In this thesis study, a fast implicit iteration scheme called Alternating Cell Directions Imp licit method is combined with Approximate Factorization scheme. This application aims to offer a mathematically well defined version of the Alternating Cell Directions Implicit Method and increase the accuracy of the iteration scheme that is being used for the numerical solutions of the partial differential equations. The iteration scheme presented here is tested using unsteady diffusion equation, Laplace equation and advection-diffusion equation. The accuracy, convergence character and the stability character of the scheme compared with suitable iteration schemes for structured and unstructured quadrilateral grids. Besides, it is shown that the proposed scheme is applicable to triangular and hybrid polygonal grids. A transonic full potential solver is generated using the current scheme. The flow around a 2-D cylinder is solved for subcritical and supercritical cases. Axi-symmetric flow around cylinder is selected as a benchmark problem since the potential flow around bodies with a blunt leading edge is a more challenging problem than slender bodies. Besides, it is shown that, the method is naturally appropriate for parallelization using shared memory approach without using domain decomposition applications. The parallelization that is performed here is partially line, partially point parallelization. T he performance of the application is presented for a 3-D unsteady diffusion problem using Cartesian cells and 2-D unsteady diffusion problem using both structured and unstructured quadrilateral cells.