There seem to be some mistakes in your equation. You use y in some place and z in others. I suppose you are dealing with 2d flow.
You need a maximum principle. At the PDE level, the maximum principle holds if the velocity is divergence-free.
Assuming a divergence free velocity field, write an upwind scheme for convection terms and central scheme for diffusion. This is explicit in time scheme. Under a restriction in time step, you can show maximum principle. This will be only first order accurate though.
You can also use a central scheme for convection, in which case you will need a mesh Peclet number to be small enough to get maximum principle.