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The fundamental difference between Finite Difference and Finite Volume Methods 

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January 13, 2016, 15:37 
The fundamental difference between Finite Difference and Finite Volume Methods

#1 
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eu sou cfd
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Hey.
I have a rudimentary question about the differences between finite difference (FDM) and finite volume methods (FVM). Bear with me here. In FDM we concentrate on the nodes (points) in space while in FVM we concentrate in the volumes enclosed by faces. Correct me if I am wrong here. When the Gauss Divergence theorem is applied on the integral equation of the FVM, we do not calculate the variables on the volume as a whole but on the faces enclosing the volume/cell. And thus we will be calculating on the points in space (the point being on the midpoint of the face generally). I understand the conservative character is lost during this process. So now my question is, how is this different from a Finite Difference Method, where we would be calculating the variables on points as well. Consider a staggered grid, where the point of calculation of the variables could lie exactly on the midpoints of the faces being used for FVM? I am starting to believe that the algebraic formulation will be exactly the same, if you do it using an FVM or FDM method. Am I right in assuming this? Please point out any inconsistencies in my observations made above. 

January 13, 2016, 16:42 

#2  
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Michael Prinkey
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FDM makes NO similar constructionit only enumerates a number of points in space where the discrete form is exactly satisfied. It says absolutely nothing about what happens in between the finite number of collocation points. Socalled conservative finite difference methods are formulated exactly like the Finite Volume Method and, in fact, are the precursors to what eventually became the FV method. Quote:


January 13, 2016, 17:10 

#3 
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eu sou cfd
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Thanks Michael for the detailed answer. It really put things into perspective.


January 14, 2016, 04:13 

#4 
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Filippo Maria Denaro
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just to add some suggestions to the good Michael's answers.
1) FD is the discretization method applied on the differential for of the equations, FV is the counterpart of the discretization applied on the integral form. Note that this gives reason of the fact that the integral form (and FV) is suitable to be applied on problems producing singularity (shock). 2) As a consequence, the FD method works on the discretization of derivatives while FV works on the discretization of the fluxes 3) the FD method works for resolving pointwise functions, FV method works to resolve the volumeaveraged functions. I suggest a reading of some books such as Peric&Ferziger and LeVeque that treat the FD and FV methods 

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fdm, finite difference method, finite volume method, fundamentals, fvm 
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