82
4. Finite DifIerence Methods
Eq. (4.1.32), are shown in Figure 4.4, where the numerical solution of Eq.
(4.1.33) is obtained by the implicit scheme. As shown in the figure, the front
of the analytic solution is vertical without any transition zone, but the numerical solution has one. Such a transition zone does not exist physically. It
is created by the error of numerical calculation, and is therefore generally
called the numerical dispersion. Let us analyze its origins. Suppose the backward difference approximation is used for oC/ox as in Eq. (4.1.33). According
to Taylor's formula, the major part of the truncation error caused by the
approximation should be [(V~x)/2J (0 2 C/ox 2 ). This term is equivalent to a
physical dispersion term with dispersion coefficient D = (V~x)/2. It is in fact
the numerical dispersion generated by the finite difference approximation.
When oC/ox in (4.1.33) is approximated by the central difference, it seems
that the numerical dispersion term is not in existence. However, since the
difference approximation is also used in replacing oC/ot on the left-hand side
ofEq. (4.1.33), another truncation error (~t/2)(02C/ot2) will be created. Using
Eq. (4.1.33), we have
~t 02C V2~t 02C
2 ot2 - -2- ox2 .
(4.1.34)
Therefore, a numerical dispersion with dispersion coefficient D = [V 2 MJ/2 is
created.
As stated above, when finite difference approximations are used to Eq.
(4.1.31), additional numerical dispersion may be produced by the truncation
error because of the existence of the first order partial derivative. The values
of numerical dispersion depend on ~x, ~t, and velocity V. When the order of
magnitude ofnumerical dispersion is the same as that ofthe real physical dispersion, the accuracy of the numerical solution will be significantly decreased.
As shown in Figures 4.2 and 4.3, when the Peclet number is large, the
oscillation phenomenon occurs in the numerical solution of finite difference.
At the upstream of the front, the calculated concentrations at some internal
nodes may be greater than the concentration Co on boundary x = O. This is
an unreasonable result. At the downstream ofthe front, there are some nodes
where calculated concentrations are negative. This is physically impossible.
These phenomena are called "overshoot." It is another kind of calculation
error.
To a certain extent, the finite difference method contain two kinds of
calculation errors-"numerical dispersion" and "overshoot." If the concentration front is not sharp, their effects are small; Otherwise, they may introduce large errors into the numerical solutions. There is a close relationship
between numerical dispersion and overshoot. As stated by Pinder and Gray
(1977), when a numerical scheme is developed to minimize the numerical
dispersion, overshoot is encountered; but when overshoot is controlled, it is
generally at the expense of increased numerical dispersion. This problem will
be discussed in detail in Chapter 6.
4. Finite DifIerence Methods
Eq. (4.1.32), are shown in Figure 4.4, where the numerical solution of Eq.
(4.1.33) is obtained by the implicit scheme. As shown in the figure, the front
of the analytic solution is vertical without any transition zone, but the numerical solution has one. Such a transition zone does not exist physically. It
is created by the error of numerical calculation, and is therefore generally
called the numerical dispersion. Let us analyze its origins. Suppose the backward difference approximation is used for oC/ox as in Eq. (4.1.33). According
to Taylor's formula, the major part of the truncation error caused by the
approximation should be [(V~x)/2J (0 2 C/ox 2 ). This term is equivalent to a
physical dispersion term with dispersion coefficient D = (V~x)/2. It is in fact
the numerical dispersion generated by the finite difference approximation.
When oC/ox in (4.1.33) is approximated by the central difference, it seems
that the numerical dispersion term is not in existence. However, since the
difference approximation is also used in replacing oC/ot on the left-hand side
ofEq. (4.1.33), another truncation error (~t/2)(02C/ot2) will be created. Using
Eq. (4.1.33), we have
~t 02C V2~t 02C
2 ot2 - -2- ox2 .
(4.1.34)
Therefore, a numerical dispersion with dispersion coefficient D = [V 2 MJ/2 is
created.
As stated above, when finite difference approximations are used to Eq.
(4.1.31), additional numerical dispersion may be produced by the truncation
error because of the existence of the first order partial derivative. The values
of numerical dispersion depend on ~x, ~t, and velocity V. When the order of
magnitude ofnumerical dispersion is the same as that ofthe real physical dispersion, the accuracy of the numerical solution will be significantly decreased.
As shown in Figures 4.2 and 4.3, when the Peclet number is large, the
oscillation phenomenon occurs in the numerical solution of finite difference.
At the upstream of the front, the calculated concentrations at some internal
nodes may be greater than the concentration Co on boundary x = O. This is
an unreasonable result. At the downstream ofthe front, there are some nodes
where calculated concentrations are negative. This is physically impossible.
These phenomena are called "overshoot." It is another kind of calculation
error.
To a certain extent, the finite difference method contain two kinds of
calculation errors-"numerical dispersion" and "overshoot." If the concentration front is not sharp, their effects are small; Otherwise, they may introduce large errors into the numerical solutions. There is a close relationship
between numerical dispersion and overshoot. As stated by Pinder and Gray
(1977), when a numerical scheme is developed to minimize the numerical
dispersion, overshoot is encountered; but when overshoot is controlled, it is
generally at the expense of increased numerical dispersion. This problem will
be discussed in detail in Chapter 6.