Volume 6 • Issue 2 • PP: 11 –17 • 2026
A Discrete Nonlinear Advection–Diffusion–Reaction Scheme with Vortex Transport: Stability, Convergence, and Algorithmic Complexity
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© 2026 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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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.
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References
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