The two-dimensional Benard convection process is numerically simulated using a modified Marker and Cell finite difference method where the temperature equation is solved implicitly at every time step. Effects of the initial conditions, end-wall boundary conditions, Rayleigh number, Prandtl number, and aspect ratio on the development of the convection process and steady-state solution have been investigated. It is found that the Benard convection process is an initial value problem in the sense that the steady-state solution depends on the initial perturbation. If small scale perturbations are added in the initial temperatures, flow development and steady-state convergence is quicker. It is also found that when an appropriate aspect ratio is chosen, an infinite horizontal extent Benard system can be successfully and economically simulated using a small computational domain with periodic end-wall boundary conditions. The critical Rayleigh numbers are also predicted which agree closely with other published values. The critical Rayleigh number for infinite horizontal extent system is found to be independent of the fluid Prandtl number. The flow behavior and patterns at higher Rayleigh numbers are also investigated.