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August 28, 2006, 11:45 
consistency and conservativity

#1 
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Hi,
I have two problems and I am hopefully that somebody might help me. 1. What is the difference between consistency and conservativity? 2. I am looking for a paper (mathematical explanation) to prove why finitedifference methods may be not conservative (compared to finitevolume methods)! Thanks a lot. 

August 28, 2006, 11:59 
Re: consistency and conservativity

#2 
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Dear Myway,
1. Consistency refers to the fact that as deltax and deltat tends to zero, the modified pde tends to the pde. Consistency is a reflection of the behaviour of the error, if the error diminishes with grid refinement, the procedure is consistent. Conservation is a reflection of Reynolds' transport theorem; the rate of change of any quantity \phi in a control volume is just the net flux of \phi cutting across the control surface. Thus, in a oned situation, the integral of \phi dx will have dependence only on the boundary values, due to flux cancellations in the domain interior(also referred to as Telescopic collapse). Also refer to Randall LeVeque, "Numerical solutions to conservation laws" 2. It is not true that finite difference methods are not conservative. It is possible to have conservative finite difference formulations, it is just that not all finite difference formulations are conservative, but finite volume is. See Conservative FiniteDifference Methods on General Grids by Mikhail Shashkov Hope this helps Regards, Ganesh 

August 29, 2006, 01:45 
Re: consistency and conservativity

#3 
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Hi,
I am a little bit confused. Discretizing with FVM, there are 4 rules to accomplish. The first one is consistency on the boundaries of a volume. This means that the flux, leaving volume V1 via the area A, must be expressed by the same expression as the flux that enters volume V2 (via A). (V1 and V2 are adjacent volumes!) If this would not be the case, there would be sources/sinks. I cannot see that my explanation coincides with yours....? Unfortunately, I have no chance to get the book by Mikhail Shashkov. Moreover, I have to get it within the next 36 hours, otherwise it will be useless to me. Could you send me these pages by mail? Regards. 

August 29, 2006, 02:29 
Re: consistency and conservativity

#4 
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Dear Myway,
The fact that the flux leaving V1 must enter V2, so that source/sinks do not exist is a statement on conservation and not consistency. Consistency demands that the numerical flux function reduce to the exact flux function when the states are constant. Unfortunately, nor do I have Shashkov's book, only a few pages are available online in google books. Regards, Ganesh 

August 29, 2006, 10:03 
Re: consistency and conservativity

#5 
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Dear Ganesh,
I think I start to understand what you mean. What did you mean with "states are constant"? Would you say that a discretization using a parabolic profile for phi cannot be conservative (because the gradient of a right/left slope can be different on the face)? In this case QUICK would not be conservative? For what kinds of problems is QUICK a good approach (high Reynoldsnumber problems because of the two upstream nodes)? Thanks. 

August 30, 2006, 03:09 
Re: consistency and conservativity

#6 
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Dear Myway,
The idea of "constant states" is used to clarify the consistency of the flux function. If Ul and Ur are the left and right states of the inteface and F and f are the numerical and exact flux functions, then F(Ul,Ur)=f(U), when Ul=Ur=U. This idea of consistency and more detials are there in LeVeque's book. The use of a parabolic profie for phi, merely reconstructs solution variation in a cell, and constructs the Riemann problem at the interface, and as long as the system is in divergence form (or 'conservative' form) as is given by FVM, conservation is guaranteed. The parabolic profile guarantees a higher order of accuracy, as one would desire. Note that conservation and consistency are bothe properties that are essential. Also, note that consistency can also be looked into in the perspective of the local truncation error, as in finite difference schemes. In this case, consistency means mpde tends to the pde, as the steps in space and time approach zero. However, it is also possible to construct apparently inconsistent schemes, that converge to the physical solution, and this happens thanks to the conservation property. If you are interested in these issues in a greater detail, refer to Prof. Bernanado's Cockburn's works. I am not familiar with the QUICK scheme, so I am not in a position to comment on the same. Regards, Ganesh 

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