The fundamental difference between Finite Difference and Finite Volume Methods
 

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 mid-point 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 mid-points 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.

This is right. FDM only enforces the discrete form of the transport equation at a finite number of points. FVM enforces the transport equation only in an integral sense over a finite number of control volumes.

This is precisely wrong. The divergence theorem turns integrals of divergence terms (diffusion, convection, etc) over of the control volume into surface integrals over the faces. The surface integral over a face between two neighbor cells is the same for both--what flows out of one cell flows into the other. FVM is globally conservative by definition. The real effective difference between FDM and FVM is that convective/pressure terms involve interpolation to faces and diffusion terms involve first derivative approximations at faces for FVM where FDM would have 1st and 2nd derivative approximations respectively.

Whether the grid is staggered or not is of no consequence. For any Euler/NS formulation, you will need to interpolate from cell centroids (or reconstruct from cell average values) the face averaged values of at least some terms. Normally, FVM interpolates (aka, reconstructs) these values at the face centroid and then approximates the surface integral over the face by multiplying that face centroid value times the area. Again, this operation is the same for both cells sharing the face---so the mass/momentum/energy/species balance is perfectly conservative. The entire solution volume is covered by these paired control volumes, so the whole simulation is conservative.

FDM makes NO similar construction--it 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. So-called 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.

FVM will coincide with FDM only by accident, generally under assumption of evenly spaced cartesian grids and simple transport equations. Just consider the complexity of introducing a source term in FDM vs FVM. In FD, you'd just add the point value of the source term at that collocation point. In FV, you need to integrate the source term over the entire control volume. Those two numbers will ONLY be the same if the source term varies linearly with location. If it is anything more complicated, the schemes diverge and the FD scheme will in no way conserve the source input.

Thanks Michael for the detailed answer. It really put things into perspective.

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 volume-averaged functions.

I suggest a reading of some books such as Peric&Ferziger and LeVeque that treat the FD and FV methods