phi-FD, a second order finite difference scheme for geometries defined by a level-set function: the Neumann case
Year: 2026
Type: preprint
Abstract: We extend the phi-FD finite-difference scheme, which handles domains described by a level-set function on a Cartesian grid, to Neumann boundary conditions. The condition is relaxed near the boundary through the level-set and an auxiliary field, which is eliminated by writing the relaxed condition at a boundary node and at its nearest interior node. The scheme uses only interior nodes and the level-set: no ghost values, flux reconstruction or cut-cell volumes. We prove optimal consistency and show second-order convergence in the L2, H1 and Linf norms on a non-convex domain. Two elementary precautions, an individual normalization of the boundary equations and a rule for the choice of the coupled interior node, make the condition number O(h^-2) and independent of the smallest cut cell, without any stabilization term.
Source: arXiv (Cornell University)
Institutions: Centre National de la Recherche Scientifique, Laboratoire des Sciences de l'Ingénieur, de l'Informatique et de l'Imagerie, Université de Strasbourg, Université de Montpellier, Institut Montpelliérain Alexander Grothendieck, Laboratoire de Mathématiques de Besançon, Université Marie et Louis Pasteur, Institut de Recherche Mathématique Avancée, Université de Bordeaux
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214230379