Convergence and Error Analysis of Explicit Finite Difference Schemes Applied to the Burgers–Advection–Diffusion–Reaction Coupling

This work presents a numerical study of the coupling between the Burgers equation with a source term and the advection–diffusion–reaction equation in one spatial dimension, employing an explicit finite difference scheme. The formulation considers the Burgers equation as the governing model for the evolution of the velocity field, whose solution is subsequently used in the transport equation as the convective velocity at the current time step, thereby characterizing a sequential coupling between nonlinear advection and diffusion phenomena. The adopted discretization is based on the forward Euler method for the transient term, combined with second-order centered approximations for the first- and second-order spatial derivatives. The study focuses on the investigation of the consistency, stability, and convergence properties of the numerical scheme, with a particular emphasis on the influence of the spatial (dx) and temporal (dt) discretization steps on solution accuracy. The error analysis is conducted using standard norm-based metrics, enabling the quantification of convergence rates and the identification of regimes in which temporal and spatial truncation errors dominate the solution. Stability constraints associated with convective and diffusive terms are examined, along with the effects of inter-equation coupling on the propagation of numerical errors. The results demonstrate the sensitivity of the explicit scheme to the discretization parameters and highlight conditions under which the coupled system preserves stable and convergent behavior. This study contributes to a deeper understanding of the performance of explicit schemes in nonlinear coupled systems relevant to transport phenomena, simplified fluid dynamics, and reactive process modeling.