Geoscience Reference
In-Depth Information
There are some types of differential equations, known as elliptic equations, for which explicit
methods are not suitable since the effective wave speed is theoretically infinite. In hydrology,
flow in a saturated soil or aquifer, where the effective specific moisture capacity or storage
coefficient is very small, results in a quasi-elliptic equation which is one reason why some
physically based models have had significant problems with stability of solutions.
A more robust method arises if some form of averaging of the estimates of the spatial differ-
entials at time steps
j
and
j
+
1 is used, e.g.
∂
∂x
i
+1
,j
+1
−
)
i
+
1
,j
−
i
−1
,j
+1
2
x
i
−1
,j
2
x
i,j
≈
+
(1
−
(B5.3.3)
where
is a weighting coefficient in time. An
explicit solution
has
0. Any solution al-
gorithm with
>
0 is known as an
implicit
scheme (Figure B5.3.2b). A central difference, or
Crank-Nicholson scheme, has
=
=
0
.
5 , while a backward difference or fully implicit scheme
has
1. For linear problems, all implicit algorithms with
>
0
.
5 can be shown analytically
to be unconditionally stable, but no such general theory exists for nonlinear equations. The
central difference approximation is superior to the backward difference scheme in principle,
since the truncation error of the approximation is smaller. However, the backward difference
scheme has been found to be useful in highly nonlinear problems. Details of a four-point im-
plicit finite difference scheme to solve the one dimensional St. Venant channel flow equations
are given in Box 5.6; a simpler implicit scheme to solve the one-dimensional kinematic wave
equation is given in Box 5.7.
With an implicit scheme, the solution for node
i
at time step
j
=
+
1 involves values of the
dependent variable at other surrounding nodes, such as at
i
1,
which are themselves required as part of the solution (as shown in Figure B5.3.2b). Thus it is
necessary to solve the problem as a system of simultaneous equations at each time step. This is
achieved by an iterative method, in which an initial guess of the values at time
j
+
1 and
i
−
1 at time step
j
+
1 is used to
evaluate the spatial differentials in the implicit scheme. The system of equations is then solved
to get new estimates of the values at time
j
+
1 which are then used to refine the estimates
of the spatial differentials. This iterative procedure is continued until the solution converges
towards values that change by less than some specified tolerance threshold between successive
iterations. A successful algorithm results in a stable solution with rapid convergence (a small
number of iterations) at each time step. At each iteration, the system of equations is assembled
as a matrix equation, linearised at each iteration for nonlinear problems in the form:
+
[
A
]
{
}={
B
}
(B5.3.4)
where [
A
] is a two-dimensional matrix of coefficients known at the current iteration,
{
}
is
a one-dimensional vector of the unknown variable and
is a one-dimensional vector of
known values. Explicit schemes can also be expressed in this form but are solved only once at
each time step. For a large number of nodes, the [
A
] matrix may be very large and
sparse
(i.e.
having many zero coefficients) and many algorithms use special techniques, such as indexing
of non-zero coefficients, to speed up the solution.
Solution methods may be direct, iterative or a combination of the two. Direct methods carry
out a solution once and provide an exact solution to the limit of computer round-off errors.
The problem with direct methods is that, with a large sparse [
A
] matrix, the computer storage
required may be vast. Examples of direct methods include Gaussian elimination and Cholesky
factorisation.
Iterative methods attempt to find a solution by making an initial estimate and then refining
that estimate over successive iterations until some error criterion is satisfied. For implicit solu-
tions of nonlinear problems, the nonlinear coefficients can be refined at each iteration. Iterative
methods may not converge for all nonlinear problems. Steady state problems, for example, of-
ten pose greater problems in obtaining a solution than transient problems and convergence
may depend on making a good initial guess at the solution. This is often easier in the transient
{
B
}
