# space discretization of convection term using first order upwind

$\int_{\Gamma}\rho\underline{v}.\underline{n}d\Gamma$

where $\rho$ and $\underline{v}$ both vary. I know that it would be something like this in a 1D case:

$\int_{\Gamma}\rho\underline{v}.\underline{n}d\Gamma=\left(\rho\underline{v}.\underline{n}\right)_{e}-\left(\rho\underline{v}.\underline{n}\right)_{w}$

where the small w and e represents the west and east cell faces. I'm not sure then how to find the $\left(\rho\underline{v}.\underline{n}\right)$ at e and w! This is probably very basic and I've been stuck on it for ages! Please help asap.

Regards!

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What do you mean, how to find $\rho\mathbf v\cdot\mathbf n$? Do you not have $\rho$ and $\mathbf v$ that you can evaluate at the faces? In one dimension, the value of $\mathbf n$ is $1$ at the positive (east) face and $-1$ at the negative (west) one, assuming you want the outward normal. –  Rahul Jul 2 '12 at 21:52
I know what n is. I want to write these terms in terms of values at the cell centres. I know for instance if $v$ was a constant you would have: $\underline{v}\left(\rho\right)_{e}=max(v,0)\rho_{P}+min(v,0)\rho_{E}$ but in this case both v and $\rho$ vary and I'm not sure how to use the same logic! –  Hooman Jul 2 '12 at 22:03
I think I have an idea of what to do. Please don't reply questions with a question! –  Hooman Jul 3 '12 at 7:24