Abstract

The advection–diffusion equation is one of the fundamental partial differential equations used for describing the simultaneous transport and spreading of a physical quantity. It occurs in problems involving heat transfer, pollutant movement, solute transport, groundwater contamination, atmospheric dispersion and many other engineering and environmental processes. Although analytical solutions are available for a limited number of simplified problems, practical transport problems generally require numerical treatment. The present study investigates the numerical solution of a one-dimensional advection–diffusion transport equation using finite difference methods. Particular attention is given to explicit, implicit and Crank–Nicolson discretization’s and to the influence of spatial and temporal step sizes on stability, accuracy and numerical error. The study considers the mathematical formulation of the transport equation, finite-difference approximation of the first- and second-order derivatives, boundary conditions and stability requirements. The numerical methodology is designed around classical test problems for which analytical solutions can be obtained, allowing the numerical results to be compared directly with reference solutions. The investigation emphasizes truncation error, numerical diffusion, stability and convergence.  

Author

Priyanka and Vakul Bansal  ( Pages 469-482 )
Email:rpdhawan@gmail.com
Affiliation: Department of Physics, CMJ University, G.S Road, Jorabat, Ri-Bhoi District, Meghalaya       DOI: https://doi.org/10.58517/ITMSC.2011.4230

Keyword

Advection–diffusion equation, finite difference method, numerical analysis, Crank–Nicolson method, upwind scheme, stability, convergence, numerical diffusion, transport equation

References

1. Anderson, D.A., Tannehill, J.C. and Pletcher, R.H. (1984). Computational Fluid Mechanics and Heat Transfer. New York: McGraw-Hill.

2. Burden, R.L. and Faires, J.D. (2001). Numerical Analysis. 7th ed. Pacific Grove: Brooks/Cole.

3. Chapra, S.C. and Canale, R.P. (2006). Numerical Methods for Engineers. 5th ed. New York: McGraw-Hill.

4. Crank, J. and Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proceedings of the Cambridge Philosophical Society, 43, 50–67.

5. Devkota, B. (2009). Lagrangian modeling of advection-diffusion transport in open channel flow. Water Resources Research, 45.