conservative finite differences and finite volumes

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 January 23, 2014, 16:26 conservative finite differences and finite volumes #1 Senior Member   Joachim Join Date: Mar 2012 Location: Paris, France Posts: 145 Rep Power: 14 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

 January 23, 2014, 16:41 #2 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,738 Rep Power: 71 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

 January 23, 2014, 16:46 #3 Senior Member   Joachim Join Date: Mar 2012 Location: Paris, France Posts: 145 Rep Power: 14 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?

January 23, 2014, 17:10
#4
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Filippo Maria Denaro
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Quote:
 Originally Posted by Joachim 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

 January 23, 2014, 17:12 #5 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,738 Rep Power: 71 just as note, you can looking for some similar posts on CFD Online

 January 23, 2014, 17:14 #6 Senior Member   Joachim Join Date: Mar 2012 Location: Paris, France Posts: 145 Rep Power: 14 so solving the integral form of the equations written in generalized coordinates does not make the scheme finite volume?

January 23, 2014, 17:30
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Filippo Maria Denaro
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Quote:
 Originally Posted by Joachim 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

 January 23, 2014, 17:32 #8 Senior Member   Joachim Join Date: Mar 2012 Location: Paris, France Posts: 145 Rep Power: 14 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...