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In this paper, we employed linear algebra and functional analysis todetermine necessary and sufficient conditions for oscillation-free and stablesolutions to linear and nonlinear parabolic partial differential equations. Weapplied singular value decomposition and Fourier analysis to various finitedifference schemes to extract patterns in the eigenfunctions (sampled by theeigenvectors) and the shape of their eigenspectrum. With these, we determinedhow the initial and boundary conditions affect the frequency and long termbehavior of numerical oscillations, as well as the location of solution regionsmost sensitive to them.

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