172
6. Numerical Solutions of Advection-Dominated Problems
The UWMCBMs have been extended to solve multiple layer and threedimensional advection-dominated transport problems, which can be seen in the
papers written by Sun et al. (1984) and Wang et al. (1986).
All the upstream weighted FDMs and FEMs introduced in this section
contain undetermined upstream weighting factors, e.g., IX in Eqs. (6.2.8) and
(6.2.18), and IXi' IXi' IXk in Eq. (6.2.25). In the UWMCBM, the upstream weights
are added directly to basis functions and the values of the weighting factors
are associated with the coefficients of the equations. However, there is still an
undetermined upstream weighting factor A, as shown in Eq. (6.2.58). An
appropriate selection of weighting factors may result in an optimal compromise between controlling oscillation of the numerical solution and decreasing
numerical dispersion, but in practice it is not easy to find the optimal factors.
Herrera (1985a,b), Celia and Herrera (1987), Celia et al. (1989), and
Kindred and Celia (1989) proposed an optimal weighting function method
for determining the weighting coefficients. The optimal weighting functions
to be derived are completely determined by the coefficients of the governing
equations.
Let us consider the following one-dimensional advective-difJusive-reactive
transport equation:
(6.2.59)
where retardation factor R, velocity V, diffusion coefficient D and reaction
coefficient Kare assumed to be constants, and Q is the sourcejsink term.
With different proportions of these coefficients, various cases of Eq. (6.2.59)
will occur, such as advection-dominated, diffusion-dominated and reactiondominated cases. Obviously, the same weighting function cannot yield satisfactory results for all different cases.
The optimal weighting function designed by Celia et al. (1989) depends on
the coefficients of Eq. (6.2.59). It can automatically adapt to various cases,
such as advection-dominated, diffusion-dominated, and reaction-dominated
equations, without any undetermined weighting factors.
6.3 Moving Co ordinate System and Moving
Point Methods
6.3.1 M oving Coordinate System M ethods
Let us return to the canonical one-dimensional advection-dispersion equation:
ae
ae
a 2 e
at + V ax - D ax2 = O.
(6.3.1)
If the advection-dispersion phenomena are observed in a coordinate system
~ moving with the velocity V, only dispersion will be seen. Using the coordi-
6. Numerical Solutions of Advection-Dominated Problems
The UWMCBMs have been extended to solve multiple layer and threedimensional advection-dominated transport problems, which can be seen in the
papers written by Sun et al. (1984) and Wang et al. (1986).
All the upstream weighted FDMs and FEMs introduced in this section
contain undetermined upstream weighting factors, e.g., IX in Eqs. (6.2.8) and
(6.2.18), and IXi' IXi' IXk in Eq. (6.2.25). In the UWMCBM, the upstream weights
are added directly to basis functions and the values of the weighting factors
are associated with the coefficients of the equations. However, there is still an
undetermined upstream weighting factor A, as shown in Eq. (6.2.58). An
appropriate selection of weighting factors may result in an optimal compromise between controlling oscillation of the numerical solution and decreasing
numerical dispersion, but in practice it is not easy to find the optimal factors.
Herrera (1985a,b), Celia and Herrera (1987), Celia et al. (1989), and
Kindred and Celia (1989) proposed an optimal weighting function method
for determining the weighting coefficients. The optimal weighting functions
to be derived are completely determined by the coefficients of the governing
equations.
Let us consider the following one-dimensional advective-difJusive-reactive
transport equation:
(6.2.59)
where retardation factor R, velocity V, diffusion coefficient D and reaction
coefficient Kare assumed to be constants, and Q is the sourcejsink term.
With different proportions of these coefficients, various cases of Eq. (6.2.59)
will occur, such as advection-dominated, diffusion-dominated and reactiondominated cases. Obviously, the same weighting function cannot yield satisfactory results for all different cases.
The optimal weighting function designed by Celia et al. (1989) depends on
the coefficients of Eq. (6.2.59). It can automatically adapt to various cases,
such as advection-dominated, diffusion-dominated, and reaction-dominated
equations, without any undetermined weighting factors.
6.3 Moving Co ordinate System and Moving
Point Methods
6.3.1 M oving Coordinate System M ethods
Let us return to the canonical one-dimensional advection-dispersion equation:
ae
ae
a 2 e
at + V ax - D ax2 = O.
(6.3.1)
If the advection-dispersion phenomena are observed in a coordinate system
~ moving with the velocity V, only dispersion will be seen. Using the coordi-