A Comparative Study of Analytic and FDM Solutions for the Advection-Diffusion Equation

The advection-diffusion equation plays a pivotal role in modeling transport phenomena in various scientific and engineering disciplines, such as fluid dynamics, heat transfer, and pollutant dispersion. This study presents a comparative analysis of analytic and numerical solutions for the advection-diffusion equation, focusing on the finite difference method (FDM) as a numerical approach. A Gaussian initial condition is employed to ensure a smooth and well-defined profile for solution comparison. The analytical solution, derived using separation of variables, serves as a benchmark for assessing the accuracy of the numerical results. The FDM is applied with appropriate discretization schemes to solve the equation over a one-dimensional spatial domain. The comparison highlights the ability of the FDM to closely replicate the analytical solution, demonstrating its accuracy and stability for varying time steps and spatial resolutions. The study also explores the impact of diffusion coefficients and boundary conditions on solution behavior. The findings validate the finite difference method as a reliable computational tool for solving advection-diffusion problems and provide insights into its applicability for practical scenarios where analytical solutions are infeasible.