Let us first consider a purely advective process. In this case, the transport is governed by the advection Eq. 11.11 and by the balance Eq. 11.6. Associating these two equations, one obtains:
(VTC). ffluld + C = 0 (11.45)
diff v ‘
which is a hyperbolic differential equation. It cannot be solved by the finite element or finite difference problem, but by characteristic methods. The idea is to follow the movement of a pollutant particle by simply integrating step by step the fluid velocity field. This integration has to be accurate enough, as errors are cumulated from one step to the next.
On the other hand, if advection is very small compared to diffusion, then the finite element and finite difference methods are really efficient.
For most practical cases, an intermediate situation holds. It can be checked by Peclet’s number, Eq. 11.13, which is high for mainly advective processes and low for mainly diffusive ones. As diffusion has to be taken into account, the numerical solution must be based on the finite element method (the finite difference one may also be used but will not be discussed here). However, numerical experiments show that the classical Galerkin’s formulation gives very poor results with high spatial oscillations and artificial dispersion. Thus, new solutions have been proposed (Zienkiewicz & Taylor, 1989, Charlier & Radu, 2001). A first solution is based on the use in the weighted residual method of a weighting function that differs from the shape one by an upwind term, i. e. a term depending in amplitude and direction on the fluid velocity field. The main advantage of this method is the maintenance of the finite element code formalism. However, it is never possible to obtain a highly accurate procedure. Numerical dispersion will always occur.
Other solutions are based on the association of the characteristic method for the advection part of the process and of the finite element method for the diffusive part (Li et al., 1997). The characteristic method may be embedded in the finite element code, which has a strong influence on the finite element code structure. It is also possible to manage the two methods in separated codes, as in a staggered procedure (cf. Section 11.3.4).
