The field of Computational Fluid Dynamics (CFD) is in a continual state of advancement due to new numerical techniques, optimization of existing codes, and the increase in memory and processing speeds of computers. In this thesis, the solution technique for a pre-existing Navier-Stokes flow solver is adapted from an explicit Runge Kutta method to a Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit time integration method. Explicit time integration methods were originally used in CFD codes because these methods require less memory. Information needed to advance the flow in time is localized to each grid point. These explicit methods are, however, restricted by time step sizes due to stability criteria. In contrast, implicit methods are unaffected by a large time step sizes but are restricted by memory requirements due to the complexities of unstructured grids. The implementation of LU-SGS performs grid re-ordering for unstructured meshes because of the coupling of grid points in the integration method's solution. The explicit and implicit flow solvers were tested for inviscid flows in incompressible, compressible, and transoinc flow regimes. The results found by comparing the implicit and explicit algorithms revealed a significant speed up in convergence to steady state by the LU-SGS method in terms of iteration number and CPU time per iteration.