Geoscience Reference
In-Depth Information
TABLE 6.1
Example of a Time Evolution in a 1D Channel Computed Using Explicit
Central Differences, a Unitary Courant Number, and No Diffusion
Grid Point Number
Time Step
Total Amount
+
+
+
i - 3
i - 2
i - 1
i
i
1
i
2
i
3
0
0
0
0
1
0
0
0
1
1
0
0.00
-0.50
1.00
0.50
0.00
0
1
2
0
0.25
-1.00
0.5
1.00
0.25
0
1
3
0
0.75
-1.13
-0.50
1.13
0.75
0
1
4
0
1.31
-0.50
-1.63
0.50
1.31
0
1
5
0
1.56
0.97
-2.13
-0.97
1.56
0
1
6
0
1.08
2.81
-1.16
-2.81
1.08
0
1
7
0
-0.33
3.93
1.66
-3.93
-0.33
0
1
8
0
-2.29
2.94
5.59
-2.94
-2.29
0
1
9
0
-3.76
-1.00
8.52
1.00
-3.76
0
1
10
0
-3.26
-7.14
7.52
7.14
-3.26
0
1
11
0
0.31
-12.54
0.38
12.54
0.31
0
1
diffusion. In that case Equation (6.2) becomes
t
t
+
t
t
CC
t
U CC
x
i
i
=
i
1
i
+
1
2
(6.14)
Ut
x
Ut
x
1 2
1 2
t
+
t
t
t
t
C
=
CC
+
C
i
i
1
i
i
1
Ut
x
=
where is the Courant number representing the ratio between the path length
of a particle during a time step and the grid size. This is a critical parameter for
most discretizations. Let us consider the case of a channel where initial conditions
are zero everywhere except in a generic point i . Table 6.1 shows the temporal
evolution along 11 time steps (0 to 11) for the case of a unitary Courant number
( C r
C r
=
1) and Table 6.2 shows the corresponding solution for the case of C r
=
2.
3 represent the boundary conditions (zero
outside of the modeling area) and total amount stands for the total amount of matter
inside the channel. Both solutions are unrealistic.
In such conditions, one would expect the contaminated water to move forward
and, after a certain time, the entire channel should have a concentration equal to
zero because the water entering the model area has concentration zero. The value
of the total amount of matter inside the channel should remain constant until the
matter reaches the outflow boundary, and then drop to zero while it leaves the domain.
In both tables, columns i - 3 and i
+
6.2.5.2
Stability
A model is said to be unstable if errors generated inside the modeling area are
amplified. This is what has happened in both the calculations. As time increased,
 
 
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