011 Posts: 12 Rep Power: 6 Yes you are right. limitedLinear is using the Sweby limiter [1]. If you want further information about TVD Schemes and different limiters, I think [2] might be interesting to get a first brief impression. About linearUpwind I don't know much. Till now I just thought it is something like linear upwind differencing but reading your post seems like its only half the story. [1] P.K. Sweby. High resolution schemes using flux limiters for hyperbolic conservation laws.SIAM Journal on Numerical Analysis, (Vol. 21):pp. 995–1011, 1984. [2] H. K. Versteeg and W. Malalasekera. An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson, 2007.

 September 29, 2011, 14:37 #4 Senior Member     Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 553 Rep Power: 18 @caramelo : I found a mistake in my first post concerning the linearUpwind definition and just corrected it (sorry about that). I did have a chance to look at the Sweby paper and also a few other sources on TVD schemes, very interesting. Thanks for the sources. I see throughout the forum that the linearUpwind is bounded, but Ferziger and Peric' note that second order upwind is unbounded. My question is that if one uses Gauss linear as a grad scheme for linearUpwind, will this approach the traditional second order upwind and therefore unbounded? Also, is Gauss linear just the arithmetic average of the two cells or is it the distance weighted average similar to the linear divScheme for advection? Lastly, is the linearUpwind scheme only bounded if limited versions of the gradient scheme are used, i.e. limited linear, cellLimited, faceMDLimited etc.? Dan

October 1, 2011, 15:51
[Solved] A few simple questions about linearUpwind and limitedLinear
#5
Senior Member

Daniel P. Combest
Join Date: Mar 2009
Location: St. Louis, USA
Posts: 553
Rep Power: 18
Quote:
 Originally Posted by chegdan @caramelo : I found a mistake in my first post concerning the linearUpwind definition and just corrected it (sorry about that). I did have a chance to look at the Sweby paper and also a few other sources on TVD schemes, very interesting. Thanks for the sources. I see throughout the forum that the linearUpwind is bounded, but Ferziger and Peric' note that second order upwind is unbounded. My question is that if one uses Gauss linear as a grad scheme for linearUpwind, will this approach the traditional second order upwind and therefore unbounded? Also, is Gauss linear just the arithmetic average of the two cells or is it the distance weighted average similar to the linear divScheme for advection? Lastly, is the linearUpwind scheme only bounded if limited versions of the gradient scheme are used, i.e. limited linear, cellLimited, faceMDLimited etc.? Dan
Sometimes I think I need to put the hash tag #dumbquestion or #thinking_out_loud on my comments to let everyone know that I'm thinking out loud about something. For my question:

Quote:
 Also, is Gauss linear just the arithmetic average of the two cells or is it the distance weighted average similar to the linear divScheme for advection
The answer to that one is that it is the regular Gauss linear interpolation scheme that we use all the time.

In the end I think that I have the answer that I was looking for.

Dan
__________________
Dan

October 3, 2011, 03:45
#6
Senior Member

Florian Krause
Join Date: Mar 2009
Location: Munich
Posts: 103
Rep Power: 8
Hi Dan,

Quote:
 When is the sfCorr used to correct the value of the face? I guess I am just interested in generating a discussion and confirming or clarifying my thoughts
did you find an answer to this? I also had a look at the source and couldn't find the part where the pieces are put together, namely cell centered phi and gradient of phi.

Apart from this, as far as I understand, linearUpwind scheme is unbounded as long as you don't specify one of the limiters cellLimited, faceLimited etc.

Best,
Florian

October 3, 2011, 12:55
#7
Senior Member

Daniel P. Combest
Join Date: Mar 2009
Location: St. Louis, USA
Posts: 553
Rep Power: 18
Quote:
 When is the sfCorr used to correct the value of the face? I guess I am just interested in generating a discussion and confirming or clarifying my thoughts
From what I can see in the Doxygen information for the limitedSurfaceInterpolationScheme class, everything is pieced together there. linearUpwind returns the weighting factors and then its all put together nicely by the limitedSurfaceInterpolationScheme class.

and of course:
Quote:
 Apart from this, as far as I understand, linearUpwind scheme is unbounded as long as you don't specify one of the limiters cellLimited, faceLimited etc
This is what I have found as well...so far. Thanks for the reply and if anyone has a correction, please respond.

Dan
__________________
Dan

October 18, 2013, 23:49
How can you find out which file contains the source code implementing the 'linearUpwi
#8
Member

yehanyu
Join Date: Mar 2012
Location: Beijing, China
Posts: 36
Rep Power: 5
Quote:
 Originally Posted by chegdan I am a writing about the schemes that I used in my simulations in my thesis and could use some clarification on the linearUpwind and limitedLinear schemes. I have read second order schemes and Alberto did a good explanation, but I need a little more clarification. linearUpwind: I know linearUpwind is second order and bounded by a limiter (e.g. cellMDLimited from cellMDLimited vs. cellLimited). This is different than the limiters used in limitedLinear. In Fluent documentation, the definition of second order upwind is given as: where and are the face value of phi and the cell centered values of phi in the upstream cells respectively. I am reading through the code of linearUpwind and I see: Code:  GeometricField& sfCorr = tsfCorr(); const surfaceScalarField& faceFlux = this->faceFlux_; const labelList& owner = mesh.owner(); const labelList& neighbour = mesh.neighbour(); const volVectorField& C = mesh.C(); const surfaceVectorField& Cf = mesh.Cf(); GeometricField ::type, fvPatchField, volMesh> gradVf = gradScheme_().grad(vf); forAll(faceFlux, facei) { if (faceFlux[facei] > 0) { label own = owner[facei]; sfCorr[facei] = (Cf[facei] - C[own]) & gradVf[own]; } else { label nei = neighbour[facei]; sfCorr[facei] = (Cf[facei] - C[nei]) & gradVf[nei]; } } I see the mesh.C and mesh.Cf and I am led to believe that through quickly reading the code, these are the cell centroid and face centroid, thus producing an r vector that is dotted with the gradient between the owner and neighbor cells (&gradVf) controlled by gradSchemes. I then see some similar code to take care of the boundaries. When is the sfCorr used to correct the value of the face? I guess I am just interested in generating a discussion and confirming or clarifying my thoughts. limitedLinear This again is second order and is bounded using a sweby limiter (second order schemes post 16). I am just a little fuzzy on what the final simplified form of the equation is for limitedLinear (I admit I haven't looked at it as much as linearUpwind). I am out of my office and will look through Richard Pletcher, John Tannehill and Dale Anderson when I get a chance. Laslty, correct me if I'm wrong but the parameter following a definition of a limitedLinear declaration is passed to the limiter, where 1 uses the limiter and 0 does not (with anything in between being a blend). Sorry for the massive post, and any clarification would be fantastic and if I find an answer, I will post the result here. And for those interested on how I did the math in the forum, look at (Guide: Writing Equations in LaTeX on the CFD Online Forums) Dan
How can you find out which file contains the source code implementing the 'linearUpwind' scheme? Please teach me. Thank you very much.

 October 21, 2013, 09:57 #9 Senior Member     Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 553 Rep Power: 18 Linux commands to search for file names and within files are your friend. To look for the linearUpwind source, try Code: cd $FOAM_SRC grep -R 'linearUpwind' . and then look for references to linearUpwind (you will see many). From that you can go through source code files. If you already know the name of the source file (in this case its easy since we want linearUpwind.C or linearUpwind.H), you can do a Code: cd$FOAM_SRC find . -name 'linearUpwind.C' and that will tell you where a specific file lives. After a while you will learn where different source code resides and be able to find things without searching like this. Good luck. hua1015 and seav like this. __________________ Dan Find me on twitter @dancombest and LinkedIn

 October 26, 2013, 22:44 #10 Member   yehanyu Join Date: Mar 2012 Location: Beijing, China Posts: 36 Rep Power: 5 Thank you very much.

January 10, 2014, 18:51
#11
Senior Member

Anne Gerdes
Join Date: Aug 2010
Location: Hamburg
Posts: 154
Rep Power: 7
Quote:
 Originally Posted by chegdan I am a writing about the schemes that I used in my simulations in my thesis and could use some clarification on the linearUpwind and limitedLinear schemes. I have read second order schemes and Alberto did a good explanation, but I need a little more clarification. linearUpwind: I know linearUpwind is second order and bounded by a limiter (e.g. cellMDLimited from cellMDLimited vs. cellLimited). This is different than the limiters used in limitedLinear. In Fluent documentation, the definition of second order upwind is given as: where and are the face value of phi and the cell centered values of phi in the upstream cells respectively.
Dear Dan, Dear Foamers,

I found your post which is very interesting for me.
Could you please tell me how one can derive the equation above?
I found this equation implemented in OpenFOAM, and in Fluent it is obviously the same.
But I have problems in understanding this equation.
For LUDS (linear upwind) we discretize the face value as

How is the implemented formula connected to that?

Anne

April 10, 2015, 03:47
#12
Member

Join Date: Jun 2012
Posts: 65
Rep Power: 5
Quote:
 Originally Posted by Anne Lincke Dear Dan, Dear Foamers, I found your post which is very interesting for me. Could you please tell me how one can derive the equation above? I found this equation implemented in OpenFOAM, and in Fluent it is obviously the same. But I have problems in understanding this equation. For LUDS (linear upwind) we discretize the face value as How is the implemented formula connected to that? Thank you very much for an answer in advance! Anne
I am a bit late to the discussion but I am studying interpolation schemes a bit and I think I can answer your question. I am still fairly new to the topic so there might be an error so feel free to correct me.

Taylor Expansion around P gives you this formula:

Now for upwind the flux over the east surface of a 2D CV for a flow from west:

The rest of the equation is the truncation error, thus first order.
If you also calculate the first truncated term your scheme will become second order and you will get this expression for your flux over face e:

For

You get:

Hope that helps

Last edited by Bazinga; April 13, 2015 at 03:41.

 April 10, 2015, 04:54 #13 Senior Member   Anne Gerdes Join Date: Aug 2010 Location: Hamburg Posts: 154 Rep Power: 7 Thank you very much for your answer! Meanwhile I have also been able to understand the formula, but it is nice to see the answer here as well. Have a nice day! Anne

 October 10, 2015, 11:52 #14 Member   Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 58 Rep Power: 3 Isnt the correct way like this?: Such that, if : If : And if , we get the Linear Upwind Differencing (LUD):

October 11, 2015, 02:29
#15
Member

Join Date: Jun 2012
Posts: 65
Rep Power: 5
Quote:
 Originally Posted by davibarreira Isnt the correct way like this?: Such that, if : If : And if , we get the Linear Upwind Differencing (LUD):
Just looked through it briefly. I think you are right. Thanks. The last part of my last reply should be:

" which does not equal the proposed equations."

 October 11, 2015, 10:17 #16 Member   Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 58 Rep Power: 3 One thing got me confused though, in the OpenFOAM website (http://cfd.direct/openfoam/user-guide/fvschemes/), it's written "Some TVD/NVD schemes require a coefficient ψ,0 ≤ ψ ≤ 1 where ψ = 1 corresponds to TVD conformance, usually giving best convergence and ψ = 0 corresponds to best accuracy. Running with ψ = 1 is generally recommended." So this implies that using ψ = 0 would give the higher order scheme (central difference), which is the opposite stated in dr. Jasak's thesis. I created a new thread with some more questions related to the topic. TVD schemes, questions about limitedLinear Any thoughts?

 October 13, 2015, 03:59 #17 Senior Member   Anne Gerdes Join Date: Aug 2010 Location: Hamburg Posts: 154 Rep Power: 7 Dear Bazinga and Davi, I try to derive the implemented formula once again: Two cell centers in upwind flow direction ( and ) are taken into account. The Taylor series leads to the approximation of at cells and and, subsequently, at face where denotes the distance from to . The first order upwind approximation is numerically very stable, as the upwind term stabilizes the diagonal of the matrix. Therefore, we use the implicit term of an upwind discretization as a matrix coefficient and add the remaining terms as explicit correction terms on the right-hand side of the equation system. The numerical approximation for the LUDS discretization thus transforms to The second term is the explicit correction term. This equals the implemented formula.

 Tags limitedlinear, linearupwind

 Thread Tools Display Modes Linear Mode

 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 OffTrackbacks are On Pingbacks are On Refbacks are On Forum Rules