# discretization of a gradient term in FVM

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 December 17, 2012, 11:42 discretization of a gradient term in FVM #1 Senior Member   Join Date: Jun 2010 Posts: 111 Rep Power: 9 Hi, In the finite volume method, are there any other ways of discretizing a dv/dx term other than central differencing i.e. for instance (dv/dx) at east face = (v_E - v_p )/(x_E -x_p ), where p and E are the central and east nodes of a structured grid. I am looking for a method of the same order as the CDS method above. Thanks!

December 17, 2012, 13:49
#2
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Filippo Maria Denaro
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Quote:
 Originally Posted by Hooman Hi, In the finite volume method, are there any other ways of discretizing a dv/dx term other than central differencing i.e. for instance (dv/dx) at east face = (v_E - v_p )/(x_E -x_p ), where p and E are the central and east nodes of a structured grid. I am looking for a method of the same order as the CDS method above. Thanks!
I think that you are focusing on an issue that is often misleading in the cfd community...

If you use FV approach, then you must discretize the surface integral of the normal component of the flux. That means:

Integral [S] n . Grad f dS

can be written in a FV manner, for example on a 2D structured Cartesian grid as:

Int [-dy/2, + dy/2] (df/dx_i+1/2 - df/dx_i-1/2) dy +
Int [-dx/2, + dx/2] (df/dy_j+1/2 - df/dy_j-1/2) dx

Now you can discretize the integral with the mean value formula and the derivative with second order central formula, getting a global second order of accuracy.
If you want to increase the accuracy, you can use the Simpon rule for the integral and a third degree polynomial interpolation for the derivatives, as we explained in http://adsabs.harvard.edu/abs/2003IJNMF..43..431I

 December 17, 2012, 13:59 #3 Senior Member   Join Date: Jun 2010 Posts: 111 Rep Power: 9 I don't want to use any higher order method that involve cells other than the neighbouring cells.

December 17, 2012, 15:06
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Filippo Maria Denaro
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Quote:
 Originally Posted by Hooman I don't want to use any higher order method that involve cells other than the neighbouring cells.

ok, then you have two choice, Lagrangian linear interpolation for a second order accuracy or implicit Padè interpolation for higher

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