A space-time extension of a conservative two-fluid cut-cell method for moving diffusion problems

TL;DR

The paper develops a conservative space-time cut-cell framework for moving two-phase diffusion on a fixed Cartesian grid, enforcing sharp interfacial conditions via phase-restricted space-time control volumes and time-integrated geometric moments. By deriving Reynolds’ transport theorem-based bulk balances, a geometric conservation law, and space-time gradient/divergence operators, it achieves exact mass conservation across topology changes (fresh/dead cells) and arbitrary coefficient jumps, while preserving the algebraic structure of the static method. The approach is validated in 2D and 3D moving-domain benchmarks, showing super-linear spatial convergence, robustness to repeated topological events, and accurate two-fluid coupling across Γ(t). The results establish a solid numerical foundation for evolving multiphase transport and set the stage for future free-boundary extensions such as Stefan-type phase changes and coupled sharp-interface Navier–Stokes problems.

Abstract

We present a space-time extension of a conservative Cartesian cut-cell finite-volume method for two-phase diffusion problems with prescribed interface motion. The formulation follows a two-fluid approach: one scalar field is solved in each phase with discontinuous material properties, coupled by sharp interface conditions enforcing flux continuity and jump laws. To handle moving boundaries on a fixed Cartesian grid, the discrete balance is written over phase-restricted space-time control volumes, whose geometric moments (swept volumes and apertures) are used as weights in the finite-volume operators. This construction naturally accounts for the creation and destruction of cut cells (fresh/dead-cell events) and yields strict discrete conservation. The resulting scheme retains the algebraic structure of the static cut-cell formulation while incorporating motion through local geometric weights and interface coupling operators. A series of verification and validation tests in two and three dimensions demonstrate super-linear accuracy in space, robust behavior under repeated topology changes and conservation across strong coefficient jumps and moving interfaces. The proposed space-time cut-cell framework provides a conservative building block for multiphase transport in evolving geometries and a foundation for future free-boundary extensions such as Stefan-type phase change.

Figures (9)

• Figure 1: Fixed background domain $\Omega$ with a prescribed moving interface $\Gamma(t)$ partitioning $\Omega$ into $\Omega^-(t)$ and $\Omega^+(t)$. Normals $\mathbf n^\pm$ point outward of their respective phases.
• Figure 2: Geometry definitions for a moving embedded interface on a Cartesian cell. \ref{['eq:boundary_partition_moving']}.
• Figure 3: Space-time control volume over $[t_{n},t_{n+1}]$ illustrating the phase-restricted sub-volumes $V^+_{i,j}(t_{n})$ and $V^+_{i,j}(t_{n+1})$ and the swept interface within the time slab.
• Figure 4: Space-time cell highlighting a time-integrated face aperture $\mathcal{A}^{1+}_{n+\frac{1}{2},i-\frac{1}{2},j}$ and an auxiliary measure of the second kind $\mathcal{B}^{1+}_{n+\frac{1}{2},i,j}$ used by the discrete operators.
• Figure 5: Space-time cut-cell taxonomy over one slab $[t_n,t_{n+1}]$: persistent cut cell (interface intersects the cell at both slab endpoints), fresh cell for phase $+$ (the $+$ phase appears during the slab) and dead cell for phase $+$ (the $+$ phase disappears during the slab).