  We analyze operator splitting methods applied to scalar equations with a
nonlinear advection operator, and a linear (local or nonlocal) diffusion
operator or a linear dispersion operator. The advection velocity is determined
from the scalar unknown itself and hence the equations are so-called active
scalar equations. Examples are provided by the surface quasi-geostrophic and
aggregation equations. In addition, Burgers-type equations with fractional
diffusion as well as the KdV and Kawahara equations are covered. Our main
result is that the Godunov and Strang splitting methods converge with the
expected rates provided the initial data is sufficiently regular.
