Revisiting numerical artifacts in the generalized porous medium equation with continuous coefficients: Does averaging really matter ?

Abstract

The Generalized Porous Medium Equation (GPME) with continuous coefficients is a degenerate parabolic equation and the finite volume solutions to this equation are known to exhibit numerical artifacts depending on how the non-linear diffusion coefficient $k\left(p\right)$ is computed at the faces. While arithmetic averaging is known to lead to reasonably accurate solutions for the degenerate diffusion equation, the use of harmonic averaging results in temporal oscillations and non-physical locking, neither of which can be eliminated by grid refinement. In this work, we propose an explicit finite volume discretisation of the GPME based on a novel approach to compute the diffusive fluxes referred to as the $\alpha$-damping (AD) flux scheme. The $\alpha$-damping flux scheme may be interpreted as a conservative “flux correction” approach which makes the averaging irrelevant to the numerical solution. Using theoretical analysis and numerical experiments in both one and two dimensions, we show that the new scheme is second-order accurate, applies to any temporal discretisation and that the solutions are independent of the choice of averaging while being free of numerical artifacts.