  The degenerate parabolic Generalized Porous Medium Equation (GPME) poses
numerical challenges due to self-sharpening and its sharp corner solutions. For
these problems, we show results for two subclasses of the GPME with
differentiable $k(p)$ with respect to $p$, namely the Porous Medium Equation
(PME) and the superslow diffusion equation. Spurious temporal oscillations, and
nonphysical locking and lagging have been reported in the literature. These
issues have been attributed to harmonic averaging of the coefficient $k(p)$ for
small $p$, and arithmetic averaging has been suggested as an alternative. We
show that harmonic averaging is not solely responsible and that an improved
discretization can mitigate these issues. Here, we investigate the causes of
these numerical artifacts using modified equation analysis. The modified
equation framework can be used for any type of discretization. We show results
for the second order finite volume method. The observed problems with harmonic
averaging can be traced to two leading error terms in its modified equation.
This is also illustrated numerically through a Modified Harmonic Method (MHM)
that can locally modify the critical terms to remove the aforementioned
numerical artifacts.
