184
6. Numerical Solutions of Advection-Dominated Problems
represents the concentration vector of the single step reverse point at time t k ,
which may be determined from the known concentration vector e k by means
of interpolation. CHI can then be computed using Eq. (6.4.25). This is the
single step reverse F EM, which is also called the modified characteristics
FEM.
For each node i, ct+ 1 - C~. represents the difference between the total
concentration and the concent~ation due to pure advection, i.e., the concentration variation caused by dispersion. It is written as C;(t H1 ), and we have
(6.4.26)
o
Since Cf+1 has been solved by Eq. (6.4.25), C;(tHd is also known. Using finite
element interpolation,
o
N 0
C;(x, Y, tHd = L C;(tHdtp;(x, y),
(6.4.27)
;=1
we can obtain the concentration change generated by the dispersion effect for
an arbitrary point (x, y) at time t H1 .
In order to increase the accuracy in simulating the steep front, we may
place some moving points in the domain to trace the development of the
front. It is assumed that there are v moving points, and their coordinates are
(x m , Ym), m = 1, 2, ... , v. If their concentrations C! (m = 1,2, ... ,v) at time t k
are known, their concentrations at time t H1 should be the sum of advection
and dispersion effects:
o
C~+l = C~ + Ci(xm ,Ym,tH1 )
(6.4.28)
Neuman (1984) proposed a smooth criterion for controlling the number of
moving points in the elements. If the concentrations of all moving points
calculated from Eq. (6.4.28) in an element are between the maximum and
minimum concentrations of the element, then the smooth criterion is considered to be satisfied in the element. If an element satisfies the smooth criterion
for several time steps in succession, the moving points within the ~:lement
may be eliminated to increase the efficiency of the solution process. If the
criterion is not satisfied, we can update the nodal concentrations by projecting the moving points onto the nodes and introducing new moving points.
The concentrations of moving points are defined by equations similar to
Eq. (6.4.13). Furthermore, we can use
N
CHI = '" C~+I.1..(X y)
m
~''''l m' m
;=1
to define the concentrations of the nodes.
(6.4.29)
This method has advantages of both the Eulerian and the Lagrangian
methods. It is applicable for any Peclet number, from 0 to 00, and even if the
6. Numerical Solutions of Advection-Dominated Problems
represents the concentration vector of the single step reverse point at time t k ,
which may be determined from the known concentration vector e k by means
of interpolation. CHI can then be computed using Eq. (6.4.25). This is the
single step reverse F EM, which is also called the modified characteristics
FEM.
For each node i, ct+ 1 - C~. represents the difference between the total
concentration and the concent~ation due to pure advection, i.e., the concentration variation caused by dispersion. It is written as C;(t H1 ), and we have
(6.4.26)
o
Since Cf+1 has been solved by Eq. (6.4.25), C;(tHd is also known. Using finite
element interpolation,
o
N 0
C;(x, Y, tHd = L C;(tHdtp;(x, y),
(6.4.27)
;=1
we can obtain the concentration change generated by the dispersion effect for
an arbitrary point (x, y) at time t H1 .
In order to increase the accuracy in simulating the steep front, we may
place some moving points in the domain to trace the development of the
front. It is assumed that there are v moving points, and their coordinates are
(x m , Ym), m = 1, 2, ... , v. If their concentrations C! (m = 1,2, ... ,v) at time t k
are known, their concentrations at time t H1 should be the sum of advection
and dispersion effects:
o
C~+l = C~ + Ci(xm ,Ym,tH1 )
(6.4.28)
Neuman (1984) proposed a smooth criterion for controlling the number of
moving points in the elements. If the concentrations of all moving points
calculated from Eq. (6.4.28) in an element are between the maximum and
minimum concentrations of the element, then the smooth criterion is considered to be satisfied in the element. If an element satisfies the smooth criterion
for several time steps in succession, the moving points within the ~:lement
may be eliminated to increase the efficiency of the solution process. If the
criterion is not satisfied, we can update the nodal concentrations by projecting the moving points onto the nodes and introducing new moving points.
The concentrations of moving points are defined by equations similar to
Eq. (6.4.13). Furthermore, we can use
N
CHI = '" C~+I.1..(X y)
m
~''''l m' m
;=1
to define the concentrations of the nodes.
(6.4.29)
This method has advantages of both the Eulerian and the Lagrangian
methods. It is applicable for any Peclet number, from 0 to 00, and even if the