Geoscience Reference
In-Depth Information
TABLE 6.3
Example of a Time Evolution in a 1D Channel Computed Using
Explicit Upstream Differences,
C
r
= 1.0, and No Diffusion
Grid Point Number
+
+
+
Time Step
i
- 3
i
- 2
i
- 1
i
i
1
i
2
i
3
Total Amount
0
0
0
0
1
0
0
0
1
1
0
0
0
0
1
0
0
1
2
0
0
0
0
0
1
0
1
3
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
9
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
6.2.5.3
The Need for a Fine Resolution Grid
The reason why the upstream scheme with
C
r
0.5 gives such poor results is the
coarse discretization used. In this case, matter travels only half of the grid size and
consequently the matter contained in cell
i
at
t
=
=
0 is distributed between two
1. Because the concentration is computed as the mass
divided by the volume, its value is reduced to
1
/
2
. This result is obtained because
the initial hypothesis that “the grid cell is small enough to allow the concentration
computing cells at time
t
=
TABLE 6.4
Example of a Time Evolution in a 1D Channel Computed Using Explicit
Upstream Differences,
C
r
=
0.5, and No Diffusion
Grid Point Number
+
+
+
Time Step
i
- 3
i
- 2
i
- 1
i
i
1
i
2
i
3
Total Amount
0
0
0
0
1
0
0
0
1
1
0
0.00
0.00
0.50
0.50
0.00
0
1
2
0
0.00
0.00
0.25
0.50
0.25
0
1
3
0
0.00
0.00
0.13
0.38
0.38
0
0.88
4
0
0.00
0.00
0.06
0.25
0.38
0
0.69
5
0
0.00
0.00
0.03
0.16
0.31
0
0.5
6
0
0.00
0.00
0.02
0.09
0.23
0
0.34
7
0
0.00
0.00
0.01
0.05
0.16
0
0.23
8
0
0.00
0.00
0.00
0.03
0.11
0
0.14
9
0
0.00
0.00
0.00
0.02
0.07
0
0.09
10
0
0.00
0.00
0.00
0.01
0.04
0
0.05
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