phi-FD, a second order finite difference scheme for geometries defined by a level-set function: the Neumann case
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.
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BibTeX
@misc{duprez2026phifd,
title = {{phi-FD, a second order finite difference scheme for geometries defined by a level-set function: the Neumann case}},
author = {Michel Duprez and Vanessa Lleras and Alexei Lozinski and Vincent Vigon and Lisl Weynans},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28313},
eprint = {2609.28313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28313},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - phi-FD, a second order finite difference scheme for geometries defined by a level-set function: the Neumann case
AU - Michel Duprez
AU - Vanessa Lleras
AU - Alexei Lozinski
AU - Vincent Vigon
AU - Lisl Weynans
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28313
UR - https://arxiv.org/abs/2609.28313
AB - 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.
ER -