Geology Reference
In-Depth Information
numerical models. Numerical models use approximations (e.g. finite
differences, or finite elements) to solve the differential equations describing
groundwater flow or solute transport. The approximations require that the
model domain and time be discretized. In this discretization process, a network
of grid cells or elements represents the model domain, and time steps represent
the time of the simulation.
The accuracy of numerical models depends upon the accuracy of the
model input data, the size of the space and time discretization (the greater
the size of the discretization steps, the greater the possible error), and the
numerical method used to solve the model equations.
In addition to complex three-dimensional groundwater flow and solute
transport problems, numerical models may be used to simulate very simple
flow and transport conditions, which may just as easily be simulated using
an analytical model. However, numerical models are generally used to simulate
problems, which cannot be accurately described using analytical models.
Depending upon the numerical techniques employed in solving the
mathematical model, there exists several types of numerical models such as
Finite Difference Models, Finite Element Models, Boundary Element Models,
Particle tracking models, Method of characteristic models, Random walk
models and Integrated Finite Difference Models.
The main features of the various numerical models are:
The solution is sought for the numerical values of state variables only at
specified points in the space and time domains defined for the problems.
The partial differential equations that represent balances of considered
extensive quantities are replaced by a set of algebraic equation.
The solution is obtained for a specified set of numerical values of the
various model coefficients.
Because of the large number of equations that must be solved simultaneously,
a computer programme is prepared.
Although a number of numerical techniques exist, only finite difference
and finite element techniques have been widely used.
Finite Difference Method: L.F. Richardson, in his classic 1910 paper,
introduced finite difference as a method to calculate approximately the solution
of partial differential equations. A vast body of knowledge has been built up
over the years concerning both the theoretical and applied aspects of these
methods.
Fundamental to both the finite element and finite difference approaches
to solving partial differential equations is the concept of discretization wherein
a continuous domain is represented as a number of sub areas.
The basic idea of these finite difference methods is to replace derivatives
at a point by ratios of the changes in appropriate variables over a small but
finite interval.
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