Numerical study of optimal pesticide remediation in the Santiago River with a 2D model

In this work, we numerically study the optimal removal of the commonly used pesticides malathion, aldrin, and glyphosate in a 2D model of a section of the highly polluted Santiago River basin near El Salto, Jalisco, Mexico. The proposed remediation strategy utilizes cheap adsorbents. Particularly residual leaves, eggshells, montmorillonite, and woody biochar. We model the system using the well-known shallow-water equations for hydrodynamic simulation, coupled with a system of PDEs to study the dispersion and pesticide-adsorption kinetics, incorporating the Langmuir and Freundlich isotherms. An optimal control problem is formulated to determine both, the optimal quantity and the optimal release point of the adsorbent to maximize remediation. The PDEs are solved using the Finite Element Method (FEM) via the FEniCS library, while the optimization problem is tackled through a combined strategy of exhaustive search with a global gradient-free optimizer, and local refinement with the L-BFGS-B algorithm. Our results demonstrate the effectiveness of each low-cost adsorbent in reducing pesticide concentration, suggesting a viable and sustainable remediation strategy that balances environmental efficiency with economic feasibility.