  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.
