Are the Finite Difference and Finite Volume Methods different after the application of the Gauss Divergence theorem on the FVM?

Question

I have a rudimentary question about the differences between finite difference (FDM) and finite volume methods (FVM).

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.

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.

Answer 1

Yes, many methods can be derived in different ways. For example, some finite volume methods for porous media flow can also be derived as staggered grid finite difference methods, as well as mixed finite element methods using particular quadrature rules.

This is, maybe, not all that surprising given that all try to approximate the underlying physics in some way. At the same time, it may not always be entire obvious how, for example, one might have to choose the quadrature rule to derive a finite difference stencil from a finite element method. But, if you succeed in doing so, this may open up a new and sometimes simpler way of analyzing the convergence of methods because you suddenly have the option of analyzing a finite volume method through the perspective of the finite element method, plus a perturbation that results from using a particular quadrature.