This paper describes the general convection-diffusion equation in 2D domain based on a particular fourth order finite difference method. The current fourth-order compact formulation is implemented for the first time, which offers a semi-explicit method of solution for the resulting equations. A nine point finite difference scheme with uniform grid spacing is also put into action for discretization purpose. The proposed numerical model is based on the Navier-Stokes equations in a stream function-vorticity formulation. The fast convergence characteristic can be mentioned as an advantage of this scheme. It combines the enhanced Fournié's fourth order scheme and the expanded fourth order boundary conditions, while offering a semi-explicit formulation. To accomplish this, some coefficients which do not influence the solutions are also omitted from Fournié's formulation. Consequently, very accurate results can be acquired with a relatively coarse mesh in a short time. The robustness and accuracy of the proposed scheme is proved using the benchmark problems of flow in a driven square cavity at medium and relatively high Reynolds numbers, flow over a backward-facing step, and flow in an L-shaped cavity.