A Cartesian Cut-Cell Two-Fluid Method for Two-Phase Diffusion Problems

TL;DR

This paper introduces a Cartesian cut-cell finite-volume method for accurately solving two-phase diffusion problems with sharp interfaces in static geometries, ensuring conservation and handling complex embedded boundaries.

Contribution

The paper presents a novel cut-cell method that discretizes two-phase diffusion equations on a fixed Cartesian grid with minimal geometric information, enabling robust and accurate interface coupling.

Findings

Demonstrates superlinear convergence in benchmarks

Achieves sharp enforcement of interfacial laws

Maintains excellent conservation properties

Abstract

We present a Cartesian cut-cell finite-volume method for sharp-interface two-phase diffusion problems in static geometries. The formulation follows a two-fluid approach: independent diffusion equations are discretized in each phase on a fixed Cartesian grid, while the phases are coupled through embedded interface conditions enforcing continuity of diffusive flux and a general jump law. Cut cells are treated by integrating the governing equations over phase-restricted control volumes and surfaces, yielding discrete divergence and gradient operators that are locally conservative within each phase. Interface coupling is achieved by introducing a small set of interfacial unknowns per cut cell on the embedded boundary; the resulting algebraic system involves only bulk and interfacial averages. A key feature of the method is the use of a reduced set of geometric information based solely on low-order moments (trimmed volumes, apertures and interface measures/centroids), allowing robust implementation without constructing explicitly cut-cell polytopes. The method supports steady (Poisson) and unsteady (diffusion) regimes and incorporates Dirichlet, Neumann, Robin boundary conditions and general jumps. We validate the scheme on one-, two- and three-dimensional single-phase and two-phase benchmarks, including curved embedded boundaries, Robin conditions and strong property/jump contrasts. The results demonstrate a superlinear convergence behavior, sharp enforcement of interfacial laws and excellent conservation properties. Extensions to moving interfaces and Stefan-type free-boundary problems are natural perspectives of this framework.