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

TL;DR

The paper tackles sharp-interface diffusion problems on fixed Cartesian grids by introducing a conservative Cartesian cut-cell two-fluid discretization in which each phase carries its own bulk unknowns and cut cells host interfacial unknowns. A reduced geometric moment description enables fluxes and gradients to be computed without explicitly constructing trimmed polyhedra, while interfacial jump conditions and flux balance are enforced sharply through moment-weighted operators. The approach is validated across one-, two-, and three-dimensional benchmarks for both single-phase and two-phase diffusion, showing near-second-order convergence, strong local conservation, and robust performance even under large interfacial jumps. The framework provides a solid foundation for moving-interface extensions and multiphysics coupling, with potential applications to Stefan problems and diffusion-advection problems in complex geometries.

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.

Figures (15)

• Figure 1: Fixed domain $\Omega$ with interface $\Gamma$, partitioning into $\Omega ^ -$ and $\Omega ^ +$. Normals $\mathbf n ^ \pm$ point out of their respective phases.
• Figure 2: Mesh cell notations
• Figure 3: Mesh cell $\Omega_{i, j}$ with interface $\Gamma_{i, j}$ separating dark $\Omega_{i, j}^+$ and light regions $\Omega_{i, j}^-$. Intersections of mesh faces with phases' domains is also represented
• Figure 4: Cut cell with the domains $\Omega_{i, j}^+$ (dark) and $\Omega_{i, j}^-$ (light), with bulk centroids $(x_{i, j}^\pm,y_{i, j}^\pm)$ and their associated $V_{i, j}^\pm$
• Figure 5: Cut cell $\Omega_{i, j}$ with the domains $\Omega_{i, j}^+$ (dark) and $\Omega_{i, j}^-$ (light), with face‐areas $A^{1\pm}_{i\pm\frac{1}{2},j}$, $A^{2\pm}_{i,j\pm\frac{1}{2}}$.