Filtration processes of liquid solutions are fundamental in many natural and industrial applications, such as environmental protection, chemical engineering, water purification, and petroleum extraction. This article develops a mathematical model for describing filtration processes and explores various numerical methods for solving the governing equations. The model is based on Darcy’s law, continuity equation, and constitutive relations of liquid solutions in porous media. Numerical methods, including finite difference, finite element, and finite volume approaches, are discussed with applications to various filtration scenarios. We also provide analysis of the stability, convergence, and efficiency of these methods.