Document Type

Thesis - Open Access

Award Date

2026

Degree Name

Master of Science (MS)

Department / School

Mathematics and Statistics

First Advisor

Jung-Han Kimn

Abstract

This work operates on two fronts, focusing on interesting physical phenomena before turning our attention to an interesting numerical math problem. First, using the spacetime finite element method (FEM), we investigate a PDE system consisting of two Klein Gordon equations, which are coupled nonlinearly through the potential energy. The system contains a ghost (negative kinetic energy term). Systems such as these are generally deemed physically unstable, resulting in infinite energy in finite time. However, recent work has shown that this is not always the case. We investigate multiple scenarios arising from different initial conditions to characterize if/when a ghost system exhibits physical stability. Second, with the spacetime FEM used to discretize hyperbolic PDEs, we work towards finding a suitable parallel in time preconditioner (PC) that will allow for iterative KSP methods to solve the linear system posed by the discretization. We build off of previous work which successfully implements a time-decomposition additive Schwarz method for the wave equation. Our parallel implementation culminates in a 2-level PC which combines a restricted additive Schwarz method (RAS) with an additive field-split method on the RAS subdomains. Our 2-level method is more effective in terms of GMRES iteration count, computational memory requirements, and CPU time than that of a 1-level standard additive Schwarz method. In the final chapter, we explore the eigenvalue spectrum of the RAS system and find that a finer temporal mesh reduces the percentage of undesirable eigenvalues and aids in the performance of the preconditioner. We also discover a reasonable starting point for future work interested in furthering the scalability of the method.

Publisher

South Dakota State University

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Rights Statement

In Copyright