  For a class of convection-diffusion equations with variable diffusivity, we
construct third order accurate discontinuous Galerkin (DG) schemes on both one
and two dimensional rectangular meshes. The DG method with an explicit time
stepping can well be applied to nonlinear convection-diffusion equations. It is
shown that under suitable time step restrictions, the scaling limiter proposed
in [Liu and Yu, SIAM J. Sci. Comput. 36(5): A2296{A2325, 2014] when coupled
with the present DG schemes preserves the solution bounds indicated by the
initial data, i.e., the maximum principle, while maintaining uniform third
order accuracy. These schemes can be extended to rectangular meshes in three
dimension. The crucial for all model scenarios is that an effective test set
can be identified to verify the desired bounds of numerical solutions. This is
achieved mainly by taking advantage of the flexible form of the diffusive flux
and the adaptable decomposition of weighted cell averages. Numerical results
are presented to validate the numerical methods and demonstrate their
effectiveness.
