Date of Award

Spring 2019

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

School

Mathematics and Natural Sciences

Committee Chair

James Lambers

Committee Chair School

Mathematics and Natural Sciences

Committee Member 2

Haiyan Tian

Committee Member 2 School

Mathematics and Natural Sciences

Committee Member 3

Zhifu Xie

Committee Member 3 School

Mathematics and Natural Sciences

Committee Member 4

Huiqing Zhu

Committee Member 4 School

Mathematics and Natural Sciences

Abstract

Depending on the type of equation, finding the solution of a time-dependent partial differential equation can be quite challenging. Although modern time-stepping methods for solving these equations have become more accurate for a small number of grid points, in a lot of cases the scalability of those methods leaves much to be desired. That is, unless the timestep is chosen to be sufficiently small, the computed solutions might exhibit unreasonable behavior with large input sizes. Therefore, to improve accuracy as the number of grid points increases, the time-steps must be chosen to be even smaller to reach a reasonable solution.

Krylov subspace spectral (KSS) methods are componentwise, scalable, methods used to solve time-dependent, variable coefficient partial differential equations. The main idea behind KSS methods is to use an interpolating polynomial with frequency dependent interpolation points to approximate a solution operator for each Fourier coefficient.

This dissertation will discuss two techniques that were developed to eliminate error in the low frequency components of the solution computed using KSS methods. These two methods are a multigrid inspired technique (coarse grid residual correction) and developing a step size controller using the residual as an error approximation (adaptive time stepping).

ORCID ID

0000-0003-2852-3914

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