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FVM vs. FDM Advantages?

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Old   June 28, 2000, 09:15
Default FVM vs. FDM Advantages?
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What are the advantages of using a finite-volume method over a finite-difference method?

I found many answers talking about differences among the methods but I am really interested to know the advantages!

Thank you

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Old   June 28, 2000, 11:41
Default Re: FVM vs. FDM Advantages?
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i) FV schemes satisfy discrete conservation laws. They are better if one requires good conservation properties.

ii) Unstructured meshes / Mixed unstructured-structured formulations.

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Old   June 28, 2000, 21:09
Default Re: FVM vs. FDM Advantages?
John C. Chien
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(1). They work on different principles, that is, based on different formulations of conservation law. (2). The finite-difference formulation is based on the differential equation, the most common form. (3). The finite-volume formulation is based on the integral equation, by applying the conservation laws over a control volume. (4). These two formulations are identical, when applied to an infinitesimal fluid particle. (5). For the finite size mesh, the finite-difference solution is affected by the truncation errors. (6). On the other hand, although the finite-volume solution will still satisfy the control volume conservation law, the distribution of variables across the control volume (cell) no longer represent the real solution.(only the integrated global variables are conserved) (7). In most cases, one can isolate a single finite volume and apply the conservation law. So, in those cases, cells can be programmed independently. Thus it make the unstructured mesh practical and simple(does not means that it is easy to program). (8). For finite-difference method, in most cases, a general coordinate system is needed (Cartesian coordinates system is an exception), therefore, the solution is more sensitive the the mesh quality. Although, the complex geometry can be handled by using the multi-block mesh, the bolck size is normally much larger than the cell size of the finite-volume method. (9). To address the accuracy of one method against another,we must first obtain the mesh independent solution. And this is always problem dependent. I would say that it still depend upon the ability to obtain the mesh for the mesh independent solution. (10). For example, the pure tet/tri (finite-volume) mesh (unstructured by nature) will have much harder time to obtain accurate solution for boundary layer type flows.(this is a difficult problem, but is being improved by adaptive schemes) (11). I would say that it is case dependent. And the evaluation should be based on the comparison between the mesh independent solutions. (not based on the solutions obtained with the same grid or cell sizes)
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Old   June 28, 2000, 22:34
Default Re: FVM vs. FDM Advantages?
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I think that in most cases they differ just in their interpretation, e.g. what you are going to compute (cell-average or point value?). So, once you discretise the equations, there is no difference. In one dimension, this is often true (if you forgot how you discretised the equations, how do you know which method you're using by looking at the code or numerical solutions?).

In higher dimensions too, I think this is true. Even if you think you're solving Euler on a trianguler grid by FVM, i.e. computing cell- averages, the discretized equations could be thought of as derived by FDM where you're computing point values at centroids of triangles with perhaps a very complicated FD discretisation.

Advantage? Well, FVM does have an advantage. It is simpler and easier to interpret what you're doing and to formulate the problem on general mesh.

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