  The present work proposes a second-order time splitting scheme for a linear
dispersive equation with a variable advection coefficient subject to
transparent boundary conditions. For its spatial discretization, a dual
Petrov--Galerkin method is considered which gives spectral accuracy. The main
difficulty in constructing a second-order splitting scheme in such a situation
lies in the compatibility condition at the boundaries of the sub-problems. In
particular, the presence of an inflow boundary condition in the advection part
results in order reduction. To overcome this issue a modified Strang splitting
scheme is introduced that retains second-order accuracy. For this numerical
scheme a stability analysis is conducted. In addition, numerical results are
shown to support the theoretical derivations.
