6.4. The Modified Methods of Characteristics
185
Courant number exceeds 1, satisfactory results will still be obtained. Since
the moving points are placed only near the steep front, the computational
efficiency is rather high. Another advantage of this method is that the coefficient matrix in the finite element equation remains symmetrie. This also
contributes to the efficiency of the method. Some applications of this method
were given by Neuman (1984), Cady and Neuman (1987), and Huang Kangle
(1985). Instead of using the FEM, Bentley and Pinder (1992) used the
least squares collocation method to combine with the Eulerian-Lagrangian
method.
Yeh (1990) proposed an Eulerian-Lagrangian method, in wh ich the technique of self-adaptive element partition is used instead of moving points. In
each element there are K hidden no des. When the concentration front in an
element becomes steep, the hidden nodes will emerge as common nodes. The
element is then subdivided into many finer elements in order to reduce the
local Peclet number and increase the accuracy of the solution. Conversely,
when the concentration front in the element becomes smooth, these nodes
will return to the hidden state for the sake of reducing the amount of computation. Yeh (1990) called this method the Lagrangian-Eulerian method with
zoomable hidden jine-grid (LEZOOM). Its basic steps are similar to those of
the hybrid moving points FEM (FDM), but the nodes which can hide or
emerge are used to substitute for the moving points that may be added or
removed. This approach is more convenient in programming and more accurate for depicting the front than the moving points method.
A new approach, the Eulerian-Lagrangian localized adjoint method
(ELLAM), for solving the advection-dispersion equation was developed by
Celia et al. (1990) and Russell (1990). ELLAM can consistently treat boundary fluxes and maintain the mass balance. Thus, the problem of large mass
balance error associated with characteristic methods may be avoided. Recently, Healy and Russell (1993) combined ELLAM with an integrated finite
difference discretization. Both global and local mass balances are guaranteed.
When the flow velocity changes with time and space, the key to increasing
the accuracy of Lagrangian-Eulerian methods is to improve the calculation
of the velocity field. Generally, the groundwater flow problem is solved first
for obtaining the head distribution, and then Darcy's law is employed to
compute the distribution of velocities. During this process, the solution accuracy is greatly affected by the numerical differentiation. Segol et al. (1975),
Ewing et al. (1983), and Russell and Wheeler (1983) suggested the use of the
mixed FEM, in which Darcy's velocities are taken as the solutions of the
first-order partial differential equations to be solved simultaneously with
the distribution of the water heads. This improves the accuracy in computing
the velocity field. We shall give an introduction to this method in combination with the discussion of salt intrusion problems in Seetion 8.2. Chiang
et al. (1989) presented a numerical method for solving the problems of mass
transport in groundwater, in which the modified characteristics method
(MMOC) is combined with the mixed finite element (MFE) method.
185
Courant number exceeds 1, satisfactory results will still be obtained. Since
the moving points are placed only near the steep front, the computational
efficiency is rather high. Another advantage of this method is that the coefficient matrix in the finite element equation remains symmetrie. This also
contributes to the efficiency of the method. Some applications of this method
were given by Neuman (1984), Cady and Neuman (1987), and Huang Kangle
(1985). Instead of using the FEM, Bentley and Pinder (1992) used the
least squares collocation method to combine with the Eulerian-Lagrangian
method.
Yeh (1990) proposed an Eulerian-Lagrangian method, in wh ich the technique of self-adaptive element partition is used instead of moving points. In
each element there are K hidden no des. When the concentration front in an
element becomes steep, the hidden nodes will emerge as common nodes. The
element is then subdivided into many finer elements in order to reduce the
local Peclet number and increase the accuracy of the solution. Conversely,
when the concentration front in the element becomes smooth, these nodes
will return to the hidden state for the sake of reducing the amount of computation. Yeh (1990) called this method the Lagrangian-Eulerian method with
zoomable hidden jine-grid (LEZOOM). Its basic steps are similar to those of
the hybrid moving points FEM (FDM), but the nodes which can hide or
emerge are used to substitute for the moving points that may be added or
removed. This approach is more convenient in programming and more accurate for depicting the front than the moving points method.
A new approach, the Eulerian-Lagrangian localized adjoint method
(ELLAM), for solving the advection-dispersion equation was developed by
Celia et al. (1990) and Russell (1990). ELLAM can consistently treat boundary fluxes and maintain the mass balance. Thus, the problem of large mass
balance error associated with characteristic methods may be avoided. Recently, Healy and Russell (1993) combined ELLAM with an integrated finite
difference discretization. Both global and local mass balances are guaranteed.
When the flow velocity changes with time and space, the key to increasing
the accuracy of Lagrangian-Eulerian methods is to improve the calculation
of the velocity field. Generally, the groundwater flow problem is solved first
for obtaining the head distribution, and then Darcy's law is employed to
compute the distribution of velocities. During this process, the solution accuracy is greatly affected by the numerical differentiation. Segol et al. (1975),
Ewing et al. (1983), and Russell and Wheeler (1983) suggested the use of the
mixed FEM, in which Darcy's velocities are taken as the solutions of the
first-order partial differential equations to be solved simultaneously with
the distribution of the water heads. This improves the accuracy in computing
the velocity field. We shall give an introduction to this method in combination with the discussion of salt intrusion problems in Seetion 8.2. Chiang
et al. (1989) presented a numerical method for solving the problems of mass
transport in groundwater, in which the modified characteristics method
(MMOC) is combined with the mixed finite element (MFE) method.