FVM vs FDM vs Conservative form vs Non conservative form

Question

My question is regarding solving the conservative form and the non-conservative form of the governing-equations (GE), like continuity or the navier stokes equation, using finite difference method (FDM) and finite volume method (FVM).

When reading about the differences between the conservative form and the non-conservative forms of the GE it was said that since in the conservative form the dependent variables are the fluxes (and not the primitive variables) they are better conserved and "physically" correct. (among other things)

When reading about the differences between the FDM and FVM it was said that the finite volume was better at conserving the fluxes which are an advantage. (among other things)

Here are my questions:

1. When using FVM do we only solve equations in conservative form?

2. When solving the conservative form of the GE using FDM are the fluxes conserved or does this only apply when solving the equations using FVM.

3. If we solve the GE in the conservative form using FDM and then using FVM what will be the difference? ie will the "conservativeness" be the same for both FDM and FVM methods?

Answer 1

• When using FVM do we only solve equations in conservative form?

Usually yes. You may also have a pde which has both conservative and non-conservative/source terms and you can apply fvm to such problems. There are methods for non-conservative systems which may be called finite volume methods, as they are based on calculating some fluctuations at cell faces. These are based on idea of path conservative schemes [1].

• When solving the conservative form of the GE using FDM are the fluxes conserved or does this only apply when solving the equations using FVM.

There are fdm methods that can be applied to conservative systems, and are conservative. The one I know best are finite difference WENO schemes applied to hyperbolic conservation laws [2].

• If we solve the GE in the conservative form using FDM and then using FVM what will be the difference? ie will the "conservativeness" be the same for both FDM and FVM methods?

The differences arise beyond second order accuracy. FVM requires accurate flux integral computation at higher orders and hence can be more complex and expensive, since the quadrature requires more flux computations. FDM approximates derivatives and can be less expensive [3]; you apply the 1-d finite difference along each coordinate direction.

[1] https://doi.org/10.1137/050628052

[2] https://doi.org/10.1006/jcph.1996.0130

[3] https://doi.org/10.1137/070679065

Answer 2

I will try to give my view on this.

1. When using FVM do we only solve equations in conservative form?

The finite volume method is described trough weighted residual unifying framework as collocation on subdomain approach. This means that the residual of the equation is exactly zero on each subdomain separately. This is general approach which is applied to any PDE. If the equation in question is describing conservation law (of mass, liner momentum, energy) the requirement to have residual equal to zero on a subdomain (control volume) means that the quantity of interest is conserved within subdomain and that the approximate solution satisfies conservation law on given decomposition of the domain. Approximate solution may not be smooth enough but it satisfies conservation law.

This view can be extrapolated to other two points (2 and 3) without losing conservativeness of the answer, but it requires special treatment (FDM pun intended).