Analytical and Numerical Solutions of Modified Convection-Diffusion Equation by Explicit Finite Difference Method

Abstract

This paper investigates the analytical and numerical solutions of the convection-diffusion equation, emphasizing its theoretical foundations, solution methodologies, and applications in engineering and environmental sciences. The study highlights the roles of convection (advection and buoyancy) and diffusion in transport phenomena, derives analytical solutions for simplified cases, and introduces numerical methods for addressing more complex scenarios.
The convection velocity u(t, x) in the Convection-Diffusion Equation (CDE) is computed by solving the viscous Burgers' equation using consistent numerical schemes. The stability conditions for these schemes are analytically derived, demonstrating that the FTCS (Forward Time Central Space) scheme outperforms the FTBSCS (Forward Time Backward Space Central Space) scheme in terms of time step selection. These stability conditions are further validated through numerical verification. Numerical simulations are conducted for various parameters, and the results are presented to illustrate the behavior of the solutions. Additionally, error comparisons between the two schemes are provided to evaluate the accuracy of the numerical solutions.