A few simple questions about linearUpwind and limitedLinear

chegdan · September 28, 2011, 14:58

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 http://www.cfd-online.com/Forums/ope...r-schemes.html 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 http://www.cfd-online.com/Forums/ope...lllimited.html). This is different than the limiters used in limitedLinear. In Fluent documentation, the definition of second order upwind is given as:

$\phi_f = \phi + \nabla\phi\cdot\vec{r}$

where $\phi_f$ and $\phi$ 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<Type, fvsPatchField, surfaceMesh>& 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
        <typename outerProduct<vector, Type>::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 (http://www.cfd-online.com/Forums/ope...r-schemes.html 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 (http://www.cfd-online.com/Forums/sit...ne-forums.html)

Dan

chegdan · September 28, 2011, 20:05

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:

$\phi_f = (\phi_f)_{UD} + \Psi(r)\left[(\phi_f)_{HO}-(\phi_f)_{UD}\right]$

where $(\phi_f)_{HO}$ 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 ( $\Psi(r)$ ) 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

caramelo · September 29, 2011, 05:48

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.

chegdan · September 29, 2011, 15:37

@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

chegdan · October 1, 2011, 16:51

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

florian_krause · October 3, 2011, 04:45

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

chegdan · October 3, 2011, 13:55

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

yhy20081016 · October 19, 2013, 00:49

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 http://www.cfd-online.com/Forums/ope...r-schemes.html 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 http://www.cfd-online.com/Forums/ope...lllimited.html). This is different than the limiters used in limitedLinear. In Fluent documentation, the definition of second order upwind is given as:

$\phi_f = \phi + \nabla\phi\cdot\vec{r}$

where $\phi_f$ and $\phi$ 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<Type, fvsPatchField, surfaceMesh>& 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
        <typename outerProduct<vector, Type>::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 (http://www.cfd-online.com/Forums/ope...r-schemes.html 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 (http://www.cfd-online.com/Forums/sit...ne-forums.html)

Dan

How can you find out which file contains the source code implementing the 'linearUpwind' scheme? Please teach me. Thank you very much.

chegdan · October 21, 2013, 10:57

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.

yhy20081016 · October 26, 2013, 23:44

Thank you very much.

Anne Lincke · January 10, 2014, 18:51

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 http://www.cfd-online.com/Forums/ope...r-schemes.html 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 http://www.cfd-online.com/Forums/ope...lllimited.html). This is different than the limiters used in limitedLinear. In Fluent documentation, the definition of second order upwind is given as:

$\phi_f = \phi + \nabla\phi\cdot\vec{r}$

where $\phi_f$ and $\phi$ 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

$\phi_f = \phi(C) + 0.5 (\phi(C) - \phi(U))$

How is the implemented formula connected to that?

Thank you very much for an answer in advance!

Anne

Bazinga · April 10, 2015, 04:47

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

$\phi_f = \phi(C) + 0.5 (\phi(C) - \phi(U))$

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:
$\phi(x_e)=\phi(x_P)+\left ( x_e-x_P \right )\left(\frac{\partial \phi}{\partial x} \right)_P+\frac{\left ( x_e-x_P \right )^2}{2!}\left(\frac{\partial^2 \phi}{\partial x^2} \right)_P+ \cdots+h.o.t$

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

$\phi(x_e)=\phi(x_P)$

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:

$\phi(x_e)=\phi(x_P)+\left ( x_e-x_P \right )\left(\frac{\partial \phi}{\partial x} \right)_P$

CDS for the gradient gives:

$\left(\frac{\partial \phi}{\partial x} \right)_P=\frac{\phi(x_E)-\phi(x_P)}{(x_E-x_P)}$

For $2\left ( x_e-x_P \right ) = (x_E-x_P)$

You get:
$\phi(x_e)=\phi(x_P)+\frac{\phi(x_E)-\phi(x_P)}{2}$

That equals your expression:

$\phi_f = \phi(C) + 0.5 (\phi(C) - \phi(U))$

Hope that helps

Anne Lincke · April 10, 2015, 05:54

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

davibarreira · October 10, 2015, 12:52

Isnt the correct way like this?:

$\phi_f=\phi(U)+\psi(\phi(C)-\phi(U))$

Such that, if $\psi=1$ :
$\phi_f=\phi(C)$

If $\psi=0$ :
$\phi_f=\phi(U)$

And if $\psi=0$ , we get the Linear Upwind Differencing (LUD):

$\phi_f=\phi(U)+0.5(\phi(C)-\phi(U))$

Bazinga · October 11, 2015, 03:29

Quote:

Originally Posted by davibarreira

Isnt the correct way like this?:

$\phi_f=\phi(U)+\psi(\phi(C)-\phi(U))$

Such that, if $\psi=1$ :
$\phi_f=\phi(C)$

If $\psi=0$ :
$\phi_f=\phi(U)$

And if $\psi=0$ , we get the Linear Upwind Differencing (LUD):

$\phi_f=\phi(U)+0.5(\phi(C)-\phi(U))$

Just looked through it briefly. I think you are right. Thanks. The last part of my last reply should be:

" $\phi_f = \phi(U) + 0.5 (\phi(C) - \phi(U))$ which does not equal the proposed equations."

davibarreira · October 11, 2015, 11:17

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.
http://www.cfd-online.com/Forums/ope...tedlinear.html
Any thoughts?

Anne Lincke · October 13, 2015, 04:59

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

$u_C= u_f - u_f' \frac d 2 + u_f'' \frac {d^2}{8} - \cdots$
$u_U= u_f - u_f'\frac{3d}2 + u_f'' \frac{d^2 9 }{8} - \cdots$
$u_f= \frac{ 3 u_C - u_U} {2} - HOT,$

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

$u_f = u_C + \frac {u_C - u_U}{2}$

The second term is the explicit correction term.
This equals the implemented formula.

alimea · January 12, 2018, 05:08

Quote:

Originally Posted by Anne Lincke

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

$u_C= u_f - u_f' \frac d 2 + u_f'' \frac {d^2}{8} - \cdots$
$u_U= u_f - u_f'\frac{3d}2 + u_f'' \frac{d^2 9 }{8} - \cdots$
$u_f= \frac{ 3 u_C - u_U} {2} - HOT,$

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

$u_f = u_C + \frac {u_C - u_U}{2}$

The second term is the explicit correction term.
This equals the implemented formula.

Yes, I think what you have written is correct. Because Bazinga mentioned that:

CDS for the gradient gives:

$\left(\frac{\partial \phi}{\partial x} \right)_P=\frac{\phi(x_E)-\phi(x_P)}{(x_E-x_P)}$

However, this isn't CDS, that was upwind. Also he/she used downwind point (E) for upwind scheme. so I think that was wrong

makaveli_lcf · May 8, 2018, 07:04

Hi guys!

I am making currently some testing with limitedLinear(V) schemes, and I totally see no difference between limiter 0..1!

I would expect with limitedLinear 0 to act as linear divergence scheme,am I write?
But the results are totally different from "Gauss linear" scheme.

I would really appreciate your feedback on this topic.

Cheers,
Alex

alimea · May 8, 2018, 08:00

Quote:

Originally Posted by makaveli_lcf

Hi guys!

I am making currently some testing with limitedLinear(V) schemes, and I totally see no difference between limiter 0..1!

I would expect with limitedLinear 0 to act as linear divergence scheme,am I write?
But the results are totally different from "Gauss linear" scheme.

I would really appreciate your feedback on this topic.

Cheers,
Alex

Hi Alexander

I haven't tested this, but theoretically yes. you are right. According to this sentence of cfd.direct:
limitedLinear: linear scheme that limits towards upwind in regions of rapidly changing gradient; requires a coefficient, where 1 is strongest limiting, tending towards linear as the coefficient tends to 0.

https://cfd.direct/openfoam/user-guide/fvschemes/

September 28, 2011, 14:58	A few simple questions about linearUpwind and limitedLinear	#1
chegdan Senior Member Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 621 Rep Power: 0	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 http://www.cfd-online.com/Forums/ope...r-schemes.html 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 http://www.cfd-online.com/Forums/ope...lllimited.html). This is different than the limiters used in limitedLinear. In Fluent documentation, the definition of second order upwind is given as: $\phi_f = \phi + \nabla\phi\cdot\vec{r}$ where $\phi_f$ and $\phi$ 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<Type, fvsPatchField, surfaceMesh>& 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 <typename outerProduct<vector, Type>::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 (http://www.cfd-online.com/Forums/ope...r-schemes.html 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 (http://www.cfd-online.com/Forums/sit...ne-forums.html) Dan mm.abdollahzadeh, zhulianhua, skeptik and 15 others like this. Last edited by chegdan; September 29, 2011 at 14:24.

September 28, 2011, 20:05		#2
chegdan Senior Member Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 621 Rep Power: 0	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: $\phi_f = (\phi_f)_{UD} + \Psi(r)\left[(\phi_f)_{HO}-(\phi_f)_{UD}\right]$ where $(\phi_f)_{HO}$ 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 ( $\Psi(r)$ ) 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 Pagoda, rdbisme, konangsh and 1 others like this.

September 29, 2011, 05:48		#3
caramelo New Member Chris Join Date: Jun 2011 Posts: 12 Rep Power: 15	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. emjay and Pagoda like this.

September 29, 2011, 15:37		#4
chegdan Senior Member Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 621 Rep Power: 0	@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 rdbisme likes this.

October 21, 2013, 10:57		#9
chegdan Senior Member Daniel P. Combest Join Date: Mar 2009 Location: St. Louis, USA Posts: 621 Rep Power: 0	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. makaveli_lcf, kaifu, hua1015 and 3 others like this.

October 26, 2013, 23:44		#10
yhy20081016 Member yehanyu Join Date: Mar 2012 Location: Beijing, China Posts: 48 Rep Power: 14	Thank you very much.

April 10, 2015, 05:54		#13
Anne Lincke Senior Member Anne Gerdes Join Date: Aug 2010 Location: Hamburg Posts: 168 Rep Power: 16	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, 12:52		#14
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	Isnt the correct way like this?: $\phi_f=\phi(U)+\psi(\phi(C)-\phi(U))$ Such that, if $\psi=1$ : $\phi_f=\phi(C)$ If $\psi=0$ : $\phi_f=\phi(U)$ And if $\psi=0$ , we get the Linear Upwind Differencing (LUD): $\phi_f=\phi(U)+0.5(\phi(C)-\phi(U))$

October 11, 2015, 11:17		#16
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	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. http://www.cfd-online.com/Forums/ope...tedlinear.html Any thoughts? minh khang likes this.

October 13, 2015, 04:59		#17
Anne Lincke Senior Member Anne Gerdes Join Date: Aug 2010 Location: Hamburg Posts: 168 Rep Power: 16	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 $u_C= u_f - u_f' \frac d 2 + u_f'' \frac {d^2}{8} - \cdots$ $u_U= u_f - u_f'\frac{3d}2 + u_f'' \frac{d^2 9 }{8} - \cdots$ $u_f= \frac{ 3 u_C - u_U} {2} - HOT,$ 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 $u_f = u_C + \frac {u_C - u_U}{2}$ The second term is the explicit correction term. This equals the implemented formula. alimea likes this.

May 8, 2018, 07:04		#19
makaveli_lcf Senior Member Dr. Alexander Vakhrushev Join Date: Mar 2009 Posts: 256 Blog Entries: 1 Rep Power: 19	Hi guys! I am making currently some testing with limitedLinear(V) schemes, and I totally see no difference between limiter 0..1! I would expect with limitedLinear 0 to act as linear divergence scheme,am I write? But the results are totally different from "Gauss linear" scheme. I would really appreciate your feedback on this topic. Cheers, Alex Tobi likes this. __________________ Best regards, Dr. Alexander VAKHRUSHEV Christian Doppler Laboratory for "Metallurgical Applications of Magnetohydrodynamics" Simulation and Modelling of Metallurgical Processes Department of Metallurgy University of Leoben http://smmp.unileoben.ac.at

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