Agriculture Reference
In-Depth Information
set of
N
equations having
N
unknown concentrations. The form of the
N
equations is
+
+
=
aC
b
Cu
C
e
++
(3.71)
ij
,
i
-1,
j
+
1
ij
,
ij
,
+
1
ij
,
i
1,
j
1
ij
,
where
N
is the number of incremental distances in the soil (
N
=
L
/Δ
x
). By
including the appropriate initial and boundary conditions in their finite-
difference forms, Equation 3.33 can be written in matrix-vector notation as:
→
→
n
+
1
=
(3.72)
Bh
(
)
w
where
B
is a tridiagonal real matrix and
h
and
w
denote the associated real
column vectors (the arrows indicate vectors). The matrix
B
may be written as:
n
n
b
u
1
1
n
n
n
a
b
u
2
2
2
a
n
b
n
u
n
3
3
3
_
(3.73)
−
−
n
n
n
a
b
u
N
−
1
N
−
1
N
−
1
n
n
a
b
N
N
The coefficients of the main diagonal of the matrix
B,
in absolute values, are
greater than the raw sum of the off-diagonal coefficients. Hence, the matrix
B
is strictly diagonally dominant. Therefore, the matrix is nonsingular, and
there exists a solution
n
+ 1 for the matrix vector equation (Equation 3.71) that
is unique. The tridiagonal system of equations was solved by an adaptation
of the Gaussian algorithm. Therefore, for any time step,
n
+ 1, where all vari-
ables are assumed to be known at time step
n,
Equation 3.71 can be solved
sequentially until a desired time
t
is reached.
The coefficients
a
,
b
,
u
, and
e
are the associated set of equation parameters.
The above
N
equations were solved simultaneously, for each time step,
using the Gaussian elimination method (Carnahan, Luther, and Wilkes,
1969) in order to obtain the concentration
C
at all nodal points (
i
) along the
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