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S. Winkler:
"Comparative mathematical modelling of groundwater pollution";
Supervisor: F. Breitenecker; Institut für Analysis und Scientific Computing, 2014; final examination: 2014-08-20.

The theme of this master thesis was motivated by a benchmark of the Federation of European
Simulation Societies, EUROSIM. The Benchmark deals with the problem of groundwater pollution.
A two-dimensional domain filled with water may be considered. In the middle of this
area a pollution source is located. This source emits solid constantly or at certain points in
time. The distribution of this pollution on the regarded domain is analyzed. For simulation
some mathematical background is summarized. The basic equation for pollution distribution
is the diffusion equation. Different kinds of this equation are used. In chemistry as well as in
biology the reaction-diffusion equation plays a very important role. Of course there are also
physical applications of the diffusion, e.g. the heat equation. However, diffusion is also used
to foresee the behavior of buyers of stocks in the financial market. In this work the focus is on
the convection-diffusion equation. Using this equation the distributive behavior of the pollution
influenced by a velocity field is described. Several approaches, ranging from analytical solutions
to some chaotic particle movement, are used for realization.
In the first part of this thesis the problem definition is restricted to a one-dimensional domain.
The second part deals with the already mentioned two-dimensional analysis. In both parts there
are three different kinds of simulations used. Analysis always starts with an analytical solution
of the equation using certain conditions. In reality it is not always possible to find such analytical
solutions. Therefore the second approach covers two commonly used numerical methods, the
finite differences and finite element method. Alternative implementations are given using the
principle of microscopic particle movements, also known as Brownian motion, as well as the
random walk. These processes can be described using stochastic theory including probability
theory.
An important part of this work is the comparison of these different approaches regarding efficiency,
accuracy and implementation. All the disadvantages and advantages will be shown.
Also the similarities and differences between them are lined out. The interaction of the different
parameters and their influences regarding simulation time and results are examined. This work
also includes methods which are not appropriate to simulate diffusion in a useable way. For most
of the used conditions an analytical solution can be given. Therefore an exact prototype for the
perfect approximation is given and can be used for comparison.
In the end all the different used approaches and their properties are summarized. Also an outlook
to other possible implementations is given.

http://publik.tuwien.ac.at/files/PubDat_248837.pdf