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arxiv: 2512.18422 · v2 · submitted 2025-12-20 · 🧮 math.NA · cs.NA· physics.flu-dyn

A pressure-projection formulation in a least-squares meshfree method for the incompressible Navier-Stokes equations using a staggered-variable arrangement

Takeharu Matsuda , Satoshi Ii This is my paper

Pith reviewed 2026-05-16 21:04 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.flu-dyn
keywords meshfree methodincompressible Navier-Stokespressure projectionstaggered variablesprimal-dual griddivergence-free velocityleast-squares approximation
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The pith

A staggered primal-dual arrangement in a least-squares meshfree method ensures consistent pressure projection and locally divergence-free velocities for incompressible Navier-Stokes flows.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a meshfree numerical method for the incompressible Navier-Stokes equations that overcomes the usual difficulty of satisfying incompressibility at the discrete level. Strong-form local evaluations and collocated velocity-pressure variables normally produce a zero-energy mode and inconsistent operators in the pressure Poisson equation of projection methods. By inserting a local primal-dual grid whose virtual dual cell is built only from node connectivity, the method obtains matching discrete operators on both sides of the Poisson equation while keeping the meshfree character intact. Velocity is evolved on the cell interfaces through a time-conversion step to maintain proper coupling. Numerical tests confirm that the resulting velocity remains divergence-free up to an arbitrarily small discrete error, recovers the design convergence rate, and reproduces flow structures across a broad range of Reynolds numbers.

Core claim

By constructing a virtual dual cell solely from the local connectivity among nodes within the mesh-constrained discrete point method, consistent discrete operators are obtained on both sides of the pressure Poisson equation. Time evolution converting is then applied to evolve the velocity on cell interfaces, producing a staggered-variable arrangement that enforces a local divergence-free velocity field up to an arbitrarily small discrete error while retaining the meshfree nature of the discretization.

What carries the argument

The virtual dual cell defined solely from local connectivity among nodes in a staggered primal-dual grid, which supplies matching discrete divergence and gradient operators for the pressure Poisson equation.

If this is right

  • Local divergence-free velocity is ensured up to an arbitrarily small discrete error.
  • The expected spatial convergence order is recovered in the computed solutions.
  • Flow features are accurately reproduced across a wide range of Reynolds numbers.
  • Arbitrary geometries can be treated without constructing a global mesh system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local-connectivity construction could be reused in other projection-based meshfree schemes to remove operator inconsistency without adding global data structures.
  • Because the dual cell is rebuilt from instantaneous node links, the method may adapt more readily to problems with moving or deforming boundaries than fixed-grid staggered schemes.
  • Extension of the time-evolution conversion step to higher-order Runge-Kutta integrators would test whether the staggered coupling remains stable under stronger time-step restrictions.

Load-bearing premise

The virtual dual cell constructed from local connectivity produces consistent discrete operators on both sides of the pressure Poisson equation without introducing new inconsistencies or accuracy loss.

What would settle it

Numerical experiments in which the discrete divergence of the velocity field fails to approach zero under successive refinement, or in which the observed spatial convergence order falls below the design rate, would show that the operator consistency has not been achieved.

Figures

Figures reproduced from arXiv: 2512.18422 by Satoshi Ii, Takeharu Matsuda.

Figure 1
Figure 1. Figure 1: Illustrations of the local primal grid (left) and [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of applying the normal component of any vector (or tensor) quantity as a boundary [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Staggered-variable arrangement for the 2D linear acoustic equation on local primal–dual grids. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Staggered-variable arrangement for the 2D Navier–Stokes equations on local primal–dual grids. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Snapshots of the pressure field for N = 129 at five times: (a) t = 0, (b) t = 7.5 × 10−5 , (c) t = 1.5 × 10−4 , (d) t = 2.25 × 10−4 , and (e) t = 3 × 10−4 , calculated using the staggered-variable arrangement. −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.5 0.6 0.7 0.8 0.9 1 p x Reference (FDTD) N = 17 N = 33 N = 65 N = 129 N = 257 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.5 0.6 0.7 0.8 0.9 1 p x (a) (b) [PITH_FULL_IMAGE:figures… view at source ↗
Figure 6
Figure 6. Figure 6: Comparisons of the pressure p along the x-axis for x ∈ [L/2, L] using the MCD methods with the (a) staggered arrangement and (b) collocated arrangement. The black solid curves indicate the reference solution calculated with the FDTD method. The purple solid, green dashed, yellow dotted, blue dash-dot, and red dash-dot-dot curves correspond to resolutions N = 17, 33, 65, 129, and 257, respectively. (a) (b) … view at source ↗
Figure 7
Figure 7. Figure 7: Snapshots of pressure fields (N = 129) at t = 3 × 10−4 in the presence of an obstacle, calculated using (a) the staggered-variable arrangement and (b) collocated arrangement. (c) Comparison of the pressure distributions along the x-axis, where the orange solid and black dashed lines correspond to the staggered and collocated arrangements, respectively. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the fluid flow problem around a cylinder. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Snapshots at t = 100 of pressure and vorticity fields around a cylinder at Re = 200 for ∆t = ∆t ∗ . (a) Proposed staggered arrangement with β = 0.99, (b) staggered arrangement with β = 0, and (c) conventional collocated arrangement. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Snapshots at t = 100 of velocity divergence around a cylinder at Re = 200 for ∆t = ∆t ∗ . (a) Proposed staggered arrangement with β = 0.99, (b) staggered arrangement with β = 0, and (c) conventional collocated arrangement. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Pressure distribution around a circular cylinder at [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Pressure distribution around a circular obstacle ( [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Computational domain of the Taylor–Green vortex ( [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Spatial convergence of L1, L2, and L∞ error norms at t = 0.1 for (a) the x component of velocity, (b) the y component of velocity, and (c) the pressure for a uniform DP arrangement. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Streamline visualizations of the lid-driven cavity flow calculated using the proposed staggered [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Centerline velocity profiles of the lid-driven cavity flow compared with the reference data of Ghia [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Numerical results for the pressure (left) and vorticity (right) isolines around an oscillating cylinder. [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparisons of x and y components of velocities for four cross-sections: x = −0.6D (red curve/squares), x = 0 (green dashed curve/triangles), x = 0.6D (blue dash-dotted curve/diamonds), and x = 1.2D (purple dotted curve/circles). The phase angles of the cylinder position are (a) 180°, (b) 210°, and (c) 330°. Experimental results are taken from [53]. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Comparison of the time variation in the drag force coefficient and its components. [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
read the original abstract

Incompressible flow solvers based on strong-form meshfree methods represent arbitrary geometries without the need for a global mesh system. However, their local evaluations make it difficult to satisfy incompressibility at the discrete level. Moreover, the collocated arrangement of velocity and pressure variables tends to induce a zero-energy mode, leading to decoupling between the two variables. In projection-based approaches, a spatial discretization scheme based on a conventional node-based moving least-squares method for the pressure causes inconsistency between the discrete operators on both sides of the Poisson equation. Thus, a solenoidal velocity field cannot be ensured numerically. In this study, a numerical method for the incompressible Navier-Stokes equations is developed by introducing a local primal-dual grid into the mesh-constrained discrete point method, enabling consistent discrete operators. The constructed virtual dual cell is defined solely from the local connectivity among nodes, and thus the method retains its meshfree nature. To achieve a consistent coupling between velocity and pressure variables under the primal-dual arrangement, time evolution converting is applied to evolve the velocity on cell interfaces. For numerical validation, a linear acoustic equation is solved to confirm the effectiveness of the staggered-variable arrangement based on the local primal-dual grid. Then, incompressible Navier-Stokes equations are solved, and the proposed method is demonstrated to ensure a local divergence-free velocity field up to an arbitrarily small discrete error, achieve the expected spatial convergence order, and accurately reproduce flow features over a wide range of Reynolds numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a pressure-projection formulation for the incompressible Navier-Stokes equations within a least-squares meshfree framework. By introducing a virtual dual cell defined from local node connectivity to create a staggered arrangement, the method aims to ensure consistent discrete gradient and divergence operators in the pressure Poisson equation, thereby achieving a locally divergence-free velocity field up to arbitrarily small discrete error while preserving the meshfree nature of the discretization. Validation includes a linear acoustic problem and Navier-Stokes flows demonstrating expected convergence rates and accurate flow reproduction across Reynolds numbers.

Significance. If the consistency of the primal-dual operators is rigorously established, this approach would represent a meaningful contribution to meshfree methods for incompressible flows, addressing a common limitation in strong-form discretizations without introducing global meshing requirements. The ability to handle arbitrary geometries with high-fidelity incompressibility enforcement could be valuable for engineering applications.

major comments (1)
  1. [Construction of virtual dual cell and discrete operators] The central claim that the method ensures a local divergence-free velocity field up to arbitrarily small discrete error depends on the virtual dual cell yielding adjoint-consistent operators for the pressure Poisson equation. However, the manuscript does not appear to provide an explicit proof of a discrete integration-by-parts identity or adjoint property for arbitrary local connectivities in the least-squares approximation. This is load-bearing for the claim, as without it, residual divergence may remain at the order of the approximation error rather than machine epsilon, particularly on irregular node sets.
minor comments (1)
  1. [Validation section] The details of how the 'arbitrarily small discrete error' is quantified (e.g., specific tolerance or norm used) should be clarified for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and positive assessment of the manuscript's potential contribution. We address the major comment point by point below, agreeing that an explicit derivation will strengthen the theoretical claims, and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Construction of virtual dual cell and discrete operators] The central claim that the method ensures a local divergence-free velocity field up to arbitrarily small discrete error depends on the virtual dual cell yielding adjoint-consistent operators for the pressure Poisson equation. However, the manuscript does not appear to provide an explicit proof of a discrete integration-by-parts identity or adjoint property for arbitrary local connectivities in the least-squares approximation. This is load-bearing for the claim, as without it, residual divergence may remain at the order of the approximation error rather than machine epsilon, particularly on irregular node sets.

    Authors: We appreciate the referee's identification of this key theoretical point. The virtual dual cell is constructed directly from local node connectivity to enforce that the discrete divergence operator is the adjoint of the gradient operator within the least-squares framework, ensuring consistency in the pressure Poisson equation by design (analogous to staggered finite-volume schemes). Numerical results in the manuscript already demonstrate divergence errors at levels consistent with machine precision for the tested node distributions. However, we agree that an explicit proof of the discrete integration-by-parts identity for arbitrary connectivities would provide rigorous support and address potential concerns on irregular sets. In the revised manuscript, we will add a new appendix deriving this adjoint property, showing that any residual divergence is bounded by the controllable least-squares approximation error (which can be driven arbitrarily small by increasing polynomial degree or support radius). This will confirm the local divergence-free property up to this discrete tolerance. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and validation remain independent

full rationale

The derivation introduces a virtual dual cell defined from local node connectivity to produce consistent discrete gradient and divergence operators on the staggered primal-dual arrangement. This construction is presented as a direct definition that preserves meshfree character while enforcing adjoint consistency for the pressure Poisson equation; the resulting local divergence-free property is then demonstrated numerically rather than asserted by algebraic identity or fitted parameter. No step equates the target result to its own inputs by construction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The central claim therefore rests on the explicit staggered discretization and its numerical verification, not on a self-referential reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The method rests on the standard incompressible Navier-Stokes equations and the new ad-hoc construction of virtual dual cells from local node connectivity; no free parameters are explicitly fitted in the abstract.

axioms (2)
  • domain assumption Incompressible Navier-Stokes equations govern the flow
    The entire formulation is built on the standard incompressible NS equations as the governing system.
  • ad hoc to paper Local connectivity among nodes defines consistent dual cells
    The virtual dual cell is defined solely from local connectivity to retain meshfree nature while providing consistent operators.
invented entities (1)
  • local primal-dual grid no independent evidence
    purpose: To enable consistent discrete operators for the pressure Poisson equation under staggered arrangement
    New virtual dual cells constructed from local node connectivity to address inconsistency in conventional meshfree pressure discretization.

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