Geology Reference
In-Depth Information
uu
t
n1
n
u
n
2
u
n
u
n
i
i
i1
i
i1
+
O
(
t
+
x
2
)
=
(28)
2
x
t
u
n
2
u
n
u
n
u
i
n
+ 1
=
u
i
n
+
Now,
2
i1
i
i1
x
=
u
i
n
+
ru
n
2
u
n
u
n
(29)
i1
i
i1
r
=
t
/(
x
)
2
u
i
n
+ 1
=
ru
i+1
where
n
+ (1 - 2
r
)
u
i
n
+
ru
i -1
(30)
t
. This
is the Explicit method where the values unknown can be calculated from the
known values of previous levels.
This
r
will depend on interval levels which we assign for
x
and
Stability Criterion
: The Explicit method is stable if
t
x
= 1/2
2
or, 0 <
r
< 0.5
One can see from the above conditions that
t
must be very small since
x
)
2
, and the number of computations required to reach a time level
'
t
m
' are extremely large.
An Implicit Approximation:
Here instead of forward difference approximation
we use backward difference approximation. So we will have
t
$'
1/2(
n
n
u
t
2
u
x
=
where, '
n
' is the time interval and '
i
' is the space interval.
Now writing the equation of backward difference approximation at
t
=
(
n
+ 1)
2
i
i
t
, we obtain for constant
x
uu
t
n1
n
u
n1
2
u
n1
u
n1
i
i
x
2
)
=
i1
i
i1
O
(
t
+
'
(31)
2
x
Here only one value is associated with previous time level and the three
are associated with the present time level unlike Explicit method. Now
assuming that the values
u
n
i
are known at all nodes at time
n
t
, eq. (31) is
a single equation containing three unknowns
u
n+1
n+1
i1
u
n+1
i1
. We can,
however, write equations for each node in the flow domain like eq. (31).
Then since there is one unknown value of head, for time (
n
+1)
t
, at each
node, we shall have a system of equations in which the total number of
equation is equal to the total number of unknowns. Therefore now we can
,
u
, and
