Advection is sometimes confused with the more encompassing process of convection which is the combination of advective transport and diffusive transport.

Even with one space dimension and a constant velocity field, the system remains difficult to simulate. The equation becomes

Using vector calculus identities, these operators can also be expressed in other ways, available in more software packages for more coordinate systems.

This form also makes visible that the skew-symmetric operator introduces error when the velocity field diverges. Solving the advection equation by numerical methods is very challenging and there is a large scientific literature about this.