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May 25, 2002, 11:47 
transformations for finite difference why?

#1 
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generally we need to transform the grid and constitue a computational plane when using the finite difference methods . this computational plane is a uniform grid on which we solve the modified equations(modified by the transformation) .A point to point correspondence exists between computational plane and physical domain.
whereas for finite volume methods we can directly solve on the non uniform mesh. we dont need any computational plane. It is said that this is because of inherent nature of techniques. Can anyone explain this in more detail . or can you tell me any good refernce which discusses this aspect. regards, Adarsh 

May 26, 2002, 10:18 
Re: transformations for finite difference why?

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Hello Adarsh,
The transformation is used even when using FVM. Don't you think that it is difficult to implement a numerical method on a nonuniform mesh directly? If you work on a irregular cell directly, you need to compute differently for different cells. The advantage with the transformation is that each physical cell is mapped onto a regular computational cell. A Jacobian relates a physical cell and a computational cell. That way, the methodology is the same for every cell because the computations are on a regular cell. Thanks, Thomas 

May 27, 2002, 10:51 
Re: transformations for finite difference why?

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thanks for reply Thomas .... but i have read in the CFD book by anderson that "FVM can be applied to irregular mesh and hence transformations are needed only for finite difference schemes
transformations in FVM are used only for grid generation for physical plane Can u tell me any other book which gives some good concept on it . regards, Adarsh 

May 27, 2002, 15:21 
Re: transformations for finite difference why?

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Hello Adarsh,
Most of the commercial codes based by some variation of SIMPLE whether it's from FLUENT, STARCD or CFDRC, they all use FVM. They all use coordinate transformation. Thanks, Thomas 

May 27, 2002, 20:39 
Re: transformations for finite difference why?

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You are correct that finite volume schemes do not necessarily use coordinate transformations, but can deal with the physical grid points. However, the transformational equivalents show up in terms of computational work through the volumes and areas that have to be computed. In finite difference schemes coordinate transformations are used to give unit divisors in the finite differences. A code can be developed without resort to nonuniform difference approximations, which some claim gives a uniform error behavior (although the absolute error will still depend on the grid geometry to some extent). You can develop finite difference algorithms using nonuniform differences if you wish, but developmentally it is harder. The 2vol set by Hirsch is a good reference, as is the latest edition of Anderson, Tannehill, and Pletcher. Although I cannot remember the citation (and the paper is at work) Marcel Vinokur published a paper in the 1980s (?) that compared and contrasted finite difference and finite volume approaches and demonstrated the "general" equivalence between them.


May 27, 2002, 23:55 
Re: transformations for finite difference why?

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Generally, the basic difference is that FVM solves intergal version of the NSe while FDM solves the original differential form of the NSe. To well define a difference numerically, we usually have to do in a stright line, this is why we have to transform a irregular domain into rectangular domain where stright line can be readily defined. For FVM, no matter the element is regular or irregualr, one can always intergate the NSe with some efforts (for irregular element, one has much more geometric considerations) and then to assemble a linear system. FEM also deals with integral form of the NSe ( but in some different fashion). Therefore FVM and FEM are more flexiable for complex geometry.


May 28, 2002, 05:24 
Re: transformations for finite difference why?

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why then FEM is rarely used .. and FVM is most commonly used ..whats the major difference between the two... FEM and FVM. this discussion is going nicely.


May 28, 2002, 08:00 
Re: transformations for finite difference why?

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For an earlier discussion on FEM, FVM, FDM, see also,
www.cfdonline.com/Forum/main_archive_2001.cgi?read=18691 

May 28, 2002, 17:46 
Re: transformations for finite difference why?

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FDM, FVM and FEM all need transformation (mapping) from the physical plane to the computational one when dealing with complex geometries. However, the FDM requires a global mapping (say boundary fitted) while the FVM and FEM uses a local one for each cell (element). Of cause, the local mapping is much easier. Moreover, a global mapping may not exist for some tough configurations.


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