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transformations for finite difference ---why?

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,

 Thomas P. Abraham May 26, 2002 10:18

Re: transformations for finite difference ---why?

The transformation is used even when using FVM. Don't you think that it is difficult to implement a numerical method on a non-uniform 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

Re: transformations for finite difference ---why?

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 .

 Thomas P. Abraham May 27, 2002 15:21

Re: transformations for finite difference ---why?

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 co-ordinate transformation.

Thanks,

Thomas

 greg May 27, 2002 20:39

Re: transformations for finite difference ---why?

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 non-uniform 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 non-uniform differences if you wish, but developmentally it is harder. The 2-vol 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.

 Paul May 27, 2002 23:55

Re: transformations for finite difference ---why?

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.

 krishan May 28, 2002 05:24

Re: transformations for finite difference ---why?

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.

 Axel Rohde May 28, 2002 08:00

Re: transformations for finite difference ---why?

 Tony May 28, 2002 17:46

Re: transformations for finite difference ---why?

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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