Date of Award
Spring 5-11-2023
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
School
Mathematics and Natural Sciences
Committee Chair
James V. 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
Krylov subspace spectral (KSS) methods are high-order accurate, one-step explicit time-stepping methods for partial differential equations (PDEs) that also possess stability characteristic of implicit methods. Unlike other time-stepping approaches, KSS methods compute each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE, using techniques developed by Golub and Meurant for approximating bilinear forms involving matrix functions. As a result, KSS methods scale effectively to higher spatial resolution.
This dissertation will present spectral multistep methods, explicit and implicit, designed through the combination of KSS methods and Adams methods. This combination allows spectral multistep methods to achieve the efficiency of classical multistep methods while maintaining the scalability and stability of one-step KSS methods. Furthermore, while one-step KSS methods can only be applied to nonlinear PDEs through combination with exponential integrators, spectral multistep methods are applicable to both linear and nonlinear PDEs as is. A convergence analysis will be presented, and the effectiveness of spectral multistep methods will be demonstrated using numerical experiments. It will also be shown that the region of absolute stability exhibits striking behavior that helps explain the scalability of these new methods.
Copyright
2023, Bailey Rester
Recommended Citation
Rester, Bailey, "Spectral Multistep Methods for the Scalable Simulation of Time-Dependent Phenomena" (2023). Dissertations. 2128.
https://aquila.usm.edu/dissertations/2128