TVD schemes, questions about limitedLinear

davibarreira · October 10, 2015, 14:09

I've been reading about the numerical schemes in OpenFOAM, and I found (perhaps) some divergence concerning the TVD schemes and Sweby limiters.

In Dr. Jasak's thesis (pg 98), it says that:
$\phi_f=\phi_{UD}+\psi(\phi_{HO}-\phi_{UD})$

So for ψ = 1 we get a Higher Order scheme (central difference), while ψ = 0 we get Upwind.

The thing is, at OpenFOAM's website (http://cfd.direct/openfoam/user-guid...20-1140004.4.5), it is said that "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". But doesn't that mean that for ψ = 0 we get the higher order (more accurate) and ψ = 1 we get less accurate (Upwind)? And isn't it the opposite of what is written in Dr. Jasak's thesis?

I expected that limitedLinear 0.5 would be the same as linearUpwind grad(U), but when I run the same case for these two schemes, the results are different.

Also, as stated in this thread (http://www.cfd-online.com/Forums/ope...tedlinear.html), I thought that limitedLinear 0 would be equal to upwind and limitedLinear 1 would be equal to linear, but it is not the case.

Did I understand thing in a wrong way, or is there an inconsistency? Also, should these schemes be indeed the same or not?

Thanks in advance.

Btw, sorry if this is posted in the wrong section of the forum. I saw that another similar thread was posted in this section, so I did the same.

Bazinga · October 12, 2015, 09:12

Quote:

Originally Posted by davibarreira

I've been reading about the numerical schemes in OpenFOAM, and I found (perhaps) some divergence concerning the TVD schemes and Sweby limiters.

In Dr. Jasak's thesis (pg 98), it says that:
$\phi_f=\phi_{UD}+\psi(\phi_{HO}-\phi_{UD})$

So for ψ = 1 we get a Higher Order scheme (central difference), while ψ = 0 we get Upwind.

I can not really tell how it is implemented in OpenFOAM but I will try to have a look at it if I find the time.

If we use the equation of Jasak
$\phi_f=\phi_{UD}+\psi(\phi_{HO}-\phi_{UD})$
We get Upwind for $\psi= 0$ but we would not get the CDS for a value of $\psi= 1$ . I think we would need a $\psi= 0.5$ to get CDS using Jasak's equation.

In Versteeg and Malalasekra you will also find a section about TVD Schemes. Here the equation is:
$\phi_e=\phi_{P}+\frac{1}{2}\psi(\phi_{E}-\phi_{P})$
Now, we would get a CDS for $\psi= 1$ .

davibarreira · October 12, 2015, 09:28

Indeed, I just checked Versteeg and Malalasekra, and it's like you said. Thanks for the correction.
Although, the problem remains, cause using limitedLinear 0 and limitedLinear 1, none of them is equal to Upwind (when comparing the results of the simulations).

Bazinga · October 13, 2015, 09:21

Just a bit more information and a correction. Jasak's equation is actually the same as Versteeg's equation. I understood Jasak's naming convention wrong in my last reply:

Jasak:
$\phi_f=\phi_{UD}+\psi(\phi_{HO}-\phi_{UD})$
Using Versteeg's naming convention and assuming that we take CDS for as the higher-order scheme (HO) we get:
$\phi_f = \phi_e \; and \; \phi_{UD}=\phi_P \; and \; \phi_{HO} = \frac{\phi_P+ \phi_E}{2}$
If you insert these in the first equation you get exactly Versteeg's equation:
$\phi_e=\phi_{P}+\frac{1}{2}\psi(\phi_{E}-\phi_{P})$

Now my guess is that due to the implementation of the schemes in OpenFOAM the $\psi$ that the user can choose is different from the $\psi$ in the equations above. Thus making for example limitedLinear 0 not equal Upwind.

davibarreira · October 13, 2015, 11:38

That's also what I thought, since the simulation results were not matching. I tried to take a look at the code, but couldn't understand it. The thing is, I'm using limitedLinear 1 for the convection term, but I dont know how appropriate it is, since I couldn't figure out the proper implementation of the limitedLinear scheme.

jurich · October 14, 2015, 16:11

limitedLinear is used in fvSchemes as Gauss limitedLinear k

The value k you specify when activating the scheme is not the same as $\psi$ , as used in Jasak's thesis.

What you get from limitedLinear is $\psi(r) = r * \frac{2}{k}$ , with r as defined in Jasak's thesis, and $\psi$ limited between 0 and 1.

See the discussion: http://www.cfd-online.com/Forums/ope...ar-scheme.html, and lines 76 and 95 of limitedLinear.H

So if you use k = 0, $\psi$ will be fixed at 1.0, and you get the higher order scheme everywhere, in this case central differencing.

If you use k = 1, you get a limiter of $\psi(r) = 2 * r$ applied. This limiter is the edge of Sweby's TVD region; it is TVD conforming.

That's why the documentation describes it that way.

davibarreira · October 14, 2015, 19:52

Thank you, Joseph. I read the thread and the code, and it is indeed as you pointed out. The only thing that I still dont get is why using Gauss linear is not giving me the same result as Gauss linearLimited 0, since both should be central diff.

I will keep on digging, but thanks again for your info.

cabrera · October 18, 2015, 16:52

Hello Joseph and Davi,

I am also working with the limitedLinear scheme. I would be grateful if you could tell me if you agree with the following interpretation of the code.

The limiter \Psi is defined in limitedLinear.H to have the following limits (line 95)

0<= \Psi = 2r/k <=1

where 0<=k<=1 is introduced by the user. Therefore, the limitedLinear scheme is a TVD scheme ONLY when the user chose k=1 (\Psi = 2r). When the user choose whatever other value for k, the scheme does not full-fill the TVD condition, because then \Psi > 2r.

Then for k=1 the TVD condition is full-filled and stability is "assured" (within the limitations of the TVD schemes), but the accuracy depends on r. As r is a function of the gradients of \Psi it represents how convective the flow is. The more convection, the more upwinding is introduced and the accuracy decreases. For pure diffusive flows r=1 and the CD scheme is used, still fulfilling the TVD condition because k=1.

Do you agree?

Thank you very much!

Ana

Information about the TVD scheme can be found at Hrvoje Jasak PhD thesis.

Regions for a TVD second order scheme:

davibarreira · October 19, 2015, 13:08

Yes, Ana. I have pretty much the same interpretation.

jurich · October 20, 2015, 10:04

Hello Ana,

I agree with your interpretation on TVD as well.

However, r is calculated as the ratio of upwind gradient to downwind gradient. Scalar version is given starting at line 95 of NVDTVD.H. It divides the gradient at cell center projected to the face (d & gradcP) by the difference between cell value and neighbor value of phi ( phiN - phiP ). So it is calculated as a function of the local values of phi only.

I don't think that's quite the same as your interpretation of diffusion vs. convective gradients.

msuaeronautics · October 22, 2015, 16:24

For what it's worth, I've studied the limitedLinear scheme in some detail for my Master's work. Important to me because it is my divergence scheme of choice for most OF simulations. I have some figures showing the behavior of several k values on a pure scalar convection problem.

Quote:

Originally Posted by davibarreira

The only thing that I still dont get is why using Gauss linear is not giving me the same result as Gauss linearLimited 0, since both should be central diff.

I see this reflected in my results, where setting k to zero or a really small value still does not recover central differencing. Maybe I am missing something in the explanations given here that would reveal why this is. If anyone is interested in my results, I can dig through some figures and post them.

davibarreira · October 22, 2015, 16:42

Hey Eric, could you send a link to your Master thesis, so I can understand this scheme better.

msuaeronautics · October 23, 2015, 10:22

Quote:

Originally Posted by davibarreira

Hey Eric, could you send a link to your Master thesis, so I can understand this scheme better.

Here is the link.

http://pqdtopen.proquest.com/doc/171...5.html?FMT=ABS

The scheme is only a minor part of the entire document, so I have included some supplemental figures as an attachment to this post, showing the setup of the test as well as some figures. I tested the convection of a step and a Gaussian profile. In the Gaussian, you can see that smaller values of k don't necessarily recover the original central difference (my nomenclature calls this UBCD).

I finished my Master's degree before I could dig any further. I believe I have also written a program to output the value of the Phi(r) limiter at every point based on its definition in the source code, which I will try to recover.

davibarreira · October 27, 2015, 19:57

Great stuff, Eric. Havent finished, but it's already helping me.

msuaeronautics · October 28, 2015, 11:31

Quote:

Originally Posted by davibarreira

Great stuff, Eric. Havent finished, but it's already helping me.

Good to know. If you have any questions, don't hesitate to send me a message.

vnarayana · November 5, 2015, 08:03

Hello every one,
how to apply TVD/NVD schmes for divergence schmes .
i tried the following.
in fvSchmes subfolder changed
divSchemes
{
default none;
div(phi,U) Gauss NVD schemes;
}
when i am running by using the command icoFoam i got the following error
narayana@narayana-Lenovo-G50-80 ~/OpenFOAM/narayana-2.4.0/run/tutorials/incompressible/icoFoam/cavity_nvd $ icoFoam/*---------------------------------------------------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: 2.4.0 |
| \\ / A nd | Web: www.OpenFOAM.org |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
Build : 2.4.0-f0842aea0e77
Exec : icoFoam
Date : Nov 06 2015
Time : 05:55:30
Host : "narayana-Lenovo-G50-80"
PID : 5304
Case : /home/narayana/OpenFOAM/narayana-2.4.0/run/tutorials/incompressible/icoFoam/cavity_nvd
nProcs : 1
sigFpe : Enabling floating point exception trapping (FOAM_SIGFPE).
fileModificationChecking : Monitoring run-time modified files using timeStampMaster
allowSystemOperations : Allowing user-supplied system call operations

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
Create time

Create mesh for time = 0

Reading transportProperties

Reading field p

Reading field U

Reading/calculating face flux field phi

Starting time loop

Time = 0.0025

Courant Number mean: 0 max: 0

--> FOAM FATAL IO ERROR:
Unknown discretisation scheme TVD

Valid schemes are :

60
(
CoBlended
Gamma
GammaV
LUST
MUSCL
MUSCLV
Minmod
MinmodV
OSPRE
OSPREV
Phi
QUICK
QUICKV
SFCD
SFCDV
SuperBee
SuperBeeV
UMIST
UMISTV
biLinearFit
blended
clippedLinear
cubic
cubicUpwindFit
downwind
filteredLinear
filteredLinear2
filteredLinear2V
filteredLinear3
filteredLinear3V
fixedBlended
limitWith
limitedCubic
limitedCubicV
limitedLinear
limitedLinearV
limiterBlended
linear
linearFit
linearPureUpwindFit
linearUpwind
linearUpwindV
localBlended
localMax
localMin
midPoint
outletStabilised
pointLinear
quadraticFit
quadraticLinearFit
quadraticLinearUpwindFit
quadraticUpwindFit
reverseLinear
skewCorrected
upwind
vanAlbada
vanAlbadaV
vanLeer
vanLeerV
weighted
)

file: /home/narayana/OpenFOAM/narayana-2.4.0/run/tutorials/incompressible/icoFoam/cavity_nvd/system/fvSchemes.divSchemes.div(phi,U) at line 32.

From function surfaceInterpolationScheme<Type>::New(const fvMesh&, const surfaceScalarField&, Istream&)
in file lnInclude/surfaceInterpolationScheme.C at line 137.

FOAM exiting
I got answers for upwind,quikc schmes but not TVD/NVD i tried but i could not get it.
please help me .

davibarreira · November 5, 2015, 08:09

As the error message says, the way you are writing the scheme is wrong. As you must know, there are several TVD schemes (TVD is more like a class of schemes), so you must choose one. Some of them are the vanLeer, UMIDST, SuperBee... The one I'm using is the limitedLinearV 1, which is discussed in this thread. But you may try the vanLeer, which is better documented.

P.S: when you are using the divScheme of vectors, you might want to use vanLeerV, intead of vanLeer.

vnarayana · November 5, 2015, 13:32

Thank you very much. UMIDST not working.

davibarreira · November 5, 2015, 14:22

Sorry, UMIST.

vnarayana · November 6, 2015, 02:17

Thank you Davi Barreira,
In NVD schemes, other than SFCD is there any other methods. Gamma and GammaV not working?
would you please send theory on TVD/NVD schemes .
where to get each method notes ? , how works in openfoam?
Thanks & Regards
narayan

October 10, 2015, 14:09	TVD schemes, questions about limitedLinear	#1
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	I've been reading about the numerical schemes in OpenFOAM, and I found (perhaps) some divergence concerning the TVD schemes and Sweby limiters. In Dr. Jasak's thesis (pg 98), it says that: $\phi_f=\phi_{UD}+\psi(\phi_{HO}-\phi_{UD})$ So for ψ = 1 we get a Higher Order scheme (central difference), while ψ = 0 we get Upwind. The thing is, at OpenFOAM's website (http://cfd.direct/openfoam/user-guid...20-1140004.4.5), it is said that "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". But doesn't that mean that for ψ = 0 we get the higher order (more accurate) and ψ = 1 we get less accurate (Upwind)? And isn't it the opposite of what is written in Dr. Jasak's thesis? I expected that limitedLinear 0.5 would be the same as linearUpwind grad(U), but when I run the same case for these two schemes, the results are different. Also, as stated in this thread (http://www.cfd-online.com/Forums/ope...tedlinear.html), I thought that limitedLinear 0 would be equal to upwind and limitedLinear 1 would be equal to linear, but it is not the case. Did I understand thing in a wrong way, or is there an inconsistency? Also, should these schemes be indeed the same or not? Thanks in advance. Btw, sorry if this is posted in the wrong section of the forum. I saw that another similar thread was posted in this section, so I did the same. Last edited by davibarreira; October 11, 2015 at 10:15.

October 13, 2015, 09:21		#4
Bazinga Senior Member Join Date: Jun 2012 Location: Germany, Bochum Posts: 230 Rep Power: 15	Just a bit more information and a correction. Jasak's equation is actually the same as Versteeg's equation. I understood Jasak's naming convention wrong in my last reply: Jasak: $\phi_f=\phi_{UD}+\psi(\phi_{HO}-\phi_{UD})$ Using Versteeg's naming convention and assuming that we take CDS for as the higher-order scheme (HO) we get: $\phi_f = \phi_e \; and \; \phi_{UD}=\phi_P \; and \; \phi_{HO} = \frac{\phi_P+ \phi_E}{2}$ If you insert these in the first equation you get exactly Versteeg's equation: $\phi_e=\phi_{P}+\frac{1}{2}\psi(\phi_{E}-\phi_{P})$ Now my guess is that due to the implementation of the schemes in OpenFOAM the $\psi$ that the user can choose is different from the $\psi$ in the equations above. Thus making for example limitedLinear 0 not equal Upwind. davibarreira likes this. Last edited by Bazinga; October 13, 2015 at 12:36.

October 14, 2015, 16:11		#6
jurich New Member Joseph Urich Join Date: Mar 2009 Location: Pittsburgh, PA Posts: 21 Rep Power: 17	limitedLinear is used in fvSchemes as Gauss limitedLinear k The value k you specify when activating the scheme is not the same as $\psi$ , as used in Jasak's thesis. What you get from limitedLinear is $\psi(r) = r * \frac{2}{k}$ , with r as defined in Jasak's thesis, and $\psi$ limited between 0 and 1. See the discussion: http://www.cfd-online.com/Forums/ope...ar-scheme.html, and lines 76 and 95 of limitedLinear.H So if you use k = 0, $\psi$ will be fixed at 1.0, and you get the higher order scheme everywhere, in this case central differencing. If you use k = 1, you get a limiter of $\psi(r) = 2 * r$ applied. This limiter is the edge of Sweby's TVD region; it is TVD conforming. That's why the documentation describes it that way. JonW, tiam, Bazinga and 14 others like this.

October 19, 2015, 13:08		#9
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	Yes, Ana. I have pretty much the same interpretation. cabrera likes this.

October 20, 2015, 10:04		#10
jurich New Member Joseph Urich Join Date: Mar 2009 Location: Pittsburgh, PA Posts: 21 Rep Power: 17	Hello Ana, I agree with your interpretation on TVD as well. However, r is calculated as the ratio of upwind gradient to downwind gradient. Scalar version is given starting at line 95 of NVDTVD.H. It divides the gradient at cell center projected to the face (d & gradcP) by the difference between cell value and neighbor value of phi ( phiN - phiP ). So it is calculated as a function of the local values of phi only. I don't think that's quite the same as your interpretation of diffusion vs. convective gradients. cabrera likes this.

October 12, 2015, 09:28		#3
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	Indeed, I just checked Versteeg and Malalasekra, and it's like you said. Thanks for the correction. Although, the problem remains, cause using limitedLinear 0 and limitedLinear 1, none of them is equal to Upwind (when comparing the results of the simulations).

October 13, 2015, 11:38		#5
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	That's also what I thought, since the simulation results were not matching. I tried to take a look at the code, but couldn't understand it. The thing is, I'm using limitedLinear 1 for the convection term, but I dont know how appropriate it is, since I couldn't figure out the proper implementation of the limitedLinear scheme.

October 14, 2015, 19:52		#7
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	Thank you, Joseph. I read the thread and the code, and it is indeed as you pointed out. The only thing that I still dont get is why using Gauss linear is not giving me the same result as Gauss linearLimited 0, since both should be central diff. I will keep on digging, but thanks again for your info.

October 18, 2015, 16:52		#8
cabrera New Member Join Date: Jun 2015 Posts: 4 Rep Power: 10	Hello Joseph and Davi, I am also working with the limitedLinear scheme. I would be grateful if you could tell me if you agree with the following interpretation of the code. The limiter \Psi is defined in limitedLinear.H to have the following limits (line 95) 0<= \Psi = 2r/k <=1 where 0<=k<=1 is introduced by the user. Therefore, the limitedLinear scheme is a TVD scheme ONLY when the user chose k=1 (\Psi = 2r). When the user choose whatever other value for k, the scheme does not full-fill the TVD condition, because then \Psi > 2r. Then for k=1 the TVD condition is full-filled and stability is "assured" (within the limitations of the TVD schemes), but the accuracy depends on r. As r is a function of the gradients of \Psi it represents how convective the flow is. The more convection, the more upwinding is introduced and the accuracy decreases. For pure diffusive flows r=1 and the CD scheme is used, still fulfilling the TVD condition because k=1. Do you agree? Thank you very much! Ana Information about the TVD scheme can be found at Hrvoje Jasak PhD thesis. Regions for a TVD second order scheme:

October 22, 2015, 16:42		#12
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	Hey Eric, could you send a link to your Master thesis, so I can understand this scheme better.

October 27, 2015, 19:57		#14
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	Great stuff, Eric. Havent finished, but it's already helping me. msuaeronautics likes this.

November 5, 2015, 08:03	for a lid driven cavity, how to use tvd/nvd schemes for divergence schemes	#16
vnarayana New Member narayana vekamulla Join Date: Oct 2015 Posts: 15 Rep Power: 10	Hello every one, how to apply TVD/NVD schmes for divergence schmes . i tried the following. in fvSchmes subfolder changed divSchemes { default none; div(phi,U) Gauss NVD schemes; } when i am running by using the command icoFoam i got the following error narayana@narayana-Lenovo-G50-80 ~/OpenFOAM/narayana-2.4.0/run/tutorials/incompressible/icoFoam/cavity_nvd $ icoFoam/---------------------------------------------------------------------------\ \| ========= \| \| \| \\ / F ield \| OpenFOAM: The Open Source CFD Toolbox \| \| \\ / O peration \| Version: 2.4.0 \| \| \\ / A nd \| Web: www.OpenFOAM.org \| \| \\/ M anipulation \| \| \---------------------------------------------------------------------------/ Build : 2.4.0-f0842aea0e77 Exec : icoFoam Date : Nov 06 2015 Time : 05:55:30 Host : "narayana-Lenovo-G50-80" PID : 5304 Case : /home/narayana/OpenFOAM/narayana-2.4.0/run/tutorials/incompressible/icoFoam/cavity_nvd nProcs : 1 sigFpe : Enabling floating point exception trapping (FOAM_SIGFPE). fileModificationChecking : Monitoring run-time modified files using timeStampMaster allowSystemOperations : Allowing user-supplied system call operations // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // Create time Create mesh for time = 0 Reading transportProperties Reading field p Reading field U Reading/calculating face flux field phi Starting time loop Time = 0.0025 Courant Number mean: 0 max: 0 --> FOAM FATAL IO ERROR: Unknown discretisation scheme TVD Valid schemes are : 60 ( CoBlended Gamma GammaV LUST MUSCL MUSCLV Minmod MinmodV OSPRE OSPREV Phi QUICK QUICKV SFCD SFCDV SuperBee SuperBeeV UMIST UMISTV biLinearFit blended clippedLinear cubic cubicUpwindFit downwind filteredLinear filteredLinear2 filteredLinear2V filteredLinear3 filteredLinear3V fixedBlended limitWith limitedCubic limitedCubicV limitedLinear limitedLinearV limiterBlended linear linearFit linearPureUpwindFit linearUpwind linearUpwindV localBlended localMax localMin midPoint outletStabilised pointLinear quadraticFit quadraticLinearFit quadraticLinearUpwindFit quadraticUpwindFit reverseLinear skewCorrected upwind vanAlbada vanAlbadaV vanLeer vanLeerV weighted ) file: /home/narayana/OpenFOAM/narayana-2.4.0/run/tutorials/incompressible/icoFoam/cavity_nvd/system/fvSchemes.divSchemes.div(phi,U) at line 32. From function surfaceInterpolationScheme<Type>::New(const fvMesh&, const surfaceScalarField&, Istream&) in file lnInclude/surfaceInterpolationScheme.C at line 137. FOAM exiting I got answers for upwind,quikc schmes but not TVD/NVD i tried but i could not get it. please help me .

November 5, 2015, 08:09		#17
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	As the error message says, the way you are writing the scheme is wrong. As you must know, there are several TVD schemes (TVD is more like a class of schemes), so you must choose one. Some of them are the vanLeer, UMIDST, SuperBee... The one I'm using is the limitedLinearV 1, which is discussed in this thread. But you may try the vanLeer, which is better documented. P.S: when you are using the divScheme of vectors, you might want to use vanLeerV, intead of vanLeer. vnarayana likes this.

November 5, 2015, 13:32	for a lid driven cavity, how to use tvd/nvd schemes for divergence schemes	#18
vnarayana New Member narayana vekamulla Join Date: Oct 2015 Posts: 15 Rep Power: 10	Thank you very much. UMIDST not working.

November 5, 2015, 14:22		#19
davibarreira Member Davi Barreira Join Date: Apr 2014 Location: Fortaleza Posts: 76 Rep Power: 12	Sorry, UMIST.

November 6, 2015, 02:17	for a lid driven cavity, how to use tvd/nvd schemes for divergence schemes	#20
vnarayana New Member narayana vekamulla Join Date: Oct 2015 Posts: 15 Rep Power: 10	Thank you Davi Barreira, In NVD schemes, other than SFCD is there any other methods. Gamma and GammaV not working? would you please send theory on TVD/NVD schemes . where to get each method notes ? , how works in openfoam? Thanks & Regards narayan

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