Numerical artifacts in the Generalized Porous Medium Equation: Why harmonic averaging itself is not to blame

Abstract

The degenerate parabolic Generalized Porous Medium Equation (GPME) poses numerical challenges due to selfsharpening 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.

Numerical determination of pressure‐dependent effective thermal conductivity in Berea sandstone

“…For further information about these approaches, we refer to Maddix et al . (2018), Siegert et al . (2021), Kadioglu et al .…”

Numerical artifacts in the discontinuous Generalized Porous Medium Equation: How to avoid spurious temporal oscillations

“…The low frequency temporal oscillations have been studied in the literature [1,2,40,41]. It has been shown in these works that the temporal oscillations occur as the front crosses a grid cell.…”

“…The purpose of the paper is to identify the cause of the numerical artifacts reported in the literature [1][2][3] for second order finite volume discretizations of the Generalized Porous Medium Equation (GPME) with discontinuous coefficients, and to suggest a numerical approach that does not have these problems. The GPME, commonly known as the Filtration Equation, can be expressed in both conservative and integral forms as: p t = ∇ · (k(p)∇p) = ∆Φ(p), where Φ(p) = p 0 k(p)dp, k(p) = Φ (p).…”

“…(1.4). The discontinuity in k(p) does make the challenges more severe and because of this discontinuity, the Modified Equation Analysis approach in [1] for the continuous GPME is not applicable. Here, we look at the discontinuous GPME, and also refer to subclasses of the continuous GPME.…”

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

“…To solve this problem, a new mimetic scheme with a staggered discretization of diffusion coefficients was then suggested in [19] and analyzed in [26]. The authors in [23,24] presented the discrete schemes with harmonic averaging by adding a correction term for both continuous and discontinuous generalized porous medium equations. In three-temperature RHD, radiation diffusion equations are coupled with hydrodynamics that are usually simulated by Lagrange or ALE algorithms.…”