Mathematical model analysis and solution properties of nonlinear filtration processes in multidimensional domains

This paper investigates a nonlinear thermo-chemical diffusion-reaction system characterized by coupled parabolic equations that govern the spatio-temporal evolution of temperature and concentration-dependent processes. The model incorporates nonlinear diffusion coefficients, cross-coupling terms, and nonlinear source functions, which collectively describe the complex interplay between heat and mass transfer in reactive media. A robust numerical framework is developed based on an Alternating Direction Implicit (ADI) scheme of the Peaceman-Rachford type, allowing for efficient and stable time integration of the nonlinear system. The implementation ensures high computational efficiency and improved numerical stability, particularly for stiff reaction terms and strongly coupled diffusion dynamics. Comprehensive numerical experiments are conducted to validate the accuracy and stability of the proposed scheme. Error convergence analysis confirms the expected second-order spatial and first-order temporal accuracy, while the time-step sensitivity and stability tests demonstrate the robustness of the algorithm under various discretization parameters. Boundary layer behavior is also examined to capture localized gradients and nonlinear interaction patterns. The obtained results reveal that the proposed computational framework accurately reproduces the characteristic thermo-chemical diffusion phenomena and maintains stability even under extreme parameter regimes. The study provides a reliable numerical tool for analyzing multi-scale diffusion-reaction systems relevant to chemical engineering, materials processing, and thermal energy storage applications.