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

TL;DR

This paper uses eigenvector decomposition, singular value analysis, and Fourier techniques to identify conditions for oscillation-free, stable solutions in parabolic PDEs, revealing how initial/boundary conditions influence oscillations.

Contribution

It introduces a novel eigenvector-based approach combining linear algebra and Fourier analysis to analyze numerical oscillations in parabolic PDE schemes.

Findings

Eigenvector patterns correlate with oscillation presence.

Initial and boundary conditions influence oscillation locations.

Eigen spectrum analysis predicts long-term oscillation behavior.

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.