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

TL;DR

This paper investigates numerical artifacts in the generalized porous medium equation, demonstrating that harmonic averaging is not solely responsible for issues, and proposes an improved discretization approach using modified equation analysis and a Modified Harmonic Method.

Contribution

The study reveals the causes of numerical artifacts beyond harmonic averaging and introduces a modified discretization method to mitigate these issues in GPME simulations.

Findings

Harmonic averaging is not the sole cause of numerical artifacts.

Modified equation analysis identifies key error terms responsible for artifacts.

A Modified Harmonic Method effectively reduces numerical issues.

Abstract

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.