Muhammad Masudur Rahaman¹, Humaira Takia^2,, Md. Kamrul Hasan³, Md. Bellal Hossain¹, Shamim Mia⁴, Khokon Hossen²

Water pollution is one of the leading environmental issues faced especially at developing countries all over the World. However, the study of pollution movement is a necessary basis for solving water quality problems. The mathematical model for soluble and insoluble water pollutants has been formulated in light of partial differential equations. This research proposes the motion of flowing pollution by using a mathematical model in one-dimensional advection diffusion equation. Therefore, the study explored finite difference method for the numerical solutions of advection diffusion equation (ADE). Moreover, the study considered the ADE as an initial boundary value problem (IBVP) for the estimation of water pollution. By implementing a finite difference scheme for the IBVP, we examined as well as analyzed the extent of water pollution at different times and different points in an 1-dimensional spatial domain. We estimated relative error of the schemes in comparison with an exact solution of ADE and the numerical features of the rate of convergence are presented graphically. Finally, we compared our numerical solution of the ADE with solution obtained by finite element method. Numerical analysis indicates that water treatment is an essential tool for obtaining quality water for human consumption.

advection diffusion equation, finite difference schemes, water pollutants, mathematical modeling, feasibility of solution, stability analysis