Date of Award

Spring 1-1-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Bengt Fornberg

Second Advisor

Natasha Flyer

Third Advisor

Keith Julien

Fourth Advisor

Gunnar Martinsson

Fifth Advisor

Francois Meyer

Abstract

Traditional finite difference methods for solving the partial differential equations (PDEs) associated with wave and heat transport often perform poorly when used in domains that feature jump discontinuities in model parameter values (interfaces). We present a radial basis function-derived finite difference (RBF-FD) approach that solves these types of problems to a high order of accuracy, even when curved interfaces and variable model parameters are present.

The method generalizes easily to a variety of different problem types, and requires only the inversion of small, well-conditioned matrices to determine stencil weights that are applied directly to data that crosses an interface. These weights contain all necessary information about the interface (its curvature; the contrast in model parameters from one side to the other; variability of model parameter value on either side), and no further consideration of the interface is necessary during time integration of the numerical solution.

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