with Vortex Transport: Stability, Convergence, and Algorithmic

Complexity

Ali M. Abdulameer1,*

1 Jabir Ibn Hayyan University for Medical and Pharmaceutical Sciences, Baghdad, Iraq

Received: April 10, 2026 Revised: June 09, 2026 Accepted: August 08, 2026 ⋆ Corresponding author

ABSTRACT

This paper develops a two-dimensional discrete mathematical model for a nonlinear advection–diffusion–reaction

equation under a divergence-free vortex field. The formulation is treated as a finite-dimensional dynamical system

generated by an explicit finite-difference algorithm: first-order upwind operators approximate advection, secondorder

centered operators approximate diffusion, and explicit Euler time stepping advances the nonlinear reaction.

The resulting update rule is analyzed through Courant–Friedrichs–Lewy restrictions, a diffusion constraint, and a

frozen-coefficient von Neumann argument. A Gaussian initial state and homogeneous Neumann boundary conditions

define the computational problem. Numerical experiments show stable vortex-driven redistribution, diffusion-induced

smoothing, decay of total mass in the presence of the linear loss term, and localized changes caused by the quadratic

reaction. Grid-refinement, time-step-sensitivity, and analytical diffusion benchmarks support convergence of the

discrete approximation. For an Nx by Ny grid, each time step requires O(NxNy) arithmetic operations and O(NxNy)

storage, so the method provides a transparent algorithmic bridge between nonlinear PDE analysis and discrete

computation. The study emphasizes the mathematical structure, stability conditions, convergence behavior, and

computational complexity of the scheme while retaining pollutant transport as an illustrative application.

Keywords: Nonlinear partial differential equations Finite-difference algorithm Discrete dynamical systems

Advection–diffusion–reaction equation von Neumann stability Convergence analysis Computational complexity

Vortex transport

1. INTRODUCTION

Environmental pollution is one of the most serious challenges

of the modern world. Rapid industrial expansion, population

growth, and increased human activities release large quantities

of pollutants into the air, water, and soil [1, 3, 5]. The

risk of this phenomenon is that it is not localized but travels

across time and space through fluid movement, wind, and

water currents, making its behavior extremely complex to

predict [1, 3, 5].

The transport of pollutants is a multiphysical process involving

convection, diffusion, chemical reactions, and sometimes

thermal changes [1, 2, 3, 4, 5, 6, 7, 13]. Therefore,

the mathematical modeling of this phenomenon requires

powerful tools to represent this complexity through accurate

mathematical frameworks based on partial differential

equations [1, 2, 3, 4, 5, 6, 7, 13]. This field relies heavily

on partial differential equations, in particular the convection–

diffusion equation that describes mass transfer in the presence

of motion and diffusion [1, 2, 3, 4, 5, 6, 7, 13]. However,