Geology Reference
In-Depth Information
n+1
i
be able to solve the entire set of equations, obtaining the new value
u
at
each node
i
, therefore it is called Implicit method.
It has been deduced by comparing these two methods of solution, that
though implicit method requires much more work in solving the set of
simultaneous equations, it is unconditionally stable, regardless of the size of
the time step
t
.
Crank-Nicolson Method
In an Implicit method,
t
can be chosen independent of
x
. Another
refinement is possible. Generally the smaller the truncation error, the faster
is the convergence of the finite difference equations to the differential equation.
In Explicit and Implicit methods for the time derivative -
u
/-
t
the truncation
error was of the order of 0(
t
is replaced by the central difference,
the truncation error associated with the time term would be reduced from
O(
t
). If
-
u
/
-
t
)
2
}. This was done by Crank-Nicolson and later on known as
Crank-Nicolson method.
The Crank-Nicolson method is obtained by averaging the approximations
t
) to O{(
2
u
/
x
2
at
n
and
n
+1 time levels. Sometimes a weight
-
-
(0<
<1) is used
here. We then obtain value for -
u
/-
t
evaluated at (
n
+1/2)
u
n1
2
u
n1
u
n1
1
u
n
2
u
n
u
n
uu
t
n1
n
i1
i
i1
i1
i
i1
So
i
i
=
(32)
2
x
The Explicit approximation is obtained by setting = 0 in eq. (12) and
= 1 gives the
Implicit scheme
.
The Crank-Nicolson approximation is given by setting eq. (12 ) with
=½
1/ 2
u
n1
2
u
n1
u
n1
1/ 2
u
n
2
u
n
u
n
uu
t
n1
n
i1
i
i1
i1
i
i1
i
i
=
(33)
2
x
This equation can be modified to treat the problems of unsaturated flow
as well as saturated flow. Extension to more than one space dimension
causes no difficulty.
CONCLUSION
The groundwater modelling is an efficient tool for the assessment and
management of groundwater potential as well as pollution due to
anthropogenic and/or natural causes. Model may be used to predict some
future groundwater flow or contaminant transport condition. The model may
also be used to evaluate different remediation alternatives, such as hydraulic
containment, pump-and-treat or natural attenuation, and to assist with risk
evaluation. In order to perform these tasks, the model, whether it is a
