Some applications of integral equations to the solution of transient partial differential equations
| Author(s) | Hassell, Matthew E. | |
| Date Accessioned | 2018-07-10T14:35:23Z | |
| Date Available | 2018-07-10T14:35:23Z | |
| Publication Date | 2016 | |
| Abstract | This thesis studies boundary integral methods for solving time-dependent partial differential equations from continuum mechanics. The two models we analyze will be transient Stokes flow and scalar acoustic scattering by penetrable obstacles. We will see the two main flavors of analysis of Time Domain Boundary Integral Equations in these two problems: for analysis of the Stokes system we take a Laplace domain approach that dates to [7]. The analysis of the acoustic scattering and transmission problem will be carried out with the newer semigroup theory based analysis from [73] and [36]. ☐ We begin with a detailed exposition of a central tool, Convolution Quadrature, that we will use throughout the rest of the thesis for temporal discretization. We provide motivation for the method from various points of view and derive both multistep and Runge-Kutta Convolution Quadrature with an eye towards implementation of the method. From this foundation, we move on to the analysis of transient Stokes flow by way of the Laplace transform. This leads us to a detailed study the Brinkman equations. Analysis of the Brinkman Single Layer potential and operator is then used to derive stability and convergence results back in the time domain for the Stokes problem. We provide numerical experiments and simulations using various spatial discretization schemes. Finally, we study the scattering of acoustic waves by inhomogeneous penetrable obstacles. Chapter 4 presents a detailed stability and convergence analysis of a three-field boundary and finite element coupling scheme. By showing the underlying problem generates a C0-group of isometries, we are able to prove stability and convergence of the scheme directly in the time domain. In a similar vein, Chapter 5 explores computationally two alternative coupling schemes. | en_US |
| Advisor | Sayas, Francisco-Javier | |
| Degree | Ph.D. | |
| Department | University of Delaware, Department of Mathematical Sciences | |
| Unique Identifier | 1043757810 | |
| URL | http://udspace.udel.edu/handle/19716/23613 | |
| Publisher | University of Delaware | en_US |
| URI | https://search.proquest.com/docview/1840891730?accountid=10457 | |
| Keywords | Applied sciences | en_US |
| Keywords | Boundary integral methods | en_US |
| Keywords | Continuum mechanics | en_US |
| Keywords | Differential equations | en_US |
| Keywords | Scalar acoustic scattering | en_US |
| Keywords | Time Domain Boundary Integral Equations | en_US |
| Keywords | Transient Stokes flow | en_US |
| Title | Some applications of integral equations to the solution of transient partial differential equations | en_US |
| Type | Thesis | en_US |