[granzer]

Hello, my question is regarding solving the conservative form and the non-conservative form of the governing-equations (GE) 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 confusions and questions:

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

ii)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.

iii)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?

[LuckyTran]

FVM method naturally lends itself to governing equations that are already in conservative form because of the Gauss-divergence theorem. The volume integral of a divergence is easily converted into a surface integral of fluxes. Now supposing you knew (i.e. you solve for) the fluxes exactly, there is no discretization error and your original GE is exactly solvable.

In FDM, you would need to discretize (using a finite difference) the divergence itself. There is now a discretization error. At best you satisfy the balance in your discretized GE and not the original GE.

If you were to do FVM on an equation (or any term) that cannot be cast in a conservative form, you can't abuse the divergence theorem and FVM becomes less appealing and starts to look more like FDM because a bunch of things will need to be discretized (using some discretization scheme, which is usually a finite difference scheme). Actually this already happens in FVM to some extent.


Knowing that, it's kind of a no-brainer in FVM to always write your equations in conservative form in the first place if possible. Otherwise, don't do FVM!

[FMDenaro]

First of all, let me address the different forms of the equations in their continuous form.

According for example to the textbook of Hirsch, you will read:


1) Integral form

2) Differential divergent form

3) Differential quasi-linear form


1) is the basic form to be discretized in a FV formulation and is conservative by construction.
2) can be written in a FD formulation with the divergence of the fluxes that can be written in FV-like conservative form.

3) Only suitable for FD formulation and is generally not conseervative.


The dependent variables are not the fluxes but the variables that appear under time derivatives.
I suggest to search for similar questions, this topic was already addressed.

[granzer]

Originally Posted by LuckyTran

FVM method naturally lends itself to governing equations that are already in conservative form because of the Gauss-divergence theorem. The volume integral of a divergence is easily converted into a surface integral of fluxes. Now supposing you knew (i.e. you solve for) the fluxes exactly, there is no discretization error and your original GE is exactly solvable.

In FDM, you would need to discretize (using a finite difference) the divergence itself. There is now a discretization error. At best you satisfy the balance in your discretized GE and not the original GE.

If you were to do FVM on an equation (or any term) that cannot be cast in a conservative form, you can't abuse the divergence theorem and FVM becomes less appealing and starts to look more like FDM because a bunch of things will need to be discretized (using some discretization scheme, which is usually a finite difference scheme). Actually this already happens in FVM to some extent.


Knowing that, it's kind of a no-brainer in FVM to always write your equations in conservative form in the first place if possible. Otherwise, don't do FVM!

So does this mean that, while soling the GE with FDM the advantages of using the conservative form of the GE (in the differential form), is lost?

[granzer]

Originally Posted by FMDenaro

First of all, let me address the different forms of the equations in their continuous form.

According for example to the textbook of Hirsch, you will read:


1) Integral form

2) Differential divergent form

3) Differential quasi-linear form


1) is the basic form to be discretized in a FV formulation and is conservative by construction.
2) can be written in a FD formulation with the divergence of the fluxes that can be written in FV-like conservative form.

3) Only suitable for FD formulation and is generally not conseervative.


The dependent variables are not the fluxes but the variables that appear under time derivatives.
I suggest to search for similar questions, this topic was already addressed.

Referring to the book 'CFD The basics with application' by J.D. Anderson, the integral form of the equation can be written in non-conservative form too. Is my understanding here wrong?

[FMDenaro]

Originally Posted by granzer

Referring to the book 'CFD The basics with application' by J.D. Anderson, the integral form of the equation can be written in non-conservative form too. Is my understanding here wrong?

The integral form for mass, momentum and total energy is always conservative. It is written as a variation in time of the volume-averaged variables due to the surface integral of the fluxes.
If you conversely see an integral equation for some variable having a production term, this latter remains a volume-averaged term that cannot be written as a surface integral of the fluxes. But that is due to the physics of the problem not to the numerics.


From the integral form you can write the differential divergence form and then the quasi-linear (non conservative) form.
Again, similar question was already answered in other post.