First Advisor

Hannah Kravitz

Date of Award

Spring 6-2026

Document Type

Thesis

Degree Name

Bachelor of Science (B.S.) in Mathematics and University Honors

Department

Mathematics and Statistics

Language

English

Subjects

wave equation, metric graphs, semi-spectral method, spectral finite difference method, energy conservation, Fourier transform

DOI

10.15760/honors.1811

Abstract

To investigate the accuracy and long-term energy conservation of a spectral finite difference numerical method for a wave equation on metric graphs. In conservative systems, numerical methods should preserve total energy. However, explicit finite difference methods require impractically small space steps and exhibit energy drift at end points. To address these limitations, a spectral finite difference method is implemented using a Fourier transformation. This semi-spectral method improves stability at endpoints while maintaining second-order accuracy, achieving an overall error of O(∆t2). We implement the semi-spectral method on the IEEE14 metric graph and provide visuals showing the initial condition projected onto the eigenfunction basis, the decay of energy contribution per eigenfunction, energy in the system through time, and how the number of eigenfunctions used affected mean squared error. We discuss the modes that contributed the most energy and where the energy exists on the IEEE14 metric graph for those modes, noting that high energy contribution is not always associated with energy localization.

Persistent Identifier

https://archives.pdx.edu/ds/psu/44722

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