  We present a new discretization for advection-diffusion problems with Robin
boundary conditions on complex time-dependent domains. The method is based on
second order cut cell finite volume methods introduced by Bochkov et al. to
discretize the Laplace operator and Robin boundary condition. To overcome the
small cell problem, we use a splitting scheme that uses a semi-Lagrangian
method to treat advection. We demonstrate second order accuracy in the $L^1$,
$L^2$, and $L^\infty$ norms for both analytic test problems and numerical
convergence studies. We also demonstrate the ability of the scheme to handle
conversion of one concentration field to another across a moving boundary.
