Geoscience Reference
In-Depth Information
Truncating this series at the first derivative, we obtain
t
t
+
∆
t
t
CC
t
−
∂
∂
C
t
=
i
i
+
Ot
()
∆
(6.4)
∆
i
This equation states that the resolution of all the terms of the equation at time
t
allows the calculation of the variable at time
t
t
with first-order precision because
the first missing term in the series is multiplied by
+∆
t
.
Similarly, we can relate the concentration at time
t
with the concentration at
time
t
∆
+ ∆
t
:
t
+
∆
t
t
+
∆
t
t
+
∆
t
22
33
∂
∂
C
t
∆
t
∂
∂
C
t
∆
t
∂
∂
C
t
t
t
+
∆
t
CC
=
−
∆
t
+
−
i
i
2
3
2
3
!
i
i
i
t
+
∆
t
nn
∆
t
n
∂
∂
C
t
n
+
1
++
L
+
0
()
∆
t
n
!
i
Truncating this series after the first derivative as before, we obtain
t
+
∆
t
t
+
∆
t
t
CC
t
−
∂
∂
C
t
=
i
i
+
0(
∆
t
)
(6.5)
∆
i
This equation shows that in implicit methods the truncation error is also of the
first order, as in explicit methods, although processes are computed at time
t
t
.
The difference between implicit and explicit methods is their stability properties, as
described in the following.
From the above paragraph, it is expected that explicit and implicit methods
should have the same truncation error and it is also expected that the calculation of
the derivatives (or fluxes) at the center of the time step must have a smaller truncation
error. To demonstrate this, let us use the Taylor series to relate properties at time
+ ∆
t
and
t
+∆
t
with variables at
t
+∆
t
/2
.
t
+
∆
t
/
2
+
( )
∂
2
∆
t
t
+
∆
t
/
2
2
∂
∂
C
t
C
t
( )
2
2
3
∆
t
∆
t
t
+
∆
t
t
+
∆
t
/
2
CC
=
+
+
0
2
2
i
i
2
i
i
(6.6)
t
+
∆
t
/
2
+
( )
∂
2
∆
t
t
+
∆
t
/
2
2
∂
∂
C
t
C
( )
2
2
3
∆
t
∆
t
t
t
+
∆
t
/
2
CC
=
−
+
0
2
2
i
i
2
∂
t
i
i
Subtracting the second equation from the first equation, we obtain
t
+
∆
t
/2
t
+
∆
t
t
CC
t
−
∂
∂
C
t
( )
2
∆
t
=
i
i
+
0
2
∆
i
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