This study presents an innovative implicit-explicit time-stepping algorithm based on a first-order temporal accuracy method, addressing challenges in simulating all-regimes of fliud flows. The algorithm\\\\\\\\\\\\\\\'s primary focus is on mitigating stiffness inherent in the density-based \\\\\\\\\\\\\\\"Roe\\\\\\\\\\\\\\\" method, pivotal in finite volume approaches employing unstructured meshes. The objective is to comprehensively evaluate the method\\\\\\\\\\\\\\\'s efficiency and robustness, contrasting it with the explicit fourth-order Runge-Kutta method. This evaluation encompasses simulations across a broad spectrum of Mach numbers, including scenarios of incompressible and compressible flow. The scenarios investigated include the Sod Riemann problem to simulate compressible Euler equations, revealing the algorithm\\\\\\\\\\\\\\\'s versatility, and the low Mach number Riemann problem to analyze system stiffness in incompressible flow. Additionally, Navier-Stokes equations are employed to study viscous and unsteady flow patterns around stationary cylinders. The study scrutinizes two time-stepping algorithms, emphasizing accuracy, stability, and computational efficiency. Results demonstrate the implicit-explicit Runge-Kutta algorithm\\\\\\\\\\\\\\\'s superior accuracy in predicting flow discontinuities in compressible flow. This advantage arises from the semi-implicit nature of the equations, reducing numerical errors. The algorithm significantly enhances accuracy and stability for low Mach number Riemann problems, addressing increasing stiffness as Mach numbers decrease. Notably, the algorithm optimizes computational efficiency for both low Mach number Riemann problems and viscous flows around cylinders, reducing computational costs by 38% to 68%. The investigation extends to a two dimensional (2D) viscous flow over a circular cylinder, showcasing the method\\\\\\\\\\\\\\\'s proficiency in capturing complex flow behavior. Overall, this research advances the understanding of time discretization techniques in computational fluid dynamics, offering an effective approach for handling a wide range of Mach numbers while improving accuracy and efficiency

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