6.4. The Modified Methods of Characteristics
177
In the Random-Walk method, the modification of the concentrations of
moving points does not depend on a fixed grid. In other words, it is unnecessary to project the concentration of the moving points onto the nodes at
each time step. The modification of the concentration of moving points
genera ted by the dispersion efTect is determined by the rule of random walk
and so it is not necessary to project the concentration of the fixed nodes onto
the moving points either. Thus, the moving points are allowed to move freely
in the domain for many steps. Only when we wish to know the distribution
of the concentration is it necessary to project the concentrations of the
moving points onto the fixed grid. In this way, we can avoid calculation
errors resulting from repeated interpolations. The accuracy of Random-Walk
methods depends basically on the number of moving points. If the number is
insufficient, local mass balance cannot be maintained.
The techniques oi moving points are flexible and powerful in handling
of advection-dominated problems, but when the dispersion efTect increases,
solution accuracy may be decreased. As pointed out by Neuman (1981), the
convergence of these methods has not yet been proven. Although the numerical test made by Farmer (1985) showed that the method of characteristics
with quadratic interpolation is always convergent, the detail of the numerical
analysis was not given. Besides the Random-Walk methods, Raviat (1985)
proposed another moving point method that did not require a fixed grid. Some
improved forms of moving point methods will be presented in the next section.
6.4 The Modified Methods of Characteristics
6.4.1 The Single Step Reverse M ethod
Eulerian methods seem to be quite suitable for solving dispersion-dominated
problems, while Lagrangian methods are very powerful for handling
advection-dominated problems. The problem is how to combine the two
types to get satisfactory results for both large and small Peclet numbers. In
this section, we will introduce several Eulerian-Lagrangian methods.
The single step reverse method is a modified characteristics method. It was
first proposed by Hinstrup (1977), and further analyzed by Neuman and
Sorek (1982), Douglass and Russell (1982), and others. Cheng et al. (1984)
presented a finite difTerence version associated with the natural coordinate
system, which makes the method more practical.
Consider the following advection-dispersion equation:
Using the total derivative
oC
8t + V·VC = V·(DVC).
DC oC
-=-+V·VC
Dt
ot
'
(6.4.1)
(6.4.2)
177
In the Random-Walk method, the modification of the concentrations of
moving points does not depend on a fixed grid. In other words, it is unnecessary to project the concentration of the moving points onto the nodes at
each time step. The modification of the concentration of moving points
genera ted by the dispersion efTect is determined by the rule of random walk
and so it is not necessary to project the concentration of the fixed nodes onto
the moving points either. Thus, the moving points are allowed to move freely
in the domain for many steps. Only when we wish to know the distribution
of the concentration is it necessary to project the concentrations of the
moving points onto the fixed grid. In this way, we can avoid calculation
errors resulting from repeated interpolations. The accuracy of Random-Walk
methods depends basically on the number of moving points. If the number is
insufficient, local mass balance cannot be maintained.
The techniques oi moving points are flexible and powerful in handling
of advection-dominated problems, but when the dispersion efTect increases,
solution accuracy may be decreased. As pointed out by Neuman (1981), the
convergence of these methods has not yet been proven. Although the numerical test made by Farmer (1985) showed that the method of characteristics
with quadratic interpolation is always convergent, the detail of the numerical
analysis was not given. Besides the Random-Walk methods, Raviat (1985)
proposed another moving point method that did not require a fixed grid. Some
improved forms of moving point methods will be presented in the next section.
6.4 The Modified Methods of Characteristics
6.4.1 The Single Step Reverse M ethod
Eulerian methods seem to be quite suitable for solving dispersion-dominated
problems, while Lagrangian methods are very powerful for handling
advection-dominated problems. The problem is how to combine the two
types to get satisfactory results for both large and small Peclet numbers. In
this section, we will introduce several Eulerian-Lagrangian methods.
The single step reverse method is a modified characteristics method. It was
first proposed by Hinstrup (1977), and further analyzed by Neuman and
Sorek (1982), Douglass and Russell (1982), and others. Cheng et al. (1984)
presented a finite difTerence version associated with the natural coordinate
system, which makes the method more practical.
Consider the following advection-dispersion equation:
Using the total derivative
oC
8t + V·VC = V·(DVC).
DC oC
-=-+V·VC
Dt
ot
'
(6.4.1)
(6.4.2)