Modeling of Pollutant Dispersion in a Steady-State 1D Advection-Diffusion System

Pollutant dispersion in natural systems, such as rivers and atmospheric flows, represents a major environmental challenge with serious consequences for ecosystems and human health. This study focuses on the steady-state transport of a pollutant in a one-dimensional domain governed by coupled advection and diffusion processes. An exact analytical solution of the governing ordinary differential equation (ODE) is derived and complemented by a numerical solution obtained using the finite difference method, which is solved through the Gauss–Seidel iterative algorithm. The numerical implementation is carried out in Python, and the results are graphically visualized to allow a direct and reliable comparison between analytical and numerical solutions. To provide a physical interpretation, the system is modeled as a simplified river segment with clearly defined boundary conditions, illustrated by a TikZ diagram. The model assumes constant advection velocity and diffusion coefficient, an assumption that is justified under steady-state conditions typically encountered in controlled environmental studies. The results demonstrate a strong agreement between the analytical and numerical solutions, with a maximum absolute error of 0.000128 and convergence achieved after 2505 iterations for n = 100 grid points. Furthermore, second-order convergence is observed as the spatial grid is refined and parameters such as ε are varied, confirming the reliability of the numerical scheme. Finally, the implications for environmental monitoring are discussed, and future research directions are proposed, including stochastic extensions, fractional models, three-dimensional formulations, and the integration of experimental data.