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Transport Equation

  • Akpofure E. Taigbenu
Chapter

Abstract

Having gone through the earlier chapters in which derivations of the Green element method were done for linear and nonlinear Laplace/Poisson problems in homogeneous and heterogeneous media, and for the transient linear and nonlinear diffusion equations, you should by now have a good feel of the way the method is derived for a differential operator. For each of these problems, we sought a system of discrete element equations and then assembled it for all the elements to obtain the global matrix equation from which the nodal solutions were computed. The core of the Green element formulation, therefore, lies in one’s ability to correctly derive these element equations for the differential operator. When that is done, everything else can be considered straightforward. Our attention in this chapter focuses on the derivation of the Green element equations for the contaminant transport problem. Under conditions of transport in a homogenous domain with uniform ambient flow, three distinct Green element formulations are derived. However, the amount of effort involved in formulating the element equations escalates appreciably as we progress from the first formulation to the third one. At the end of the exercise of going through these three formulations, coupled with the understanding gained from the two formulations for the diffusion equation, we will be in a position to assess the relationship between computational efficiency and versatility of GEM, and the nature of the free-space Green’s function employed. Furthermore, as you must have come to observe, domain integrations do not pose any unusual computational difficulties in Green element formulations. There is a very simple reason for this. In Green element formulations, the source and field nodes are always on the same element so that domain integrations are much easier to evaluate.

Keywords

Transport Equation Amplification Factor Green Element Element Equation Courant Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Gray, W.G. and G.F. Pinder, “An Analysis of the Numerical Solution of the Transport Equation,” Water Resources research, 12, pp. 547–555, 1976.CrossRefGoogle Scholar
  2. 2.
    Gray, W.G. and D.R. Lynch, “Time-Stepping Schemes for Finite Element Tidal Model Computations,” in Surface Flow, ed. W.G. Gray, pp. 1–14, CML Pub. Ltd., Southampton UK, 1984.Google Scholar
  3. 3.
    Ogata, A. and R.B. Banks, “A solution of the differential equation of longitudinal dispersion in porous media”, U.S. Geol. Surv. Prof. Pap. 411-A., 7pp, 1976.Google Scholar
  4. 4.
    van Genuchten, M. Th. and W.J. Alves, “Analytical solutions of the One-dimensional Convective-Dispersive solute transport equation”, Technical Bulletin No. 1661, 151p., U.S. Dept. of Agric., 1982.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Akpofure E. Taigbenu
    • 1
  1. 1.Department of Civil and Water EngineeringNational University of Science and TechnologyBulawayoZimbabwe

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