  We consider solving a generalized Allen-Cahn equation coupled with a passive
convection for a given incompressible velocity field. The numerical scheme
consists of the first order accurate stabilized implicit explicit time
discretization and a fourth order accurate finite difference scheme, which is
obtained from the finite difference formulation of the $Q^2$ spectral element
method. We prove that the discrete maximum principle holds under suitable mesh
size and time step constraints. The same result also applies to construct a
bound-preserving scheme for any passive convection with an incompressible
velocity field.
