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Conservation Vs Non-conservation Forms

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Old   July 9, 2013, 18:28
Default Conservation Vs Non-conservation Forms
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Hello,
I understand mathematically how one can obtain the conservation equations in both the conservative and non-conservative forms. However, I am still confused, why do we call them conservative and non-conservative forms? can any one explain from a physical and mathematical point of view?
Many threads deal with this question(Conservative versus Non-conservative forms, conservative, non conservative form???? ), but none of them provides a good enough answer for me!
If any one can provide some hints, I will be very grateful.

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Old   July 10, 2013, 03:34
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"non-conservative" form is the actual differential equation. If you integrate that equation over volume and use the divergence theorem to exchange all volume-divergence integrals by the area integral of the fluxes you get the "conservative" form. That's all.
As I understand it, you call it "conservative" because it conserves the fluxes (also momentum fluxes) in your domain. Face flux of volume "a" and face flux of adjacent volume "b" are automatically the same.

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Also, the latter isn't always the case, when you just solve the differential equation in non-conservative form. If you have cylindrical coordinates, you will get different values for the flux from cell "a" to cell "b" in radial direction, depending on whether you calculate it in cell "a" or cell "b". However, this is not the case if you use the integral (conservative) form - also in cylindrical coordinates!
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Old   July 10, 2013, 17:36
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Quote:
Originally Posted by RodriguezFatz View Post
"non-conservative" form is the actual differential equation. If you integrate that equation over volume and use the divergence theorem to exchange all volume-divergence integrals by the area integral of the fluxes you get the "conservative" form. That's all.
As I understand it, you call it "conservative" because it conserves the fluxes (also momentum fluxes) in your domain. Face flux of volume "a" and face flux of adjacent volume "b" are automatically the same.

Edit:
Also, the latter isn't always the case, when you just solve the differential equation in non-conservative form. If you have cylindrical coordinates, you will get different values for the flux from cell "a" to cell "b" in radial direction, depending on whether you calculate it in cell "a" or cell "b". However, this is not the case if you use the integral (conservative) form - also in cylindrical coordinates!
Thanks RodriguezFatz for the interesting answer. For those who might be interested, I also obtained a differently explained answer at an other forum, and you can check it from the following links.
http://physics.stackexchange.com/que...direct=1#70540

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Old   July 11, 2013, 05:14
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Actually, there are more concepts than a mere numeric issue... I think that that stays first of any consideration about numerics.

The continuous form is conservative when it describes a balance of an estensive quantity, for example, you can see that the momentum conservative equation is

d(rho*u)/dt + Div (rho*u u) = ....

but the non-conservative form

du/dt + u Grad u = ....

expresses a balance of the accelerations, is not an evolution equation for the momentum....the same concept applies for the energy equation.

Then you can talk about conservative or non-conservative discretizations ...
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