Date of Award

Spring 5-11-2023

Degree Type


Degree Name

Doctor of Philosophy (PhD)


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


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.

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