Eigenvector decomposition to determine the existence, shape, and location of numerical oscillations in Parabolic PDEs

2020/12/29 by Ethan Jensen, Jensen, Ethan, R. Corban Harwood +1
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2012.14640

openalex publication_date 2020/12/29 · openalex created_date 2021/01/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we employed linear algebra and functional analysis to determine necessary and sufficient conditions for oscillation-free and stable solutions to linear and nonlinear parabolic partial differential equations. We applied singular value decomposition and Fourier analysis to various finite difference schemes to extract patterns in the eigenfunctions (sampled by the eigenvectors) and the shape of their eigenspectrum. With these, we determined how the initial and boundary conditions affect the frequency and long term behavior of numerical oscillations, as well as the location of solution regions most sensitive to them.

Citations

• The Discrete Cosine Transform

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