I am new to to the finite difference method and i want to understand how convection-diffusion equation is descritized in 2-D using central diffrences

$nabla .(rho vec{v} Phi)=nabla.(Gamma vec{nabla} Phi)$

I believe the descritization would yield :

$ frac{Phi_{i+1,j} – Phi_{i-1,j}}{2h}v_x+frac{Phi_{i,j+1} – Phi_{i,j-1}}{2h}v_y =frac{Gamma}{rho} frac{Phi_{i+1,j} + Phi_{i-1,j}+Phi_{i,j+1} + Phi_{i,j-1}-4Phi_{i,j}}{h^2} $

i hope i’ve got it right, now i believe this is the case for a problem with constant velocity,density etc

what i am i want to know is how can we do the same for a problem, with, a non uniform velocity for example.

edit : okay, so i worked a little more on the problem and i would really appreciate if someone could confirm or point out how the result is wrong, this is what i ended up with for a non uniform velocity:

$ frac{Phi_{i+1,j} – Phi_{i-1,j}}{2h}v_{x_{i,j}}+frac{Phi_{i,j+1} – Phi_{i,j-1}}{2h}v_{y_{i,j}} +Phi_{i,j}frac{(v_{x_{i+1,j}} -v_{x_{i-1,j}}) + (v_{y_{i,j+1}} -v_{y_{i,j-1}})}{2h} =frac{Gamma}{rho} frac{Phi_{i+1,j} + Phi_{i-1,j}+Phi_{i,j+1} + Phi_{i,j-1}-4Phi_{i,j}}{h^2} $