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FVM Flux interpolation in Curvilinear Coordinates

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Old   November 29, 2006, 05:57
Default FVM Flux interpolation in Curvilinear Coordinates
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DSS
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Hi All,

I am in process of developing a Finite Volume code. I am successsfull in a simple pure cartesian code. Now, when I attempted to extend it to curvilinear coordinates, I am struct with how to do the flux interpolation.

I was thinking if it is sane to just adopt the formulation into

dW/dx + dF/d(epsilon) + dG/d(eta) = Q

Where the flux terms contain the relevant transformations.

Is it valid? Because, physically, the equations are a bit different, as the face areas are now unity (computational space).

For the flux interpolation, if it is a simple averaging (less accurate), it is straight forward. But, if it involves some higher order interpolation schemes, how is the accuracy quantified? Wont the accuracy of the Jacobians, and terms like dx/d(epsilon) affect the accuracy of the solution altogether?

Your suggestions are greatly appreciated.
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Old   November 30, 2006, 05:15
Default Re: FVM Flux interpolation in Curvilinear Coordina
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Mar
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Starting from dQ/dt +dF/dx +dG/dy=0

You have to transform the d/dx and d/dy derivatives in terms of the derivatives in the curvilinear coordinate(let's say csi and eta) system using the rules of derivation of composite functions. After that you can resonstruct a conservative form:

dQ*/dt + dF*/dcsi + dG*/deta=0

Where Q*=Q/J F*=Fdy/deta - Gdxdeta G*=-Fdy/dcsi +Gdx/dcsi

where J is the jacobian of the transformation.

Integrating in the control volume you can recover your Finite volume approach.

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Old   November 30, 2006, 05:29
Default Re: FVM Flux interpolation in Curvilinear Coordina
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DSS
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Thank you very much for clarifying my small doubt.
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