On Dual-Finite Volume Methods for Extended Porous Medium Equations

TL;DR

This paper investigates the stability of Dual-Finite Volume Methods for nonlinear porous medium equations, revealing the need for specialized flux evaluation to ensure stability, and introduces a new isotone flux approach.

Contribution

It proposes a new isotone numerical flux for the Dual-Finite Volume Method tailored for porous medium equations, ensuring stability for nonlinear problems.

Findings

Unconditional stability does not hold for generic nonlinear equations without proper flux evaluation.

The proposed isotone flux maintains simplicity similar to conventional methods.

The new flux improves the stability and reliability of numerical solutions for PMEs.

Abstract

This article shows that the unconditional stability of the Dual-Finite Volume Method, which is at least valid for linear problems, is not true for generic nonlinear differential equations including the PMEs unless the coefficient appearing in the numerical fluxes are appropriately evaluated. This article provides a theoretically truly isotone numerical fluxes specialized for solving the PMEs presented, which is still as simple as the conventional fully-upwind counterpart.