Upstream mobility Finite Volumes for the Richards equation in heterogenous domains

TL;DR

This paper develops and analyzes finite-volume schemes with upstream mobility for the Richards equation in heterogeneous domains, proving convergence and demonstrating the benefits of local grid refinement at interfaces.

Contribution

It rigorously proves convergence of cell-centered finite-volume schemes without Kirchhoff's transform for the Richards equation in heterogeneous media.

Findings

Convergence of the scheme is established mathematically.

Local grid refinement improves accuracy at subregion interfaces.

Numerical results confirm the scheme's effectiveness in heterogeneous settings.

Abstract

This paper is concerned with the Richards equation in a heterogeneous domain, each subdomain of which is homogeneous and represents a rocktype. Our first contribution is to rigorously prove convergence toward a weak solution of cell-centered finite-volume schemes with upstream mobility and without Kirchhoff's transform. Our second contribution is to numerically demonstrate the relevance of locally refining the grid at the interface between subregions, where discontinuities occur, in order to preserve an acceptable accuracy for the results computed with the schemes under consideration.