Margot Gerritsen

Danielle Maddix, Luiz Sampaio, Margot Gerritsen

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.

Danielle Maddix, Luiz Sampaio, Margot Gerritsen

Numerical discretizations of the Generalized Porous Medium Equation (GPME) with discontinuous coefficients are analyzed with respect to the formation of numerical artifacts. In addition to the degeneracy and self-sharpening of the GPME with continuous coefficients, detailed in [1], increased numerical challenges occur in the discontinuous coefficients case. These numerical challenges manifest themselves in spurious temporal oscillations in second order finite volume discretizations with both arithmetic and harmonic averaging. The integral average, developed in [2] leads to improved solutions with monotone and reduced amplitude temporal oscillations. In this paper, we propose a new method called the Shock-Based Averaging Method (SAM) that incorporates the shock position into the numerical scheme. The shock position is numerically calculated by discretizing the theoretical speed of the front from the GPME theory. The speed satisfies the jump condition for integral conservation laws. SAM results in a non-oscillatory temporal profile, producing physically valid numerical results. We use SAM to demonstrate that the choice of averaging alone is not the cause of the oscillations, and that the shock position must be a part of the numerical scheme to avoid the artifacts.