This report presents methods for solving the steady-state incompressible and low-speed compressible fluid dynamics equations using preconditioning techniques. The incompressible equations are preconditioned by introducing artificial time derivatives, which accelerate convergence to the steady state. For the compressible equations, a generalization of the artificial compressibility method is discussed, allowing for faster convergence and a uniform treatment of both primitive and conservative variables. The resulting equations form a symmetric hyperbolic system, ensuring well-posedness. The optimal value of the preconditioning parameter β is determined for given α, and the system is reformulated in conservation form. The report also discusses the implementation of these methods in explicit and implicit schemes, including staggered grids and curvilinear coordinates. Computational results demonstrate the effectiveness of the preconditioning techniques, showing significant reduction in residuals and improved convergence rates.This report presents methods for solving the steady-state incompressible and low-speed compressible fluid dynamics equations using preconditioning techniques. The incompressible equations are preconditioned by introducing artificial time derivatives, which accelerate convergence to the steady state. For the compressible equations, a generalization of the artificial compressibility method is discussed, allowing for faster convergence and a uniform treatment of both primitive and conservative variables. The resulting equations form a symmetric hyperbolic system, ensuring well-posedness. The optimal value of the preconditioning parameter β is determined for given α, and the system is reformulated in conservation form. The report also discusses the implementation of these methods in explicit and implicit schemes, including staggered grids and curvilinear coordinates. Computational results demonstrate the effectiveness of the preconditioning techniques, showing significant reduction in residuals and improved convergence rates.