Why incompressible fluid flow is advantageous in numerical computation

Hi,

When we want to solve the Navier-Stokes equations coupled with the conservation of mass for incompressible fluids using the primitive-variable approach, we have to face to the problem that the equation for the continuity equation does not contain the pressure which leads to spurious pressure values. Moreover the compressible fluid assumption characterizes the fluid properties better. Then why don't we solve the compressible Navier-Stokes during numerical simulations?

Thanks,
Zoli