Explicit Formulas for PLIC Interface Reconstruction on a Compact Stencil

In this study, we propose the Compact Stencil Interface Reconstruction (CSIR) method, which determines interface normals for Piecewise Linear Interface Calculation (PLIC) using only face-adjacent neighbor cells. Our approach reduce stencil size compared to conventional approaches like ELVIRA. For 2D Cartesian grids, CSIR provides exact analytic solutions for linear interfaces while maintaining second-order accuracy for curved interfaces, eliminating iterative procedures. In 3D, we introduce two variants: (i) a computationally efficient explicit formula variant and (ii) CSIR $${}^{3}$$ , which achieves ELVIRA-level accuracy through localized Newton iterations for plane reconstruction. Rigorous benchmarking demonstrates that CSIR maintains comparable or superior accuracy to Parker–Youngs and ELVIRA methods, particularly on coarse grids and diffuse interfaces, while reducing communication overhead in distributed-memory parallelization. This advancement enables scalable, high-fidelity simulations of complex multiphase flows on adaptive meshes.