Geoscience Reference
In-Depth Information
case where a good initial guess will be available by extrapolating the behaviour of the solution
variable from previous time steps. Iterative methods can, however, be fast and generally require
much less computer storage than direct methods for large problems. Examples of iterative so-
lution methods include Picard iteration, Newton iteration, Gauss-Seidal iteration, successive
over-relaxation and the conjugate gradient method. Paniconi
et al.
(1991) have compared the
Picard and Newton iterations in solving the nonlinear Richards equation by a
finite element
method
. The Newton iteration method requires the calculation of a gradient matrix at each iter-
ation, but will generally converge in fewer iterations. They suggested that it was more efficient
in certain strongly transient or highly nonlinear cases (but also recommend the non-iterative
Lees method for consideration). The pre-conditioned form of the conjugate gradient method is
very popular in the solution of groundwater problems and is readily implemented on parallel
computers (Binley and Beven, 1992).
Although we have discussed the considerations of explicit and implicit schemes, direct and
iterative solutions, stability and convergence in the context of finite difference approximations,
they apply to all solution algorithms including finite element and finite volume techniques.
Note that with all these schemes, it is possible to have an algorithm that is stable and consistent
but not accurate, if an injudicious choice of space and time increments is made. It may be
very difficult to check accuracy in a practical application except by testing the sensitivity of the
solution to reducing the space and time increments. This is something that is often overlooked
in the application of numerical methods since checks on the accuracy of the solution may
often be expensive to carry out.
Finite element solutions are also commonly used in hydrological problems, such as in the
InHM and HydroGeoSphere (VanderKwaak and Loague, 2001; Therrien
et al.
, 2006). The finite
element method has an important advantage over finite difference approximations in that flow
domains with irregular external and internal boundaries are more realistically represented by
elements with straight or curved sides. No-flow and specified flux boundaries are also more
easily handled in the finite element method. The solution nodes in the finite element method
lie (mostly at least) along the boundaries of the elements. Spatial gradients are represented
by interpolating the nodal values of the variable of interest within each element using
ba-
sis functions
. The simplest form of basis function is simple linear interpolation. Higher order
interpolation can be used, but requires more solution nodes within each element. Ideally,
the form of interpolation should be guided by the nature of the problem being solved but
this is often difficult for problems involving changes over time and in which different solu-
tion variables are nonlinearly related, implying that different basis functions should be used.
A higher order finite element interpolation for time differentials can also be used.
Over the last decade, finite volume techniques have become more popular. In the finite
volume method, each node in the solution grid is surrounded by a small volume of the flow
domain. Since these need not be regularly spaced or a particular shape the finite volume tech-
nique is well suited for unstructured grids representing complex grids. Its other main advantage
is that it is inherently mass conservative. This is because divergence terms in the partial dif-
ferential equation are converted to fluxes across the boundaries of the finite volume using the
divergence theorem (see LeVeque, 2002). The change in mass in each finite volume is then cal-
culated exactly from the net balance of the surface fluxes across all the boundaries. The finite
volume approach has been used in hydrological models for surface flows in the SFV model
of Horritt
et al.
(2007), for saturated subsurface flows by Jenny
et al.
(2003) and for coupled
surface and subsurface flows by Qu and Duffy (2007), He
et al.
(2008) and Kumar
et al.
(2009).
There is a vast literature on numerical solution algorithms for differential equations. A good
detailed introduction to finite element and finite difference algorithms used in hydrology may
be found in Pinder and Gray (1977). The recent developments in finite volume methods also
appear to be promising. However, it is always important to remember that all these methods
are approximations, especially in the case of nonlinear problems, and that approximations
involve inaccuracies even if the solution is stable, convergent and mass conserving. One
form of inaccuracy commonly encountered is that of numerical dispersion. It is an important
