What are the advantages of Finite volume method (FVM) over Finite difference Method (FDM) for particularly flow simulation (CFD) ?
The Lax-Wendroff Theorem (see sec 12.20 of LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge, 2002) proves that a convergent solution of a FVM is a weak solution of the conservation law. If this is combined with an entropy condition, to exclude non-physical expansion shocks, then our weak solution will have discontinuities with the correct strength and travelling at the correct speed. i.e., a FVM ensures we capture shocks with the correct strength and shock speed.
IMO the above is why the FVM was developed and other advantages mentioned followed.
The adaptability of the finite element in the complex area is not as good as the finite volume method. Further, it is easier for the FVM in the conservation and the obvious physical concept understanding.
You can find the answers to your questions in this post:
https://www.researchgate.net/post/What_are_the_advantages_of_Finite_volume_method_FVM_over_Finite_difference_Method_FDM_for_particularly_flow_simulation_CFD
Thank you Ahmed Osama Mohamed Elhassan Mahgoub and Filippo Maria Denaro for pointing out
The main advantage is that the finite volume method is well adapted for simulating conserved quantities associated with vector fields, e.g., velocity in fluid flows. Also, scalars such as mass and energy are also preserved well. The reason is that the finite volume computes differences by summing say, mass, momentum or energy across interfaces (cell sides) in the grid. This entails a direct application of the Divergence Theorem from Calculus. I have used FVM for years and have recently applied it to 3D heat conduction problems. I have reports on ResearchGate to that effect. FVM is also quite good at addressing multiple block or domain grid systems.
The Lax-Wendroff Theorem (see sec 12.20 of LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge, 2002) proves that a convergent solution of a FVM is a weak solution of the conservation law. If this is combined with an entropy condition, to exclude non-physical expansion shocks, then our weak solution will have discontinuities with the correct strength and travelling at the correct speed. i.e., a FVM ensures we capture shocks with the correct strength and shock speed.
IMO the above is why the FVM was developed and other advantages mentioned followed.
We can notice that the Finite Differences Method are particularly well suited in the case of rectangular or parallelepipedic domains, for which one can easily define structured meshes, for example Cartesian meshes ie. whose meshes can be indexed by a natural order (i, j). In the case of more complex domains, we often mesh with the help of triangles (or tetrahedrons) and in this case the finite difference method is not easily generalized. We then use either the Finite Volumes Method, or to Finite Elements Method.
Adel Ghenaiet - FVM are conservative. FDM are not conservative by design - but may happen to be. It would be unusual to talk of a scheme being more conservative than another - what does this mean? Perhaps, scheme A loses more mass than scheme B?
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