Geoscience Reference
In-Depth Information
This equation shows that semi-implicit methods are second-order accurate, and
consequently allow for use of larger time step values. The implementation of these
methods requires the computation of all derivatives and fluxes centered in time.
Those values also can be computed with second-order accuracy, as the average
between values at time
t
and
t
+∆
t
, and can be demonstrated using expansions from
Equation 6.6:
t
t
+
∆
t
CC
+
t
+
∆
t
/
2
2
C
=
i
i
+
0
()
∆
t
i
2
This temporal semi-implicit discretization is known as the Crank-Nicholson discret-
ization. In this discretization we get
t
+
∆
t
t
t
+
∆
t
t
CC
t
−
2
2
+
=−
∂
∂
1
2
C
x
∂
∂
C
x
+−
∂
∂
1
2
C
x
∂
∂
C
x
2
i
i
U
+
ν
U
+
ν
0
()
∆
t
2
2
∆
i
i
In order to solve this equation, the spatial derivatives have to be discretized.
6.2.4.2
Spatial Discretization
Spatial discretization using the Taylor series follows an approach similar to temporal
discretization. Let us consider Taylor series developments for points on the left and
on the right of point
i
, at a distance
∆
x
at an arbitrary time level:
*
*
*
22
2
+
33
3
+
+
=+
∂
∂
C
x
+
∆
xC
x
∂
∂
∆
xC
x
∂
∂
*
*
3
CC
∆
x
0
()
∆
x
(6.7)
i
1
i
2
3
!
i
i
i
*
*
*
22
2
−
33
3
+
−
=−
∂
∂
C
x
+
∆
xC
x
∂
∂
∆
xC
x
∂
∂
()
3
*
*
CC
∆
x
0
∆
x
(6.8)
i
1
i
2
3
!
i
i
i
Subtracting Equation (6.8) from Equation (6.7), we get the so-called central differ-
ence for the first-order spatial derivative of
C
:
*
*
*
CC
x
−
∂
∂
C
x
=
2
i
+
1
i
−
1
+
0
()
∆
x
(6.9)
2
∆
i
From Equation (6.7), we obtain an expression for a noncentered derivative (right
side derivative), while from Equation (6.8), we obtain a left-side derivative, both
with a first-order truncation error:
*
*
*
CC
x
−
∂
∂
C
x
=
i
+
1
i
+
0
()
∆
x
(6.10)
∆
i
*
*
*
CC
x
−
∂
∂
C
x
=
i
i
−
1
+
0
()
∆
x
(6.11)
∆
i
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