On Options for the Numerical Modelling of the Diffusion Term in River Pollution Simulations

  • S. Neelz
  • S. G. Wallis
  • J. R. Manson


In this paper five numerical methods for modelling the diffusion term in a one-dimensional advection-diffusion equation are compared. The motivation behind the work is to find a computationally efficient method for modelling diffusion for incorporating in a semi-Lagrangian approach for advection. Three of the schemes are traditional Eulerian implicit methods (backward, Crank-Nicholson, optimised time-weighted): the other two are based on work by Teixeira (Teixeira, 1998) who proposed a method based on a discrete view of diffusion.

Numerical tests were undertaken in a non-advective uniform flow and concerned the evolution of an initial Gaussian spatial distribution of a conservative solute. Tests covered a range of initial spatial resolutions and a range of dimensionless diffusion numbers. For each test, the number of time steps required until the numerical solution agreed to within a specified tolerance of the exact solution to the problem was noted.

The results showed that an optimized time-weighted implicit scheme (used with a weighting coefficient taking values close to 0.7) was twice as efficient as the backward implicit scheme. Teixeira’s original method proved to be rather inefficient in general, but a modified form of it was as efficient as the optimum time-weighted scheme. The Crank-Nicholson scheme was the most efficient unless grid-scale oscillations were present, when it became very inefficient.


Advection Nite Cote 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abbott M B and Basco D R (1979). Computational Fluid Dynamics : an Introduction for Engineers. Longman.Google Scholar
  2. Celia M A, Russell T F, Herrera I, and Ewing R E (1990). An Eulerian-Lagrangian Localized Adjoint Method for the Advection-Diffusion Transport Equation. Advances in Water Resources 13, pp 187-206.CrossRefGoogle Scholar
  3. Einstein A (1905). Uber die von der Molekulärkinetischen Theorie der Wärme Geforderte Bewegung von in Ruhenden Flüssigkeiten Suspendierte Teilchen. Annalen der Physik 17, pp 549-.MATHCrossRefGoogle Scholar
  4. Fischer H B, List E J, Kohm R C Y, Imberger j and Brooks N H (1979). Mixing in Inland and Coastal Waters. Academic Press, New York.Google Scholar
  5. Manson J R and Wallis S G (1998). Accurate Simulation of Transport Processes in Two-Dimensional Shear Flow. Communications in Numerical Methods in Engineering 14, pp 863-869.MathSciNetMATHCrossRefGoogle Scholar
  6. Néelz S (2000). Numerical Prediction of Pollutant Fate in Streams. Licensiat Thesis, Department of Civil and Environmental Engineering, KTH, Stockholm, Sweden.Google Scholar
  7. Staniforth A and Côté J (1991). Semi-Lagrangian Integration Schemes for Atmospheric Models - a Review. Monthly Weather Review 119, pp 2206-2223.CrossRefGoogle Scholar
  8. Taylor G I (1954). The Dispersion of Matter in Turbulent Flow through a Pipe. Proceedings of the Royal Society of London A 223, pp 446-468.CrossRefGoogle Scholar
  9. Teixeira J (1998). Numerical Stability of the Physical Parametrizations in Atmospheric Models and a New Method to Solve the Diffusion Equation. Proceedings of the 6th Conference on Numerical Methods for Fluids Dynamics, ICFD, Oxford, pp 523-529.Google Scholar
  10. Van Dam G C (1994). Study of Shear Dispersion in Tidal Waters by Applying Discrete Particle Techniques. Mixing and Transport in the Environment, pp 269-. Reven K J, Chatwin P C and Millbank J H (Editors). John Wiley & Sons Ltd.Google Scholar
  11. Wallis S G and Manson J R (1997). Accurate Numerical Simulation of Advection Using Large Time Steps. International Journal for Numerical Methods in Fluids 24, pp 127-139.MATHCrossRefGoogle Scholar
  12. Wallis S G, Manson J R and Filippi L (1998). A Conservative Semi-Lagrangian Algorithm for One-Dimensional Advection-Diffusion. Communications in Numerical Methods in Engineering 14, pp 671-679.MathSciNetMATHCrossRefGoogle Scholar
  13. Wallis S G, Manson J R and Rafique S (1999). Limitations of Advection-Dispersion Calculations in Rivers. Proceedings of 28th IAHR Congress, Graz, Austria.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • S. Neelz
    • 1
  • S. G. Wallis
    • 2
  • J. R. Manson
    • 3
  1. 1.Royal Institute of TechnologyStockholmSweden
  2. 2.Heriot-Watt UniversityUK
  3. 3.Rensselaer Polytechnic InstituteTroyUSA

Personalised recommendations