  In this paper, we develop the constraint energy minimizing generalized
multiscale finite element method (CEM-GMsFEM) for convection-diffusion
equations with inhomogeneous Dirichlet, Neumann and Robin boundary conditions,
along with high-contrast coefficients. For time independent problems, boundary
correctors $\mathcal{D}^m$ and $\mathcal{N}^{m}$ for Dirichlet, Neumann, and
Robin conditions are designed. For time dependent problems, a scheme to update
the boundary correctors is formulated. Error analysis in both cases is given to
show the first-order convergence in energy norm with respect to the coarse mesh
size $H$ and second-order convergence in $L^2-$norm, as verified by numerical
examples, with which different finite difference schemes are compared for
temporal discretization. Nonlinear problems are also demonstrated in
combination with Strang splitting.
