[Martin Hegedus]

In regards to the product rule

d(xy) = d(x) y + x d(y)

is there terminology that names the term on the left side of the equal sign and the term on the right side of the equal sign?

Sometimes in CFD the left side is called the conservative term and the right side non-conservative. But, is there a more rigorous mathematical terminology?

[FMDenaro]

For example,
d(u^2/2)/dx = u*du/dx
where the expression at the RHS is called "quasi-linear".
Hope, this is what you are looking for.

[Martin Hegedus]

Hi Filippo

Thanks. I'm hoping for terminology that does not imply something about the linearity or order of the terms. Terminology that is, for lack of a better term, more neutral.

This is in regards to writing the theory web pages for my finite difference code. In regards to FD, one does not know if the LHS or the RHS of the product rule is higher or lower order, thus causing difficulties for FD. If x=z^2 and y=z^2 then the RHS is better to differentiate with FD. If x=Z^2 and y=1/Z then the LHS is better to use. So I'm trying to avoid terminology that implies linearity, order, or conservation.

Hope that made sense.

[FMDenaro]

Originally Posted by Martin Hegedus

Hi Filippo

Thanks. I'm hoping for terminology that does not imply something about the linearity or order of the terms. Terminology that is, for lack of a better term, more neutral.

This is in regards to writing the theory web pages for my finite difference code. In regards to FD, one does not know if the LHS or the RHS of the product rule is higher or lower order, thus causing difficulties for FD. If x=z^2 and y=z^2 then the RHS is better to differentiate with FD. If x=Z^2 and y=1/Z then the LHS is better to use. So I'm trying to avoid terminology that implies linearity, order, or conservation.

Hope that made sense.

Hi Martin,
sorry but I am not sure about what you are asking... your original question can be associated only to a mathematical nomenclature for the PDE since the equality is no longer true in the discrete sense.
For this reason I can only think of the simple example of Burgers where d(u^2/2)/dx is the divergence form that is mathematically equivalent (in the continuous but not in the discrete) to the quasi-linear form u*du/dx.

In order to write the equality between RHS and LHS in discrete sense, some properties are required to the discrete operators.

[Martin Hegedus]

Hi Filippo,

Yes, you hit the nail on the head. In the discrete sense the LHS and RHS are different. So I'm looking for terminology that makes the distinction of which side I'm discretizing. Personally, I just think of it as the conservative and non-conservative side, regardless of whether conservation has anything to do with it. I'll give quasi-linear more thought. Or maybe I'll just call it the non-producted and producted sides.

[Martin Hegedus]

Or maybe another question is, how did the term "conservative form" come about? If I use that term do individuals know that it is the LHS of the product rule? Or can the terminology be misleading to others?

[FMDenaro]

Originally Posted by Martin Hegedus

Hi Filippo,

Yes, you hit the nail on the head. In the discrete sense the LHS and RHS are different. So I'm looking for terminology that makes the distinction of which side I'm discretizing. Personally, I just think of it as the conservative and non-conservative side, regardless of whether conservation has anything to do with it. I'll give quasi-linear more thought. Or maybe I'll just call it the non-producted and producted sides.

Morinishi et al. defined "divergence" and "advective" the two discrete forms (Sec.2), respectively (https://pdfs.semanticscholar.org/6e9...808.1507666356).
You will find the relation between them

[FMDenaro]

Originally Posted by Martin Hegedus

Or maybe another question is, how did the term "conservative form" come about? If I use that term do individuals know that it is the LHS of the product rule? Or can the terminology be misleading to others?

"Conservative" is, in my opinion, a nomenclature suitable for the integral form of the equation where you define a unique flux for the face between two adjacent cell that guarantees the conservation of the transported variable (mass, momentum, total energy).

[Martin Hegedus]

Thanks,

I'll dig into "divergence" and "advective" and give it some thought.

[sbaffini]

I don't know if it helps but, sometimes, when you have a RHS that can be written as the LHS one says that it is "an exact differential". But, honestly, I thinks it adds nothing to the description. I prefer, in this context, to say that the RHS can be put in conservative form.

[FMDenaro]

Originally Posted by sbaffini

I don't know if it helps but, sometimes, when you have a RHS that can be written as the LHS one says that it is "an exact differential". But, honestly, I thinks it adds nothing to the description. I prefer, in this context, to say that the RHS can be put in conservative form.

Well, what you say is that for a function f(x,y) the total differential

df = dx*df/dx+dy*df/dy

which is a mathematical assesement. I suppose Martin is asking for a discrete sense of the relation, the paper of Morinishi being the one I can think is useful...

[Martin Hegedus]

Yes, this is in regards to the discritized version of the LHS and RHS of the product rule. It is also in regards to determining the volume (grid metrics) and different formulations of the dissipation matrix.

In regards to grid metrics, there is a conservation law for the area (surface) which needs to be satisfied but not one for the volume of a time independent grid, that I am aware of at least.