80
4. Finite Difference Methods
4.1.3 Numerical Dispersion and Overshoot
Through the above discussions, the reader may find that the traditional
difference methods used for solving the problems of groundwater flow can
basically be copied to solve the problems of advection-dispersion. The computational programs can also be written accordingly. In most cases, the
calculated results are satisfactory, when compared with the accurate solutions. However, for a problem with a small dispersion coefficient and relatively large velocity, common finite difference methods will not be satisfactory because the transition zone formed by the hydrodynamic dispersion will
not be calculated accurately. In other words, when the Peclet number of the
problem is large, computational difficulties will be encountered. In order
to illustrate this difficulty, let us consider the canonical one-dimensional
advection-dispersion equation as folIo ws:
ac
a 2 c
ac
at = D ax2 - V ax'
(4.1.31)
where D and V are constants, and the initial and boundary conditions are
{
C(X,o) = 0,
x> 0,
C(O,t) = Co, t;;::: 0,
C(oo,t) = 0, t;;::: 0.
(4.1.32)
The analytic solution of this problem has been obtained in paragraph
3.2.1. If finite difference methods are used, the grid Pec1et number is defined
as
V.1x
Pe =--v-.
When .1x is constant, the greater the ratio between V and D, the larger the
Pec1et number will be. Figures 4.2 and 4.3 represent the numerical solutions
of the finite difference in comparison with the analytic solutions at Pe = 0.5
and Pe = 100, respectively.
As shown in these figures, for a small Peclet number the numerical solution is very c10se to the analytic solution, but for a large Pec1et number the
transition zone of the numerical solution becomes wider, and oscillations
appear around the concentration front.
For the convenience of analysis, we shall first study the extreme case of
D = 0. In this case there is only advection without concrete physical dispersion. Thus, Eq. (4.1.31) can be reduced to
~~ = - V~~.
(4.1.33)
Its analytic and numerical solutions subject to the subsidiary conditions of
4. Finite Difference Methods
4.1.3 Numerical Dispersion and Overshoot
Through the above discussions, the reader may find that the traditional
difference methods used for solving the problems of groundwater flow can
basically be copied to solve the problems of advection-dispersion. The computational programs can also be written accordingly. In most cases, the
calculated results are satisfactory, when compared with the accurate solutions. However, for a problem with a small dispersion coefficient and relatively large velocity, common finite difference methods will not be satisfactory because the transition zone formed by the hydrodynamic dispersion will
not be calculated accurately. In other words, when the Peclet number of the
problem is large, computational difficulties will be encountered. In order
to illustrate this difficulty, let us consider the canonical one-dimensional
advection-dispersion equation as folIo ws:
ac
a 2 c
ac
at = D ax2 - V ax'
(4.1.31)
where D and V are constants, and the initial and boundary conditions are
{
C(X,o) = 0,
x> 0,
C(O,t) = Co, t;;::: 0,
C(oo,t) = 0, t;;::: 0.
(4.1.32)
The analytic solution of this problem has been obtained in paragraph
3.2.1. If finite difference methods are used, the grid Pec1et number is defined
as
V.1x
Pe =--v-.
When .1x is constant, the greater the ratio between V and D, the larger the
Pec1et number will be. Figures 4.2 and 4.3 represent the numerical solutions
of the finite difference in comparison with the analytic solutions at Pe = 0.5
and Pe = 100, respectively.
As shown in these figures, for a small Peclet number the numerical solution is very c10se to the analytic solution, but for a large Pec1et number the
transition zone of the numerical solution becomes wider, and oscillations
appear around the concentration front.
For the convenience of analysis, we shall first study the extreme case of
D = 0. In this case there is only advection without concrete physical dispersion. Thus, Eq. (4.1.31) can be reduced to
~~ = - V~~.
(4.1.33)
Its analytic and numerical solutions subject to the subsidiary conditions of