6.2. Upstream Weighted Methods
155
can eliminate the numerical dispersion, they cannot guarantee mass balance
and usually have difficulties in treating complicated practical problems.
In recent years, Lagrangian-Eulerian methods, such as the characteristicFDM, characteristic-FEM and so on, have been suggested. These methods
have the advantages, and, to some degree, the disadvantages, of both the
Lagrangian and the Eulerian methods. In the following sections, we will give
an introduction to some of the major methods that are currently under
development in this field.
6.2 Upstream Weighted Methods
6.2.1 Upstream Weighted Finite Difference Methods
In Eq. (6.1.1), if the first-order and second-order derivatives are replaced by
the central difference approximations as follows
oC I ~ Ci+l - Ci - 1 02C I ~ Ci +1 - 2Ci + Ci - 1
ox i
2Ax'
ox 2 i
(AX)2
'
then difference equation (6.1.8) at an internal node i can be derived. In
Section 4.1.6, we analyzed the relationship between truncation error and
numerical dispersion of a difference equation. If we want to obtain a satisfactory solution through this method, it is necessary to control the size of Ax
and At. Generally, the following two conditions should be satisfied:
1. The loeal Peclet number Pe< 1, i.e., VAx/D< 1. In order to satisfy this
condition, Ax < D/V should be adopted.
2. The loeal Courant number VAt/Ax< 1, which means that the average
displacement of the fluid is less than the length of one grid space in one
time step. To satisfy this condition, At < Ax/V should be adopted.
For example, suppose that the length of a one-dimensional dispersion
region is 12800, D = 2 and V = 0.5, then Ax = 4 should be used. The whole
region should be divided into 3200 grid spaces, and the time step At must not
exceed 8. Thus, it requires 1200 time steps to simulate the total time of 96000.
This is a highly inefficient method which requires an unacceptable amount of
calculation.
In order to search for more efficient methods, let us further analyze the
truncation errors and the stability of FDM. The central difference approximation of the second-order derivative may be expressed as
Ci+l - 2Ci + Ci - 1 = 02C I ~ Cl4l (A)2 HOT
(AX)2
ox 2 i + 12' x +
,
(6.2.1)
where HOT represents terms of a higher order than (AX)2 in the Taylor
expansion. Since the last two terms on the right-hand side of Eq. (6.2.1) only
involve fourth or higher order derivatives of concentration C, the finite dif-
155
can eliminate the numerical dispersion, they cannot guarantee mass balance
and usually have difficulties in treating complicated practical problems.
In recent years, Lagrangian-Eulerian methods, such as the characteristicFDM, characteristic-FEM and so on, have been suggested. These methods
have the advantages, and, to some degree, the disadvantages, of both the
Lagrangian and the Eulerian methods. In the following sections, we will give
an introduction to some of the major methods that are currently under
development in this field.
6.2 Upstream Weighted Methods
6.2.1 Upstream Weighted Finite Difference Methods
In Eq. (6.1.1), if the first-order and second-order derivatives are replaced by
the central difference approximations as follows
oC I ~ Ci+l - Ci - 1 02C I ~ Ci +1 - 2Ci + Ci - 1
ox i
2Ax'
ox 2 i
(AX)2
'
then difference equation (6.1.8) at an internal node i can be derived. In
Section 4.1.6, we analyzed the relationship between truncation error and
numerical dispersion of a difference equation. If we want to obtain a satisfactory solution through this method, it is necessary to control the size of Ax
and At. Generally, the following two conditions should be satisfied:
1. The loeal Peclet number Pe< 1, i.e., VAx/D< 1. In order to satisfy this
condition, Ax < D/V should be adopted.
2. The loeal Courant number VAt/Ax< 1, which means that the average
displacement of the fluid is less than the length of one grid space in one
time step. To satisfy this condition, At < Ax/V should be adopted.
For example, suppose that the length of a one-dimensional dispersion
region is 12800, D = 2 and V = 0.5, then Ax = 4 should be used. The whole
region should be divided into 3200 grid spaces, and the time step At must not
exceed 8. Thus, it requires 1200 time steps to simulate the total time of 96000.
This is a highly inefficient method which requires an unacceptable amount of
calculation.
In order to search for more efficient methods, let us further analyze the
truncation errors and the stability of FDM. The central difference approximation of the second-order derivative may be expressed as
Ci+l - 2Ci + Ci - 1 = 02C I ~ Cl4l (A)2 HOT
(AX)2
ox 2 i + 12' x +
,
(6.2.1)
where HOT represents terms of a higher order than (AX)2 in the Taylor
expansion. Since the last two terms on the right-hand side of Eq. (6.2.1) only
involve fourth or higher order derivatives of concentration C, the finite dif-