[minhdavid009]

In below image, I don't know hown to computing flux for split-concave cells.
Cell_697.pdf.zip
Can you help me, please.
Thank you very much!

[mprinkey]

This is firmly in the context of unstructured finite volume formulations, so you likely need to read up on those techniques. I am not sure if there are any issues with parts of unstructured FVM that requires convex cells. Concave cells are generally avoided, but off the top of my head, I can't remember if that is a preference or a requirement. Some of the TVD arguments may fall apart on convex volumes.

I would start just applying the unstructured FVM approach naively. Assuming that you have the cell average values, I'd start by building the cell gradient of those values (with Green Gauss or Least Squares), apply gradient limiting as necessary, and then interpolate to the face, say, using second-order upwinding:

phiFace = phiCell + gradPhi(Limited) . (rFace - rCell)

where rFace is the centroid of the face and rCell is the centroid of the cell. You could do that for both of the triangular faces in your example.

The flux is normally constructed as a [vector,tensor] quantity on each face and then integrated by dotting with the dA vector for the face to generate a [scalar,vector] flux. You could do that for both triangle faces and add them to get the total flux.

Good luck.

[minhdavid009]

Thank you very much!