Groundwater pollution equation: Lie's symmetry analysis and numerical consideration

The current study modeled groundwater pollution through the utilization of the advection–diffusion equation - a versatile differential equation that is capable of modeling a variety of real-life processes. Indeed, various methods of solutions were then proposed to examine the governing model after being transformed, starting with Lie's symmetry, semi-analytical, and numerical methods, including the explicit and implicit finite difference method and the finite element method. Further, the proposed methods were demonstrated on some test models; featuring forced and unforced scenarios of the forcing function. Analytically, Lie's symmetry method failed to unswervingly reveal the required solution to the problem; however, with the imposition of certain restrictions, a generalized closed-form solution for the forced model was acquired. This fact indeed triggered the quest for the deployment of more methods. Thus, semi-analytically, the adopted decomposition method swiftly gave the resultant closed-form solutions. Numerically, the efficiency of the sought methods was assessed using the L²−norm and CPU time, upon which the implicit schemes were found to win the race. All-in-all, the beseeched semi-analytical method is highly recommended for such investigation; at the same time advocating the effectiveness of the implicit finite difference schemes on advection–diffusion-related equations.