Dictyostelium discoideum (Dd) is a model organism, studied for reasons from cell movement to chemotaxis to human disease control. Creating a computer model of the life cycle of Dd has garnered great interest, one part of which is the Aggregation Stage, where thousands of amoeba gather together to form a slug. Chemotaxis is the mechanism through which this is accomplished. This thesis develops two- and three-dimensional alternating direction implicit code which solves the diffusion equation on an adaptive grid. The calculated values for both two and three dimensions are checked against the actual solution and error results are provided. Comparisons are made between the coarse grid with refinement case and a fine grid without refinement case. Also, a non-negativity condition for two dimensions is derived to give a bound on the three major parameters: the diffusion coefficient and the spatial and time discretizations.