Solving the Porous Medium Equation with the eXtreme Mesh deformation approach (X-Mesh)

TL;DR

This paper presents a novel X-Mesh approach for solving the Porous Medium Equation that conserves mass, maintains positive values, and effectively handles complex interfaces without remeshing.

Contribution

The paper introduces the X-Mesh method, a new scheme that preserves mass and positivity while accurately capturing sharp interfaces in porous medium problems.

Findings

Effective handling of complex interfaces without remeshing.

Mass conservation and positivity maintained throughout simulations.

Numerical experiments demonstrate the method's robustness.

Abstract

We introduce a new scheme for solving the non-regularized Porous Medium Equation. It is mass conserving and uses only positive unknown values. To address these typically conflicting features, we employ the eXtreme Mesh deformation approach (X-Mesh), specifically designed for problems involving sharp interfaces. The method ensures that the interface is always meshed, even in the face of complex topological changes, without the need for remeshing or altering the mesh topology. We illustrate the effectiveness of the approach through various numerical experiments.