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July 9, 2013, 18:28 
Conservation Vs Nonconservation Forms

#1 
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Hello,
I understand mathematically how one can obtain the conservation equations in both the conservative and nonconservative forms. However, I am still confused, why do we call them conservative and nonconservative forms? can any one explain from a physical and mathematical point of view? Many threads deal with this question(Conservative versus Nonconservative 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. Cheers. 

July 10, 2013, 03:34 

#2 
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Philipp
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"nonconservative" form is the actual differential equation. If you integrate that equation over volume and use the divergence theorem to exchange all volumedivergence 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 nonconservative 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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July 10, 2013, 17:36 

#3  
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Quote:
http://physics.stackexchange.com/que...direct=1#70540 Cheers. 

July 11, 2013, 05:14 

#4 
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Filippo Maria Denaro
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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 nonconservative 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 nonconservative discretizations ... 

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