  The goal of the present work is to solve a linear dispersive equation with
variable coefficient advection on an unbounded domain. In this setting,
transparent boundary conditions are vital to allow waves to leave (or even
re-enter) the, necessarily finite, computational domain. To obtain an efficient
numerical scheme we discretize space using a spectral method. This allows us to
drastically reduce the number of grid points required for a given accuracy.
Applying a fully implicit time integrator, however, would require us to invert
full matrices. This is addressed by performing an operator splitting scheme and
only treating the third order differential operator, stemming from the
dispersive part, implicitly; this approach can also be interpreted as an
implicit-explicit scheme. However, the fact that the transparent boundary
conditions are non-homogeneous and depend implicitly on the numerical solution
presents a significant obstacle for the splitting/pseudo-spectral approach
investigated here. We show how to overcome these difficulties and demonstrate
the proposed numerical scheme by performing a number of numerical simulations.
