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phi-FD, a second order finite difference scheme for geometries defined by a level-set function: the Neumann case
Michel Duprez, Vanessa Lleras, Alexei Lozinski, Vincent Vigon, Lisl Weynans
TL;DR
Cartesian-grid Neumann methods use approaches such as ghost points or cut-cell corrections, while the paper extends level-set-based φ-FD to avoid ghost values, flux reconstruction, and cut-cell volumes. On a non-convex test domain, it reports second-order convergence and conditioning independent of the smallest cut cell under two precautions, though full stability remains unproved.
Problem
Cartesian-grid methods impose Neumann conditions using modified stencils, ghost values, or volumetric corrections, including approaches that require cut-cell volumes and fluxes.
Source: §1 · p. 1Method
The scheme relaxes the Neumann condition through the level-set and an auxiliary field, then eliminates that field using a boundary node and its nearest interior node.
Source: §1 · p. 1Results
Second-order convergence in discrete L2, H1, and L∞ norms was observed on a non-convex domain, while two precautions made conditioning O(h^-2) and insensitive to the smallest cut cell.
Source: §4 · pp. 3–4Takeaways & Limitations
The tested non-convex problem supports the scheme's second-order behavior, with consistency and observed conditioning consistent with that convergence.
Source: §4 · p. 3, §5 · p. 4A full stability proof is left for future work.
Source: §5 · p. 4
Abstract
from arXiv · showhide
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.
1 Introduction
This work extends φ-FD from Dirichlet to Neumann conditions on level-set-defined domains. Its finite-difference formulation uses interior nodes and the level-set, with two precautions controlling conditioning independently of the smallest cut cell.
1 Introduction
- Prior Cartesian-grid approaches include ghost points, interface unknowns, and cut-cell flux balances; Arias et al. compute cut-cell volumes and fluxes.Weynans reports second-order accuracy, consistency, and discrete maximum principles; Arias et al. report second-order accuracy for the solution and gradient.Source: §1 · p. 1
- φ-FD imposes Neumann data by relaxing the boundary condition near the boundary and eliminating an auxiliary field using a boundary node and its nearest interior node.The method uses only the level-set and interior nodes, with no ghost values, flux reconstruction, or cut-cell volumes.Source: §1 · p. 1
- Individual normalization of boundary equations and a rule for choosing the coupled interior node yield O(h^-2) conditioning independent of the smallest cut cell, without stabilization.Source: §1 · p. 1
A PDE and a scheme:
The paper discretizes a Poisson problem with Neumann boundary conditions on a Cartesian grid over a level-set-defined domain. Its boundary treatment relaxes the condition through an auxiliary field and eliminates that field using values at a boundary node and a nearby interior node.
A PDE and a scheme:
- The problem is −∆u + u = f in Ω with ∂nu = g on ∂Ω, where Ω is represented by Ω = {φ < 0}.For consistency analysis, the paper assumes φ is C3 near the boundary with nonzero gradient there, and sufficient smoothness for u to be C4 up to the boundary.Source: A PDE and a scheme · p. 2
- The discrete solution uses unknowns indexed by grid nodes in Ωh and is obtained by solving a system of N equations.The Cartesian grid uses four signed coordinate directions; the interior-node set and its boundary-node set define Ωh.Source: A PDE and a scheme · p. 2
- Discrete gradients use centered differences when both neighbors are present, one-sided differences when only one side has enough neighbors, and an associated interior node otherwise.A boundary-isolated point lacks enough neighbors in a direction, so its associated interior node supplies the discrete derivative.Source: A PDE and a scheme · p. 2
- The Neumann condition is relaxed near the boundary using an auxiliary field, then eliminated by imposing the relaxed relation at a boundary node and its nearest interior node.The paper defines p near ∂Ω and approximates its values at both nodes by the same value before eliminating it.Source: A PDE and a scheme · p. 2
- Individually normalizing boundary equations and choosing coupled interior nodes away from the boundary prevents proportional boundary rows and rank deficiency.The interior node is selected with |φα0| ≥ 0.1h when possible; the fallback is the nearest interior node.Source: A PDE and a scheme · p. 2
- Figure 1 reports fitted error-versus-h slopes of 1.99 for relative L2, 2.02 for H1, and 1.98 for L∞.Its domain panel also marks two boundary-isolated points on the coarse grid h = 1/7.Source: Figure 1 · p. 3
3 Consistency
The analysis establishes O(h^2) consistency for interior Laplacian rows and an O(h^3) relaxed Neumann residual at regular boundary nodes, while treating isolated nodes and nonzero data explicitly.
3 Consistency
- Interior Laplacian rows have O(h^2) consistency under a smooth extension with bounded derivatives through order four.The extension assumption is used for the solution near the grid nodes.Source: §3 · p. 3
- At non-isolated boundary nodes, smoothness of p = F/φ makes the relaxed Neumann residual O(h^3).The residual combines the two node values, φ(xα)F(xα0) − φ(xα0)F(xα), with an O(h^3) remainder.Source: §3 · p. 3
- Boundary-isolated nodes have O(h^2) consistency because their gradient approximation error is O(h), and such nodes are few.The analysis notes that the non-isolated-node gradient accuracy does not apply at these points.Source: §3 · p. 3
- The same consistency argument covers nonzero Neumann data by defining F = ∇u·∇φ − G, which vanishes on the boundary.Here G = g∥∇φ∥, with g smoothly extended near the boundary.Source: §3 · p. 3
4 Numerical illustration
On a bean-shaped, slightly non-convex domain, the scheme achieves second-order convergence in three discrete norms. A geometry sweep also shows stable errors and conditioning when both boundary-equation normalization and the coupled-node rule are used.
4 Numerical illustration
- The test uses a slightly non-convex bent ellipse and the manufactured solution u = sin(Kx) cos(Ky), with K = π/0.8.Its level-set is φ(x, y) = 0.8x^2 + (y + 1.4x^2)^2 − R^2, R = 0.5.Source: §4 · p. 3
- The scheme achieves optimal second-order convergence in the discrete L2, H1, and L∞ norms on the bean-shaped test domain.The test solves −∆u + u = f with Neumann data and a manufactured solution.Source: §4 · p. 3
- Across a three-decade sweep of the smallest coupled cell s, relative L2 error varies by a factor 2.2 with no trend; Figure 2(a) shows the sweep.Using the nearest interior node instead gives a factor 2.5 variation.Source: §4 · p. 3, Figure 2 · p. 4
- With both precautions, the condition number stays between 4.09 × 10^3 and 4.56 × 10^3, a factor 1.11 across the sweep.The fitted slope against s is +0.00, and the error is unchanged.Source: §4 · p. 4
5 Conclusion
The paper extends φ-FD to Neumann conditions and reports second-order convergence on a non-convex test domain. Two boundary-equation precautions yield cut-cell-independent O(h^-2) conditioning, although a full stability proof remains future work.
5 Conclusion
- The φ-FD scheme is extended to Neumann conditions imposed directly through the level-set function.The scheme uses two elementary precautions to make conditioning independent of the smallest cut cell, without ghost-penalty stabilization.Source: §5 · p. 4
- On a non-convex test domain, the scheme achieves second-order convergence in discrete L2, H1, and L∞ norms.Source: §5 · p. 4
- The analysis establishes O(h^2) consistency, while numerical results show O(h^-2) conditioning.Source: §5 · p. 4
- A full stability proof is left to future work; a variational reformulation could make coercivity immediate but would change the scheme.Source: §5 · p. 4