We introduce a high-order compact difference scheme with multigrid algorithm to solve the convection-diffusion equations with constant coefficients. This high-order discretization scheme is shown to be more accurate and stable than the usual five-point discretization scheme. It solves the convection-diffusion equations directly without using any preconditioner or adding any artificial dissipation terms. This method is shown to converge faster than some of the existing methods and to achieve higher accuracy. Numerical experiments are presented to validate the conclusions.