Crank–Nicolson Scheme of the One-Dimensional Convection-Diffusion Equation for Pollutant Transport in Open Channels

Abstract: This study presents a numerical solution to the one-dimensional convection-diffusion equation, modeling the transport of conservative pollutants in open channel flows. Using the Crank-Nicolson finite difference scheme, the simulation captures the combined effects of advection and diffusion on pollutant dispersion. The model assumes a steady, uniform, and unidirectional flow in a straight channel with constant cross-section and no chemical reaction or lateral mixing. A tridiagonal matrix system is derived and solved using the Thomas algorithm to obtain the temporal evolution of pollutant concentration profiles. The resulting data and 3D plots highlight how pollutants gradually dilute and spread downstream, demonstrating the physical realism and numerical stability of the scheme. The study supports the effectiveness of finite difference methods in environmental modeling and opens avenues for future work on more complex scenarios involving multi-dimensional domains or reactive transport phenomena.

Keywords: Convection-diffusion, Finite difference, Pollutant transport, Numerical solution, Open channels.