Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion

Abstract

Abstract: Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study develops a two-dimensional meteorology-dependent advection-diffusion-reaction model that incorporates temperature-dependent thermodiffusion through the Soret effect and humidity-dependent modification of advective transport. The governing equation is solved numerically using the Explicit Finite Difference (FD) and Crank-Nicolson (CN) schemes to evaluate their accuracy, stability, and computational performance. Von Neumann stability analysis is employed to establish the stability conditions of both numerical methods. The analysis shows that the FD scheme is conditionally stable and requires restrictive time-step selection to satisfy the Courant-Friedrichs-Lewy condition, whereas the CN scheme remains unconditionally stable for the diffusion component and demonstrates superior numerical robustness under practical simulation conditions. Numerical experiments implemented in MATLAB reveal that the CN scheme produces smoother concentration profiles, reduced numerical diffusion, and improved solution accuracy, particularly for long simulation periods. The simulations further indicate that increased wind velocity enhances pollutant transport, higher relative humidity suppresses dispersion and increases pollutant residence time, and elevated temperatures promote stronger atmospheric mixing through thermally induced diffusion. These findings demonstrate that incorporating meteorological variables substantially improves the physical realism of atmospheric dispersion models. The study concludes that the Crank-Nicolson method provides a more reliable and efficient numerical framework for simulating pollutant transport under varying environmental conditions and offers a practical tool for urban air quality assessment, environmental planning, and evidence-based pollution control strategies.