Date of Award

5-2026

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

School

Mathematics and Natural Sciences

Committee Chair

Dr. Huiqing Zhu

Committee Chair School

Mathematics and Natural Sciences

Committee Member 2

Dr. James Lambers

Committee Member 3

Dr. Qingguang Guan

Committee Member 3 School

Mathematics and Natural Sciences

Committee Member 4

Dr. Haiyan Tian

Committee Member 4 School

Mathematics and Natural Sciences

Abstract

The numerical simulation of stiff time-dependent partial differential equations (PDEs) is a fundamental challenge in computational mathematics. Traditional explicit methods are restricted by severe stability constraints arising from the Courant–Friedrichs–Lewy (CFL) condition, while implicit methods require the repeated solution of large systems of equations, leading to high computational cost. This dissertation addresses these challenges by developing and implementing a Krylov Subspace Spectral (KSS) method combined with Chebyshev spectral collocation for diffusion–reaction type PDEs, including simplified models of the damped Lighthill–Westervelt equation. The proposed method discretizes the spatial domain using Chebyshev polynomials, which provide spectral accuracy for smooth solutions. Temporal integration is achieved using Krylov subspace polynomial approximations of the exponential solution operator, allowing stable time-stepping with step sizes far beyond classical CFL restrictions. Stability properties and accuracy behavior are analyzed for the resulting semi-discrete and fully discrete schemes, building on classical results in numerical analysis \cite{Isaacson1966} and recent advances in Krylov-based exponential integration \cite{Rester2023,Drum2024,Agress2019}. Temporal convergence is discussed in the regime of fixed spatial resolution, consistent with polynomial Krylov approximation theory. Numerical experiments are used to validate the analytical expectations for both linear and nonlinear test problems. The results demonstrate exponential decay of spatial error with increasing resolution and algebraic decay of temporal error consistent with the order of the polynomial Krylov approximation. In particular, first- and second-order temporal schemes are implemented and tested, confirming stability and accuracy in regimes where traditional explicit schemes fail. Higher-order temporal accuracy is achievable in principle through higher-degree polynomial approximations of the exponential operator. Overall, this work demonstrates that Krylov Subspace Spectral methods, when paired with Chebyshev collocation, provide a robust and computationally efficient alternative to classical time-stepping approaches for stiff PDEs. The framework developed here lays the groundwork for future extensions to multidimensional problems, adaptive time-stepping strategies, and more fully nonlinear systems.

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Available for download on Tuesday, December 01, 2026

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