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Old   March 26, 2007, 11:37
Default Conservation form?
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I'm a tad confused as to what is meant by "conservation form" in reference to finite difference equations. I'm currently solving some transonic flows in which this property is, apparently, very important, but I'm not too sure what the term means, or how to demonstrate that my scheme is actually conservative.

I'm told that if we have an equation we have to multiply by the dx & dt spacing and sum over all nodes. If the answer is zero then the scheme is not conservative. Does this then mean that for the case :

du/dx + du/dt = 0

differenced implicitly as (u(I,J+1)-u(I-1,J+1))/dx + (u(I,J+1)-u(I-1,J+1))/dt = 0

Multiplying by dx*dt and summing over all nodes and all time levels gives telescoping sums which don't cancel exactly. Some terms are remaining that originate from the boundary values and initial conditions. I'm pretty sure that I've performed the analysis correctly but I'm confused as to how to enforce conservative-ness (if that's even a word!).

To make the scheme conservative do I thus have to change my boundary conditions and risk overdetermining the problem?

I'd appreciate any advice. Pete
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Old   March 29, 2007, 12:58
Default Re: Conservation form?
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I would think that the conservation can be proved with the fact that the flux coming in should equate to flux going out. Hence when the boundary conditions are applied you should get the terms which can make it conservative!.
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