# finite volume discretizing on non-conservative eq.

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 February 1, 2010, 23:03 finite volume discretizing on non-conservative eq. #1 Member   bearcat Join Date: Jul 2009 Location: Ohio Posts: 35 Rep Power: 9 Hello. I am now working on the discretizing formulation to solve incompressible flow by pseudo-compressibility method in a velocity-pressure coupled way. From my experience, we need to linearize the eq before discretization using Jacobin matrix. for example: suppose U is matrix of primitive variables we often change ("dd" means 1st order partial differencing here) ddU/ddt+ddE/ddX+ddF/ddY=RHS (1) --conservative into ddU/ddt+A(ddU/ddX)+B(ddU/ddY)=RHS (2) (in which A=ddE/ddU, B=ddF/ddU) --non-conservative Theoretically, conservative form is needed before applying the FVM. I tried to discretize (1) without knowing E, F as functions of U first, and then applied E=U(ddE/ddU) and F=U(ddF/ddU) for the facial flux expression. I found that the final formulation will be the same with discretizing (2). Is this correct or wrong? I did this because I've to handle the pressure term in solving N-S eq. I want to include the pressure term into E and F, rather than leaving it as a separated term. I think this way may make it simpler and doesn't require other tricks to discretize the pressure term. Is my idea correct? Can it work? Thank you very much.

 February 2, 2010, 18:43 #2 Member   bearcat Join Date: Jul 2009 Location: Ohio Posts: 35 Rep Power: 9 I heard that some solvers use a technique named "momentum interpolation" to compute the pressure. Very few CFD books talk about this. Not sure if it will work as in a coupled solver using pseudo-compressibility.

 February 3, 2010, 22:40 could be right #3 Member   Join Date: Mar 2009 Posts: 33 Rep Power: 9 Hi, What you do could be right since E=AUand F=BU are exact relations for the Euler equations: a flux vector is exactly equal to the product of the Jacobian and the state vector. I remember that the Steger-Warming flux vector splitting scheme are constructed by using these relations. gory

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