Agriculture Reference
In-Depth Information
4.7 Appendix A: Derivation of Finite-Difference Equations
for CDE with a Linearly Time-Dependent Dispersivity
The governing equation for a linear dispersivity model reads (Equation 4.8):
2
∂
∂
=−
∂
c
t
c
x
∂
∂
c
2
v
+
avt
(A1)
1
2
∂
x
Denoting time and space increments by
t
and
x
, we can establish the
finite difference scheme for point
ixjt
( , )
, where
i
and
j
are integers and
denote space and time steps, respectively. Finite difference approximations
of each partial derivative in Equation A1 are as follows:
j
+
1
j
∂
∂
c
t
c
−
c
=
i
i
(A2)
t
j
+
1
j
+
1
∂
∂
c
x
c
−
c
x
=
i
+
1
i
(A3)
2
j
+
1
j
+
1
j
+
1
j
j
j
∂
∂
c
1
2
t
c
−
2
c
+
c
jt
c
−+
2
c
c
t
=
( )
j
+
i
+
1
i
i
−
1
+
i
+
1
i
i
−
1
(A4)
2
2
2
x
x
x
where
c
i
j
stands for solute concentration at node
ixjt
(
,
)
. For convenience, let
t
x
β=
Substituting Equations A2 through A4 into Equation A1 and rearranging gives:
j
+
1
j
+
1
j
+
1
Ac
+
Bc
+
Cc
=
E
(A5)
i
−
1
i
i
+
1
where
1
2
2
2
Aa vj
=
( )
+ β
(A6)
1
2
2
B
=− +β−
1
v
av j
(
+ β
1)
(A7)
1
1
2
2
2
Ca vj
=
( )
+ β−β
v
(A8)
1
1
2
(
)
j
22
j
j
j
(A9)
Ec
=− −
av j
β −+
+
c
2
cc
1
i
i
1
i
i
−
1
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