180
6. Numerical Solutions of Advection-Dominated Problems
1. Find the single step reverse point of each node at t k • For instance, the
coordinates ofnode Q are assumed to be (xQ'YQ)' so the coordinates ofits
single step reverse point P can be calculated by the following approximations:
(6.4.10)
2. Apply Eq. (6.4.8) to determine 6.s and 6.n, then find points SI' S2' NI
and N 2 along the direction of line PQ and its perpendicular direction,
respectively.
3. Use an interpolation method to obtain Ck(P), Ck(Sd, C k (S2)' Ck(Nd and
C k (N2 )·
4. Substitute these values into Eq. (6.4.9) to obtain Ci~?'
Evidently, 6.t must be sufficiently small so that the coordinates of point P
calculated from Eq. (6.4.10) can be guaranteed to be accurate enough, and the
convergence of the explicit scheme (6.4.9) can also be ensured. The key problem here is the choice of an interpolation formula for step 3 which can
maintain the accuracy of the solution. It is easy and feasible to apply the
bilinear interpolation with the neighboring four nodes taken as the basis
points. For instance, in Figure 6.15, the concentration of point P, i.e., C!' can
be determined by the bilinear interpolation based on Ct-l,j, CL-I' Ct+l,j'
and ctj+l' However, Cheng et al. (1984) pointed out that when the bilinear
interpolation formula is used, the accuracy of their method is equivalent to
the upstream FDM. Therefore, they suggested using the nine neighboring
nodes to generate a higher-order finite element interpolation.
In the single step reverse method, it is not necessary to locate and trace a
group of moving points. This is the main difference between the single step
reverse and the method of characteristics. As a result, the calculation process
of the single step reverse method is relatively simple. However, it still uses
moving points, such as P, SI' S2' NI' N2 and so on, but these moving points
are redefined in each time step.
Although we have applied the finite difference approximation to the dispersion part on the right-hand side of Eq. (6.4.5), the finite element approximation could be used instead. Neuman and Sorek (1982), Douglass and
Russell (1982) and others discussed the single step reverse techniques in
combination with FEM. Donea (1984) combined the linear Galerkin FEM
and a high-accuracy interpolation to form a single step reverse method.
6.4.2 The Hybrid Single Step Reverse-Moving
Point M ethod
Although moving points are already involved in the single step reverse method, their number may be insufficient for depicting a steep concentration front.
With few moving points, numerical dispersion will still exist near the front.
Neuman (1984) and Farmer (1985) suggested combining the single step re-
6. Numerical Solutions of Advection-Dominated Problems
1. Find the single step reverse point of each node at t k • For instance, the
coordinates ofnode Q are assumed to be (xQ'YQ)' so the coordinates ofits
single step reverse point P can be calculated by the following approximations:
(6.4.10)
2. Apply Eq. (6.4.8) to determine 6.s and 6.n, then find points SI' S2' NI
and N 2 along the direction of line PQ and its perpendicular direction,
respectively.
3. Use an interpolation method to obtain Ck(P), Ck(Sd, C k (S2)' Ck(Nd and
C k (N2 )·
4. Substitute these values into Eq. (6.4.9) to obtain Ci~?'
Evidently, 6.t must be sufficiently small so that the coordinates of point P
calculated from Eq. (6.4.10) can be guaranteed to be accurate enough, and the
convergence of the explicit scheme (6.4.9) can also be ensured. The key problem here is the choice of an interpolation formula for step 3 which can
maintain the accuracy of the solution. It is easy and feasible to apply the
bilinear interpolation with the neighboring four nodes taken as the basis
points. For instance, in Figure 6.15, the concentration of point P, i.e., C!' can
be determined by the bilinear interpolation based on Ct-l,j, CL-I' Ct+l,j'
and ctj+l' However, Cheng et al. (1984) pointed out that when the bilinear
interpolation formula is used, the accuracy of their method is equivalent to
the upstream FDM. Therefore, they suggested using the nine neighboring
nodes to generate a higher-order finite element interpolation.
In the single step reverse method, it is not necessary to locate and trace a
group of moving points. This is the main difference between the single step
reverse and the method of characteristics. As a result, the calculation process
of the single step reverse method is relatively simple. However, it still uses
moving points, such as P, SI' S2' NI' N2 and so on, but these moving points
are redefined in each time step.
Although we have applied the finite difference approximation to the dispersion part on the right-hand side of Eq. (6.4.5), the finite element approximation could be used instead. Neuman and Sorek (1982), Douglass and
Russell (1982) and others discussed the single step reverse techniques in
combination with FEM. Donea (1984) combined the linear Galerkin FEM
and a high-accuracy interpolation to form a single step reverse method.
6.4.2 The Hybrid Single Step Reverse-Moving
Point M ethod
Although moving points are already involved in the single step reverse method, their number may be insufficient for depicting a steep concentration front.
With few moving points, numerical dispersion will still exist near the front.
Neuman (1984) and Farmer (1985) suggested combining the single step re-