Numerical Artifacts in the Discontinuous Generalized Porous Medium Equation: How to Avoid Spurious Temporal Oscillations

TL;DR

This paper introduces the Shock-Based Averaging Method (SAM) for numerically solving the Discontinuous Generalized Porous Medium Equation, effectively eliminating spurious temporal oscillations by incorporating shock position into the scheme.

Contribution

The paper proposes SAM, a novel numerical method that integrates shock position calculation to prevent artifacts in discontinuous GPME simulations, improving solution accuracy.

Findings

SAM produces non-oscillatory, physically valid solutions.

Inclusion of shock position is essential to avoid temporal artifacts.

Averaging choice alone does not prevent oscillations.

Abstract

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.