154
6. Numerical Solutions of Advection-Dominated Problems
but shorter waves are not weIl damped. As a result, the front of the numerical
solution is steep but has significant oscillations. For e = 1.0, the phase angle
lag is significant, but the amplitudes of shorter waves are strongly damped.
As a result, the front of the numerical solution is rather smooth while the
oscillation is weakened. Therefore, within a reasonable range, we can find a
compromise between decreasing the oscillation and maintaining the shape of
the front through adjusting the coefficient e. When the Peclet number is
larger, the situation will become worse and a satisfactory solution may not be
obtained.
6.1.2 Eulerian and Lagrangian Reference Frames
In Section 4.2, we discussed the difference between using Eulerian and
Lagrangian reference frames for solving advection-dispersion problems. The
former uses a spatiaIly-fixed coordinate system, while the latter adopts a
coordinate system which follows the movement offluid. In order to overcome
the difficulties encountered in using the traditional numerical methods for
solving advection-dominated problems, many modified methods have been
presented. Some of them are of the Eulerian type, some are of the Lagrangian
type and others are a mix of the two.
The traditional FEM and FDM both belong to the Eulerian type and are
characterized by spatiaIly-fixed grids. The mass dissolved and mixed in the
fluid moves in the grids following the "advection" and "dispersion" rules.
The governing equation is derived under the assumption that the solute mass
in each grid square or element keeps in balance at all time. Its solution,
C(x, y, z, t), can depict the solute distribution at any moment of the whole
region through the concentration contours. The Eulerian methods applicable
for solving the advection-dominated problems are mainly composed of the
upstream weighted FEM and FDM. Their advantages are that the mass
balance can be maintained either locally or in the whole region, and the
concentration contours can be easily genera ted. The disadvantages of these
methods are that the numerical dispersion cannot be completely eliminated
and the solution oscillations can be damped only at the expense of solution
accuracy.
The Lagrangian methods are powerful for solving the problem of numerical
dispersion. When the advection-dispersion phenomena are observed in a
moving coordinate system, the advection term may be reduced or eliminated
by controlling the velocity of the moving co ordinate system. The phenomenon of advection-dispersion observed in the moving coordinate system is
always dispersion-dominated, and the effects of advection and dispersion can
be computed separately. As a result, the numerical dispersion may be reduced or eliminated. Lagrangian methods involve the moving coordinate
method, the deforming grid method, the moving point method, and the
Random-Walk method mentioned in Chapter 4. Although these methods
6. Numerical Solutions of Advection-Dominated Problems
but shorter waves are not weIl damped. As a result, the front of the numerical
solution is steep but has significant oscillations. For e = 1.0, the phase angle
lag is significant, but the amplitudes of shorter waves are strongly damped.
As a result, the front of the numerical solution is rather smooth while the
oscillation is weakened. Therefore, within a reasonable range, we can find a
compromise between decreasing the oscillation and maintaining the shape of
the front through adjusting the coefficient e. When the Peclet number is
larger, the situation will become worse and a satisfactory solution may not be
obtained.
6.1.2 Eulerian and Lagrangian Reference Frames
In Section 4.2, we discussed the difference between using Eulerian and
Lagrangian reference frames for solving advection-dispersion problems. The
former uses a spatiaIly-fixed coordinate system, while the latter adopts a
coordinate system which follows the movement offluid. In order to overcome
the difficulties encountered in using the traditional numerical methods for
solving advection-dominated problems, many modified methods have been
presented. Some of them are of the Eulerian type, some are of the Lagrangian
type and others are a mix of the two.
The traditional FEM and FDM both belong to the Eulerian type and are
characterized by spatiaIly-fixed grids. The mass dissolved and mixed in the
fluid moves in the grids following the "advection" and "dispersion" rules.
The governing equation is derived under the assumption that the solute mass
in each grid square or element keeps in balance at all time. Its solution,
C(x, y, z, t), can depict the solute distribution at any moment of the whole
region through the concentration contours. The Eulerian methods applicable
for solving the advection-dominated problems are mainly composed of the
upstream weighted FEM and FDM. Their advantages are that the mass
balance can be maintained either locally or in the whole region, and the
concentration contours can be easily genera ted. The disadvantages of these
methods are that the numerical dispersion cannot be completely eliminated
and the solution oscillations can be damped only at the expense of solution
accuracy.
The Lagrangian methods are powerful for solving the problem of numerical
dispersion. When the advection-dispersion phenomena are observed in a
moving coordinate system, the advection term may be reduced or eliminated
by controlling the velocity of the moving co ordinate system. The phenomenon of advection-dispersion observed in the moving coordinate system is
always dispersion-dominated, and the effects of advection and dispersion can
be computed separately. As a result, the numerical dispersion may be reduced or eliminated. Lagrangian methods involve the moving coordinate
method, the deforming grid method, the moving point method, and the
Random-Walk method mentioned in Chapter 4. Although these methods