Geometric-Perturbation-Robust Cut-Cell Scheme for Two-Material Flows: Exact Pressure-Equilibrium Preservation and Rigorous Analysis

TL;DR

This paper introduces a novel cut-cell scheme for two-material flows that maintains exact pressure equilibrium despite interface perturbations, ensuring high-order accuracy and robustness in complex simulations.

Contribution

The proposed geometric-perturbation-robust (GPR) cut-cell method uniquely preserves pressure equilibrium under interface errors and integrates equilibrium-compatible reconstructions for high-order accuracy.

Findings

Preserves pressure equilibrium with small interface errors.

Achieves second-order accuracy at interfaces and third-order in smooth regions.

Demonstrates robustness and stability in complex 2D tests.

Abstract

Preserving pressure equilibrium across material interfaces is critical for the stability of compressible multi-material flow simulations, yet most interface-fitted sharp-interface schemes are notoriously sensitive to interface geometry: even slight perturbations of the captured (or tracked) interface can trigger large spurious pressure oscillations. We present a cut-cell method that is geometric-perturbation-robust (GPR) for the compressible two-material flows. By construction, the scheme provably preserves exact interfacial pressure equilibrium in the presence of small interface-position errors. The key is a strict consistency between the conserved variables and the geometric moments (i.e., the integrals of monomials) of every cut cell. We formulate auxiliary transport equations, whose discrete solutions furnish evolved geometric moment, these geometric moments remain perfectly synchronized with the conserved variables -- even on a deforming mesh. Surpassing the classical geometric conservation law, our approach keeps all higher-order geometric moments consistent, thereby eliminating accuracy loss due to geometric mismatches. To prevent the reconstruction step from destroying pressure equilibrium, we introduce the notion of equilibrium-compatible (EC) reconstructions. A carefully designed modification equips any conventional weighted essentially non-oscillatory (WENO) reconstruction with the EC property; we detail a third-order EC multi-resolution WENO (EC-MRWENO) variant. The tight coupling of EC-MRWENO with the evolved moments yields the first cut-cell solver that is simultaneously provably GPR and genuinely high-order: it attains second-order accuracy precisely at material interfaces, while preserving third-order accuracy in smooth regions. Extensive two-dimensional tests confirm the framework's robustness, accuracy and stability under geometric perturbations and topology changes.