According to Zang, numerical simulation can be aided by considering the skew-symmetric form for the advection operator.
where
and  is the same as above.
Since skew symmetry implies only imaginary eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities.
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.