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conservative finite differences and finite volumes

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Old   January 23, 2014, 16:26
Default conservative finite differences and finite volumes
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Joachim
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Hey everyone!

I have a question regarding finite volume / difference methods. If someone could explain this to me, that would be really cool.

Let's say you have a structured grid. If you use generalizes coordinates, can you say that a conservative finite difference scheme IS a finite volume scheme? (computing the fluxes at the midpoints, etc).

Basically, can you use the methods for cartesian finite volumes schemes with finite difference approximations in generalized coordinates?

(U1n+1(i,j)) - U1n(i,j))/dt = F1(i+1/2,j) - F1(i-1/2,j) + G1(i,j+1/2) - G1(i,j-1/2)

where U1, F1 and G1 are defined using generalized coordinates, etc.

sorry if the question is not super clear!

Thanks!

Joachim
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Old   January 23, 2014, 16:41
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Filippo Maria Denaro
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FD is a method for discretizing the pointwise form of NS equations, conversely FV is a method for discretizing the integral form of the NS equations ...
Therefore, the methods are definitely different in general
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Old   January 23, 2014, 16:46
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hmm, you sometime end up with the same equations after using both methods though...
My question: if you try to solve the equation in generalized coordinates:

dU1/dt + dF1/dxsi = 0

then, using finite volumes, you would end up with

dU1/dt + F1(i+1/2,j) - F1(i-1/2,j) = 0

(assuming the midpoint rule)
since delta_xsi = 1. Your volume would appear as the jacobian of the cell in the equation. Then you can get the fluxes using finite differences in generalized coordinates. Can you then still say that this approach is finite volume?
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Old   January 23, 2014, 17:10
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Quote:
Originally Posted by Joachim View Post
hmm, you sometime end up with the same equations after using both methods though...
My question: if you try to solve the equation in generalized coordinates:

dU1/dt + dF1/dxsi = 0

then, using finite volumes, you would end up with

dU1/dt + F1(i+1/2,j) - F1(i-1/2,j) = 0

(assuming the midpoint rule)
since delta_xsi = 1. Your volume would appear as the jacobian of the cell in the equation. Then you can get the fluxes using finite differences in generalized coordinates. Can you then still say that this approach is finite volume?

the only case in which FV and FD produces the same algebraic equation is for linear equation discretized with second order central scheme, otherwise you get different equations.

Then, the integral equation writes as

d/dt Int [V] U dV + Int [BV] n.F dS = 0

in physical space. You can use any type of grid and write this equation in a FV manner directly in the physical space. It retains its phycial meaing of conservation equation

Conversely,

dU/dt + Div.F dS = 0

need a transformation into the computational space but that does not correspond to solve a physical integral equation in the transformed space
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Old   January 23, 2014, 17:12
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just as note, you can looking for some similar posts on CFD Online
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Old   January 23, 2014, 17:14
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so solving the integral form of the equations written in generalized coordinates does not make the scheme finite volume?
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Old   January 23, 2014, 17:30
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so solving the integral form of the equations written in generalized coordinates does not make the scheme finite volume?
In my opinion it has no sense to think that the system of coordinates can make the method finite volume or something else...
have a look to the dedicated chapter in the Ferziger & Peric book
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Old   January 23, 2014, 17:32
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I have it right there...the don't use general coordinates at all for finite volume methods.
I thought that solving the integral form of the equations would make a finite volume scheme, not the coordinates...
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