[ls99wins]

I was trying to model the velocity of a fluid in the cross section of a rectangular duct, with its source term being pressure gradient in z direction. (cross section defined in x and y), the governing equation being del^2 = -dp/dz.. w being z velocity. how do I implement the upwind scheme to find flux accross boundaries of a control volume here(question asks me to), given that there are no convective terms in the equation?

[FMDenaro]

Originally Posted by ls99wins

I was trying to model the velocity of a fluid in the cross section of a rectangular duct, with its source term being pressure gradient in z direction. (cross section defined in x and y), the governing equation being del^2 = -dp/dz.. w being z velocity. how do I implement the upwind scheme to find flux accross boundaries of a control volume here(question asks me to), given that there are no convective terms in the equation?

No, you are considering the wrong discretization ...upwind is a discretization suitable for hyperbolic equations where you have dependence and influence regions defined by real characteristic curves.

Your problem is defined by Lap w = constant (w=0 as Bc.s) that is an elliptic equation which shows imaginary characteristic curves. Therefore, there are no reasons to select particular zones of the flow as happens for the upwind. You can use FD centred discretization, for example a second order discretization for the second derivatives.

[aerosjc]

Agree with FMDenaro.
There is no mathematical support for your choice of upwind scheme on Poisson equation.