Environmental Engineering Reference
In-Depth Information
size is determined, the limitation to a Courant's number of less than one provides an
estimate of the maximum time step to be used for numerical simulation. On the
other hand, for transport in materials with low permeability, such as clay barriers,
Péclet numbers are significantly lower. In this case, diffusion is the dominant
phenomenon and Galerkin's method is well suited.
13.6.5.
Generalization in two and three dimensions
Interesting problems are not one-dimensional but two- or three-dimensional. To
solve them, the same numerical technology should be implemented. We must use
weighting functions coming from shape functions and corrected by a
decentering
in
the opposite direction to the flow, and exclusively in it [BIV 93, ZIE 89]. The
difficulty here comes from the fact that the velocity of the transporting fluid is
oblique to the elements and varies from one to another. Therefore, decentering
becomes a complex operation and there is no longer any possibility of obtaining the
exact solution. A small numerical diffusion is still present.
Finally, if we are interested in transient problems, which are the most interesting
environmental problems, it is still necessary to achieve a time discretization and to
verify its numerical stability. It is possible to simply translate the above-mentioned
concepts and to associate a standard time discretization with a Petrov-Galerkin-type
formulation, as in purely diffusive problems: linear variation of the concentration
during a time step, generalized midpoint or generalized trapeze method or full
decentering method, FUPG.
13.6.6.
The method of characteristics and Galerkin's method
The developments above clearly show the difficulties of simultaneously
modeling both the advection and diffusion effects. The method of characteristics is
ideal for modeling advection, but is not suitable for diffusion. Conversely, the finite
element method is excellent for modeling diffusion effects but it fails to properly
take pure advection into account. The combination of the two methods is likely to
solve this difficulty. In a way, it dissociates the differential operator [13.22] that
represents the problem to be solved in two successive operators, representing the
diffusion and advection aspects respectively.
Many researchers have explored this path and several procedures have been
proposed [BIV 93, LI 97, ZIE 89, ZIE 95]. They differ in the methodology of
coupling the two methods.
