  This paper investigates oscillation-free stability conditions of numerical
methods for linear parabolic partial differential equations with some example
extrapolations to nonlinear equations. Not clearly understood, numerical
oscillations can create infeasible results. Since oscillation-free behavior is
not ensured by stability conditions, a more precise condition would be useful
for accurate solutions. Using Von Neumann and spectral analyses, we find and
explore oscillation-free conditions for several finite difference schemes.
Further relationships between oscillatory behavior and eigenvalues is supported
with numerical evidence and proof. Also, evidence suggests that the
oscillation-free stability condition for a consistent linearization may be
sufficient to provide oscillation-free stability of the nonlinear solution.
These conditions are verified numerically for several example problems by
visually comparing the analytical conditions to the behavior of the numerical
solution for a wide range of mesh sizes.
