Citation: Jejeniwa, O.A.; Gidey, H.H.; Appadu, A.R. Numerical Modeling of Pollutant Transport: Results and Optimal Parameters. Symmetry 2022, 14, 2616. https:// doi.org/10.3390/sym14122616 Academic Editors: Clemente Cesarano and Juan Luis García Guirao Received: 8 September 2022 Accepted: 30 November 2022 Published: 9 December 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article Numerical Modeling of Pollutant Transport: Results and Optimal Parameters Olaoluwa Ayodeji Jejeniwa 1 , Hagos Hailu Gidey 2 and Appanah Rao Appadu 1, * 1 Department of Mathematics & Applied Mathematics, Nelson Mandela University, Gqeberha 6031, South Africa 2 Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Palapye Private Bag 16, Botswana * Correspondence: rao.appadu@mandela.ac.za or rao.appadu31@gmail.com Abstract: In this work, we used three finite difference schemes to solve 1D and 2D convective diffusion equations. The three methods are the Kowalic–Murty scheme, Lax–Wendroff scheme, and nonstandard finite difference (NSFD) scheme. We considered a total of four numerical experiments; in all of these cases, the initial conditions consisted of symmetrical profiles. We looked at cases when the advection velocity was much greater than the diffusion of the coefficient and cases when the coefficient of diffusion was much greater than the advection velocity. The dispersion analysis of the three methods was studied for one of the cases and the optimal value of the time step size k, minimizing the dispersion error at a given value of the spatial step size. From our findings, we conclude that Lax–Wendroff is the most efficient scheme for all four cases. We also show that the optimal value of k computed by minimizing the dispersion error at a given value of a spacial step size gave the lowest l 2 and l ∞ errors. Keywords: finite difference schemes; amplification factor; stability; relative phase error; optimization 1. Introduction Petroleum is used as fuel for daily human activities. Liquid petroleum has some useful advantages over other energy sources; it is concentrated and could be easily transported from one point to another. The major objective of oil spill modeling is to predict where oil is likely to go after a spill [1,2]. The use of data on ocean currents, winds, waves, and other environmental factors help in this regard [3]. Ovsienko et al. [4] developed a model to forecast the behavior and spreading of oil at sea using the particle-in-cell technique on a quasi-Eulerian adaptive grid. The fate and behavior of spilled oil can be affected by nine physical, chemical, and biological processes: advection, spreading, evaporation, dissolution, emulsification, dispersion, auto-oxidation, biodegradation, and sinking/sedimentation [5]. Cho et al. [1] analyzed the movement of oil with a numerical model that solved an advection–diffusion reaction equation with finite difference schemes. The spilled oil dis- persion model was established in consideration of tide and tidal currents, simultaneously. They obtained the velocity components in the advection–diffusion reaction equation from the shallow water equations. Another commonly used method is the split-operator ap- proach where the convection and diffusion terms are solved by two different numerical methods [6]. A one-dimensional convective diffusion equation was solved by Noye–Tan [7] using the third-order semi-implicit finite difference method. This approach was later ex- tended by Noye–Tan [8] to solve the two-dimensional convective diffusion equation but the said method had issues handling three-dimensional problems because of the large matrix inversion at each time step. The quadratic upstream interpolation convective kinematics (QUICK) method for one-dimensional unsteady flow was introduced by Leonard [9] to address the issue of numerical dispersion. This method was extended to an improved Symmetry 2022, 14, 2616. https://doi.org/10.3390/sym14122616 https://www.mdpi.com/journal/symmetry