[mihirmakwana6]

I want to apply a high order say 5th/6th order accurate UPWIND scheme for convective terms ( 2nd term in L.H.S of the momentum equation --- click the link below )

http://i.imgur.com/qDxTpi2.jpg?1

The scheme should be based on FINITE VOLUME METHOD
( as it is applied to momentum equation in conservative form)


Can someone please provide links/name of few research papers


Thanks in Advance.

Cheers !!

[FMDenaro]

I am aware of FV-based fifth order upwind in 1D, however the extension to real multidimensional high order upwind is quite complex...

[mihirmakwana6]

Originally Posted by FMDenaro

I am aware of FV-based fifth order upwind in 1D, however the extension to real multidimensional high order upwind is quite complex...

I am trying to apply FVM-based high order upwind to convective terms in 3 Dimensions.

Are there any papers on this ??

[FMDenaro]

Originally Posted by mihirmakwana6

I am trying to apply FVM-based high order upwind to convective terms in 3 Dimensions.

Are there any papers on this ??

Some authors use upwinding in 3D by factorising the problem in each direction and using 1D schemes.
It also depends on the type of flow, if compressible (possibly with shock) or not. If you search with google there are a lot of papers.

Some years ago I worked with high order upwind discretization both 1D and 3D (multidimensional) but only focusing on incompressible flows.
Maybe you can find some useful details.
https://www.researchgate.net/profile...=newest&page=2

[tt_1990]

http://www.sciencedirect.com/science...21999104002281.
I hope it's useful to you

[mihirmakwana6]

Originally Posted by FMDenaro

Some authors use upwinding in 3D by factorising the problem in each direction and using 1D schemes.
It also depends on the type of flow, if compressible (possibly with shock) or not. If you search with google there are a lot of papers.

Some years ago I worked with high order upwind discretization both 1D and 3D (multidimensional) but only focusing on incompressible flows.
Maybe you can find some useful details.
https://www.researchgate.net/profile...=newest&page=2

ok Thankyou.

Currently the code is incompressible, but later on I will modify it to Compressible.
However, I will be working in the low mach number regime.

[mihirmakwana6]

Originally Posted by tt_1990

http://www.sciencedirect.com/science...21999104002281.
I hope it's useful to you

Thankyou. I will have a look at it.

[FMDenaro]

Originally Posted by mihirmakwana6

ok Thankyou.

Currently the code is incompressible, but later on I will modify it to Compressible.
However, I will be working in the low mach number regime.

Ok, so you need only upwinding in the physical space based on the three velocity components.
I think you can find useful details in this paper (specifically based on multidimensional upwind) and its references.

https://www.researchgate.net/publica...mulation_codes

[mihirmakwana6]

Originally Posted by FMDenaro

Ok, so you need only upwinding in the physical space based on the three velocity components.
I think you can find useful details in this paper (specifically based on multidimensional upwind) and its references.

https://www.researchgate.net/publica...mulation_codes

Sir, the paper uses Collocated grid arrangement whereas I am using staggered grid.

Will the paper still be useful ?

[FMDenaro]

Originally Posted by mihirmakwana6

Sir, the paper uses Collocated grid arrangement whereas I am using staggered grid.

Will the paper still be useful ?

the general idea of the construction of upwinded region for multidimensional interpolation does not depend upon the type of colocation. However, the implementation will be different.
A higher order scheme for multidimensional upwind would really get you in trouble when using staggered colocation.

[mprinkey]

There is a great deal of literature in the the 1990s/2000s regarding LES simulations using compact differencing schemes. This is probably one of the most referenced paper. While it is formulated from a finite difference viewpoint, there is an appendix that discusses mid-point interpolation as you would need for a finite volume formulation. And for cartesian methods, Finite Difference and Finite Volume methods are more alike than different.

http://www.math.colostate.edu/~yzhou...le_1992JCP.pdf

As a general note--all of these schemes will lock you into a fully structured Cartesian mesh. Even grid spacing variations are difficult to properly address. And grid transformations must be orthogonal or nearly so to prevent missing higher cross terms from destroying the accuracy. Also, there are no TVD or other smoothness guarantees with these schemes, so you must make sure that your underlying solution is either smooth or that your model has enough artificial damping to keep the wiggles under control near sharp gradients. If you need to properly address discontinuous solutions with high order systems, take a look at WENO schemes. There are versions that are 5th-order or higher.

http://www.enu.kz/repository/2009/AIAA-2009-1612.pdf

If you are concerned with simple Cartesian Control Volume Finite Difference-type formulation, these compact or WENO schemes are useful ways to proceed, in my view. If, however, you need a fully unstructured Finite Volume-type formation that can handle tri/quad/tet/hex meshes, you need to look at Discontinuous Galerkin methods to the exclusion of everything else:

http://www.cfm.brown.edu/people/jans...s/RMMC08-I.pdf

General finite volume methods are very difficult to formulate on unstructured grids. You can see this by just reviewing basic WENO methods formulated for unstructured meshes in 2D and those are not even 5-th order:

http://www3.nd.edu/~yzhang10/RobustWENO.pdf

A great deal of formulation convenience goes away if you are unable to treat interpolation/gradients on an per-axis basis, and this difficulty is ever more amplified as you increase the accuracy order.

Good luck.

[mihirmakwana6]

Originally Posted by mprinkey

There is a great deal of literature in the the 1990s/2000s regarding LES simulations using compact differencing schemes. This is probably one of the most referenced paper. While it is formulated from a finite difference viewpoint, there is an appendix that discusses mid-point interpolation as you would need for a finite volume formulation. And for cartesian methods, Finite Difference and Finite Volume methods are more alike than different.

http://www.math.colostate.edu/~yzhou...le_1992JCP.pdf

As a general note--all of these schemes will lock you into a fully structured Cartesian mesh. Even grid spacing variations are difficult to properly address. And grid transformations must be orthogonal or nearly so to prevent missing higher cross terms from destroying the accuracy. Also, there are no TVD or other smoothness guarantees with these schemes, so you must make sure that your underlying solution is either smooth or that your model has enough artificial damping to keep the wiggles under control near sharp gradients. If you need to properly address discontinuous solutions with high order systems, take a look at WENO schemes. There are versions that are 5th-order or higher.

http://www.enu.kz/repository/2009/AIAA-2009-1612.pdf

If you are concerned with simple Cartesian Control Volume Finite Difference-type formulation, these compact or WENO schemes are useful ways to proceed, in my view. If, however, you need a fully unstructured Finite Volume-type formation that can handle tri/quad/tet/hex meshes, you need to look at Discontinuous Galerkin methods to the exclusion of everything else:

http://www.cfm.brown.edu/people/jans...s/RMMC08-I.pdf

General finite volume methods are very difficult to formulate on unstructured grids. You can see this by just reviewing basic WENO methods formulated for unstructured meshes in 2D and those are not even 5-th order:

http://www3.nd.edu/~yzhang10/RobustWENO.pdf

A great deal of formulation convenience goes away if you are unable to treat interpolation/gradients on an per-axis basis, and this difficulty is ever more amplified as you increase the accuracy order.

Good luck.

Thankyou sir for the reply.

I am using hexahedron C.V ( thus grid lines are along the Cartesian axis ) and the meshing is uniform ( with staggered grid arrangement ) in a particular Cartesian direction.


Also, the solution in the domain is smooth as there are no sharp gradients.

[FMDenaro]

Hi to all,

while I agree on the difficulty of implementing a code on unstructured grids, let me address some suggestions:

- hybrid FV/FE are best suited on unstructured. That means you can define shape functions in a FE manner but using that for computing any gradient and integral required for a FV method.
- when FV are correctly based on the discretization of the integral form of the equation, you do not get the same algebric system of equation of the FD method, neither on regular structured grid. The difference comes from the discretization of the Div(*) operator in FD methods and in (1/|V|) Int [S] n.() dS in FV methods

[mihirmakwana6]

Originally Posted by mprinkey

There is a great deal of literature in the the 1990s/2000s regarding LES simulations using compact differencing schemes. This is probably one of the most referenced paper. While it is formulated from a finite difference viewpoint, there is an appendix that discusses mid-point interpolation as you would need for a finite volume formulation. And for cartesian methods, Finite Difference and Finite Volume methods are more alike than different.

http://www.math.colostate.edu/~yzhou...le_1992JCP.pdf

As a general note--all of these schemes will lock you into a fully structured Cartesian mesh. Even grid spacing variations are difficult to properly address. And grid transformations must be orthogonal or nearly so to prevent missing higher cross terms from destroying the accuracy. Also, there are no TVD or other smoothness guarantees with these schemes, so you must make sure that your underlying solution is either smooth or that your model has enough artificial damping to keep the wiggles under control near sharp gradients. If you need to properly address discontinuous solutions with high order systems, take a look at WENO schemes. There are versions that are 5th-order or higher.

http://www.enu.kz/repository/2009/AIAA-2009-1612.pdf

If you are concerned with simple Cartesian Control Volume Finite Difference-type formulation, these compact or WENO schemes are useful ways to proceed, in my view. If, however, you need a fully unstructured Finite Volume-type formation that can handle tri/quad/tet/hex meshes, you need to look at Discontinuous Galerkin methods to the exclusion of everything else:

http://www.cfm.brown.edu/people/jans...s/RMMC08-I.pdf

General finite volume methods are very difficult to formulate on unstructured grids. You can see this by just reviewing basic WENO methods formulated for unstructured meshes in 2D and those are not even 5-th order:

http://www3.nd.edu/~yzhang10/RobustWENO.pdf

A great deal of formulation convenience goes away if you are unable to treat interpolation/gradients on an per-axis basis, and this difficulty is ever more amplified as you increase the accuracy order.

Good luck.

@Michael Prinkey

Sir, the first paper ( Compact finite difference schemes with spectral-like resolution by Sanjiva K. Lele ),
actually uses Central difference schemes.

I am actually looking for UPWIND schemes

[mprinkey]

Originally Posted by mihirmakwana6

@Michael Prinkey

Sir, the first paper ( Compact finite difference schemes with spectral-like resolution by Sanjiva K. Lele ),
actually uses Central difference schemes.

I am actually looking for UPWIND schemes

Also, very common:

http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf

[mihirmakwana6]

I found the following paper

http://www.tandfonline.com/doi/pdf/1...07799308955882

See equation 17 where a fifth-order upwind scheme called FOUB scheme is described.

Can I use this scheme for convective terms ?

[FMDenaro]

Originally Posted by mihirmakwana6

I found the following paper

http://www.tandfonline.com/doi/pdf/1...07799308955882

See equation 17 where a fifth-order upwind scheme called FOUB scheme is described.

Can I use this scheme for convective terms ?

at present, I am not able to access to T&F papers

[mihirmakwana6]

Originally Posted by FMDenaro

at present, I am not able to access to T&F papers

The paper is

THREE-DIMENSIONAL INCOMPRESSIBLE FLOW CALCULATIONS WITH ALTERNATIVE DISCRETIZATION SCHEMES

by

Panos Tamamidis & Dennis N. Assanis
( I can mail you the paper on your email-id if that is fine with you )



Here is the image where they have mentioned the fifth order upwind scheme

http://i.imgur.com/GlFRysZ.jpg?1


The paper to which they refer i.e Rai[13] is

http://i.imgur.com/Z8cEyLf.jpg?1

to which i don't have access

[mprinkey]

Originally Posted by mihirmakwana6

The paper is

THREE-DIMENSIONAL INCOMPRESSIBLE FLOW CALCULATIONS WITH ALTERNATIVE DISCRETIZATION SCHEMES

by

Panos Tamamidis & Dennis N. Assanis
( I can mail you the paper on your email-id if that is fine with you )



Here is the image where they have mentioned the fifth order upwind scheme

http://i.imgur.com/GlFRysZ.jpg?1


The paper to which they refer i.e Rai[13] is

http://i.imgur.com/Z8cEyLf.jpg?1

to which i don't have access

That formula will likely work fine. That is just a midpoint interpolation with three upwind and two downwind samples. The difficulty here will be in implementing boundary conditions--either dropping back to lower order or building lopsided stencils that retain 5th order.

Note too that this form is highly accurate, but it is "not as upwind" as QUICK or other more stable upwinding schemes that make more heavy use of upwind data. So, I suspect that solution stability is going to more tenuous.

[FMDenaro]

well, I see the eq.(17) and let me comment a couple of issues:

1) the formula is indeed an upwind for the scalar value on a face, however the upwinding on the non linear flux (uu) would be different;

2) the formula is 1D, that means you do not see in it the surface integral that is conversely in effect in 2D and 3D cases. If you use the Eq.(17) as it is, is likely you get simply a second order accuracy in space in an accuracy testing.

Biased and pure upwinding in 1D were analysed in several papers. When I worked on this issue, we analysed the distribution of the errors in physical and spectral space for FD and FV approaches. If you are interested, you can see Section 4 in https://www.researchgate.net/publica...mes_simulation

you will find some "strange" conclusions ...