Date of Award

Spring 1-1-2014

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Francois Meyer

Second Advisor

Tom Manteuffel

Third Advisor

Harihar Rajaram

Fourth Advisor

John Ruge

Fifth Advisor

Juan Restrepo

Abstract

The underlying theme of this research is using numerical methods to develop computationally efficient algorithms for three separate problems driven by diffusive processes. The problems under consideration are: contaminant dispersal through fracture networks, modelling the flow of glacial ice, and community detection on networks.

A common feature of containment facilities for nuclear waste is to use expansive geological formations as an added barrier to contaminant dispersal in the event of a leak. Although these formations are generally comprised of dense rock that is difficult to penetrate, fractures within them provide a potential means for contaminants to rapidly transport across the barrier. The typical width of such fractures is only on the order of millimeters whereas the typical scale of interest for contaminant transport is on the order of kilometers. When particle tracking methods are used to simulate the contaminant dispersal in fracture networks, this disparity of scales severely restricts maximum time step sizes because features at the millimeter scale need to be resolved. Our contribution to this problem is developing a coarse scale particle tracking method that allows for substantially larger time steps when particles are navigating straight fractures.

With global warming comes concerns as to how the changing temperature will impact glacial systems and their contribution to sea level rise. On glacial scales, ice behaves as a very slowly moving non-Newtonian fluid, and the primary problem for numerically simulating the evolution of ice masses comes with Glen's flow law for the effective viscosity. The flow law is empirically based, and its simple form has proven useful for analytical calculations. However, its simple form also allows for the effective viscosity to become unbounded in regions of low strain rate, and has proven to be very problematic for numerical simulations. Our contribution to this problem is re-examining the datasets the flow law was originally based on to develop an alternative model that fits the data with comparable accuracy, but without the problematic singularity.

When working with networks that represent real world systems, a common feature of interest is to find collections of vertices that form communities. Because the word "community" is an ambiguous term, our interpretation is that it is necessary to quantify what it means to be a community at a minimum of three scales for any given problem. These scales are at the level of: individual nodes, individual communities, and the network as a whole. Although our work focuses on detecting overlapping communities in the context of social networks, our primary contribution is developing a methodology that is highly modular and can easily be adapted to target other problem-specific notions of community.

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