can we bound a dependant variable?

Hello,

I use laminar flow plus adding a convection-diffusion equation as a pde one to NV and continuity equations(time- dependant) to solve my simulation. The problem that I have is that the magnitude of variable in convection-diffusion (u2) should be between 0 and 1 but it has overshoot and undershoot (max 1.8 ans min-0.6).

Is there any way to limit the error in each time step of iteration ? I guess these errors come from numerical errors.
As far as I know one of the ways to reduce these errors is to refine the mesh but I am wondering if we could limit the magnitude of the variable by formula in each time step(for instance if u2>1 set it to 1 and if u2<0 set it to 0)

thank you
hooman