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The degenerate parabolic Generalized Porous Medium Equation (GPME) posesnumerical challenges due to self-sharpening and its sharp corner solutions. Forthese problems, we show results for two subclasses of the GPME withdifferentiable $k(p)$ with respect to $p$, namely the Porous Medium Equation(PME) and the superslow diffusion equation. Spurious temporal oscillations, andnonphysical locking and lagging have been reported in the literature. Theseissues have been attributed to harmonic averaging of the coefficient $k(p)$ forsmall $p$, and arithmetic averaging has been suggested as an alternative. Weshow that harmonic averaging is not solely responsible and that an improveddiscretization can mitigate these issues. Here, we investigate the causes ofthese numerical artifacts using modified equation analysis. The modifiedequation framework can be used for any type of discretization. We show resultsfor the second order finite volume method. The observed problems with harmonicaveraging 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 aforementionednumerical artifacts.

Numerical Artifacts in the Generalized Porous Medium Equation: Why Harmonic Averaging Itself Is Not to Blame