  This paper is concerned with superconvergence properties of the direct
discontinuous Galerkin (DDG) method for two-dimensional nonlinear
convection-diffusion equations. By using the idea of correction function, we
prove that, for any piecewise tensor-product polynomials of degree $k\geq 2$,
the DDG solution is superconvergent at nodes and Lobatto points, with an order
of ${\cal O}(h^{2k})$ and ${\cal O}(h^{k+2})$, respectively. Moreover,
superconvergence properties for the derivative approximation are also studied
and the superconvergence points are identified at Gauss points, with an order
of ${\cal O}(h^{k+1})$. Numerical experiments are presented to confirm the
sharpness of all the theoretical findings.
