On the Reynolds-number scaling of Poisson solver complexity

\`Adel Alsalti-Baldellou; Assensi Oliva; F.Xavier Trias

arxiv: 2512.22644 · v2 · submitted 2025-12-27 · ⚛️ physics.flu-dyn · math-ph· math.MP· physics.comp-ph

On the Reynolds-number scaling of Poisson solver complexity

F.Xavier Trias , \`Adel Alsalti-Baldellou , Assensi Oliva This is my paper

Pith reviewed 2026-05-16 19:14 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math-phmath.MPphysics.comp-ph

keywords Poisson solverReynolds number scalingNavier-Stokes turbulencesolver complexitymultigrid methodsJacobi solverBurgers equationincompressible flows

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The pith

Poisson solver complexity decreases with rising Reynolds number in Navier-Stokes turbulence but increases for the Burgers equation.

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

The paper asks whether the cost of solving the Poisson equation grows or shrinks as Reynolds numbers increase in very large incompressible flow simulations. Physical scaling arguments combined with convergence theory for Jacobi and multigrid solvers predict power-law behavior at extreme Reynolds numbers and define a phase space separating decreasing-iteration regimes from increasing-iteration regimes. Numerical tests place Navier-Stokes turbulence in the decreasing regime, so the work per grid point falls as Reynolds number rises, while the one-dimensional Burgers equation falls in the opposite regime. This matters for extreme-scale computing because it suggests the relative cost of the pressure solve may improve rather than worsen for realistic turbulent flows.

Core claim

At very high Reynolds numbers the complexity of Poisson solvers follows power-law scalings derived from physical and numerical arguments. A convergence analysis of Jacobi and multigrid methods defines a two-dimensional phase space in which the number of iterations either decreases or increases with Reynolds number. Numerical experiments confirm that Navier-Stokes turbulence lies in the decreasing regime while the Burgers equation lies in the increasing regime, providing a unified framework for how solver performance scales with flow Reynolds number.

What carries the argument

The two-dimensional phase space from the convergence analysis of Jacobi and multigrid solvers, which separates Reynolds-number regimes according to whether iteration counts decrease or increase.

Load-bearing premise

The theoretical convergence rates derived for Jacobi and multigrid solvers remain valid when extrapolated to arbitrarily high Reynolds numbers.

What would settle it

A Navier-Stokes turbulence simulation at a Reynolds number ten times larger than the reported tests, checking whether the number of Poisson iterations per time step continues to follow the predicted decreasing power law.

Figures

Figures reproduced from arXiv: 2512.22644 by \`Adel Alsalti-Baldellou, Assensi Oliva, F.Xavier Trias.

**Figure 2.** Figure 2: FIG. 2. Phase space [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Energy and pressure spectra for the forced HIT simula [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Same as in Figure 3 but for the second, [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Same as in Figure 3 but for the convective term, ( [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Same as in Figure 3 but for [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Illustrative snapshots of the DNS simulations of com [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Spectrum of the temporal derivative of the pressure fi [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Spectra of the temporal derivative of the pressure fie [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Same as Figure 9, but for [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Spectra of the temporal derivative of the pressure fi [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. Energy spectra for the forced Burgers’ equation for [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Same as in Figure 12, but for the residual of the Poiss [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14. Same as in Figure 12, but for the total number of itera [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗

read the original abstract

We aim to answer the following question: is the complexity of numerically solving the Poisson equation increasing or decreasing for very large simulations of incompressible flows? Physical and numerical arguments are combined to derive power-law scalings at very high Reynolds numbers. A theoretical convergence analysis for both Jacobi and multigrid solvers defines a two-dimensional phase space divided into two regions depending on whether the number of solver iterations tends to decrease or increase with the Reynolds number. Numerical results indicate that, for Navier-Stokes turbulence, the complexity decreases with increasing Reynolds number, whereas for the one-dimensional Burgers equation it follows the opposite trend. The proposed theoretical framework thus provides a unified perspective on how solver convergence scales with the Reynolds number and offers valuable guidance for the development of next-generation preconditioning and multigrid strategies for extreme-scale simulations.

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.

Desk Editor's Note private letter to a colleague

The paper maps Poisson solver iterations onto a Re-dependent phase space and finds that NS turbulence sits in the region where complexity drops as Re rises, while Burgers goes the other way.

read the letter

The main result is a two-dimensional phase space for Jacobi and multigrid iteration counts that splits into regions where solver work either grows or shrinks with Reynolds number. The authors combine standard convergence bounds with physical scaling for the spectrum of the Poisson right-hand side, then test the picture on NS turbulence and the 1-D Burgers equation. Numerical runs show the NS case falling into the decreasing-complexity branch while Burgers follows the increasing one. That contrast is the clearest new element; it gives a compact way to think about why preconditioner choices might change at extreme Re. The framework itself is straightforward and the numerical support is presented directly, which makes the claim easy to check against other codes. The soft spot is the physical argument that keeps the effective condition number from worsening in 3-D turbulence. The analysis assumes the divergence of the convective term retains enough spectral decay even when the dissipative range is resolved, but intermittency and non-Gaussian gradients could shift the operating point across the boundary. The Burgers test does not probe that 3-D issue, so the NS scaling rests on an assumption that still needs tighter verification at the highest Re. Readers working on solver design for large-scale incompressible flows will find the phase-space picture useful for deciding where to invest effort in multigrid or preconditioning. The question is practical and the evidence is at least internally consistent, so the paper should go to a serious referee rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper investigates whether the computational complexity of solving the Poisson equation increases or decreases at very high Reynolds numbers in incompressible flow simulations. It combines physical arguments on velocity statistics with standard convergence theory for Jacobi and multigrid methods to derive power-law scalings for iteration counts. A two-dimensional phase space is defined that separates regimes where iterations decrease versus increase with Re. Numerical experiments on 3D Navier-Stokes turbulence show decreasing complexity with Re, while the 1D Burgers equation shows the opposite trend, providing a unified framework for solver behavior at extreme scales.

Significance. If the central scaling claims hold, the work supplies concrete guidance for preconditioner and multigrid design in petascale and exascale turbulence simulations, indicating that standard solvers may become relatively cheaper rather than more expensive as Re grows in 3D NS flows. The phase-space construction and the contrasting NS versus Burgers results constitute a falsifiable prediction that can be tested in future high-Re DNS.

major comments (2)

[§3] §3 (theoretical convergence analysis) and the phase-space construction: The boundary separating decreasing versus increasing iteration counts with Re is derived under the assumption that the spectral decay rate of the Poisson right-hand side (divergence of the convective term) remains independent of Re. In 3D turbulence this assumption is load-bearing for the claim of decreasing complexity, yet the manuscript provides no direct verification against resolved high-Re spectra or intermittency measures; the 1D Burgers contrast does not test the 3D case.
[Numerical results] Numerical results section (NS turbulence branch): The reported decrease in Jacobi/multigrid iterations with Re is shown only up to moderate Re; the manuscript does not demonstrate that the trend persists once the dissipative range is fully resolved and small-scale gradient intermittency becomes prominent, which could alter the effective condition number and move the operating point across the phase-space boundary.

minor comments (2)

Figure captions for the phase-space plots should explicitly state the numerical values of the two axes parameters used to locate the NS and Burgers operating points.
The definition of 'complexity' (iteration count versus wall-clock time or flop count) is used inconsistently between the theoretical scalings and the numerical tables; a single consistent metric should be adopted throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which identify key assumptions in the theoretical analysis and limitations in the numerical evidence. We address each point below, indicating where revisions will strengthen the manuscript while maintaining the integrity of the original claims.

read point-by-point responses

Referee: [§3] §3 (theoretical convergence analysis) and the phase-space construction: The boundary separating decreasing versus increasing iteration counts with Re is derived under the assumption that the spectral decay rate of the Poisson right-hand side (divergence of the convective term) remains independent of Re. In 3D turbulence this assumption is load-bearing for the claim of decreasing complexity, yet the manuscript provides no direct verification against resolved high-Re spectra or intermittency measures; the 1D Burgers contrast does not test the 3D case.

Authors: We agree that the Re-independence of the spectral decay rate for the Poisson right-hand side is a central assumption for the 3D NS branch of the phase space. This follows from standard Kolmogorov inertial-range scaling, under which the spectrum of the convective term (and thus its divergence) inherits a fixed power-law decay independent of Re once an inertial range is present. The manuscript derives the phase-space boundary from this established scaling rather than from new spectral measurements. The 1D Burgers case is included precisely to demonstrate the opposite regime when the spectral properties differ. In revision we will expand §3 with an explicit statement of this theoretical basis, supported by citations to literature on the spectral properties of the nonlinear term, and we will note that direct verification at extreme Re awaits future DNS data. This constitutes a partial revision. revision: partial
Referee: [Numerical results] Numerical results section (NS turbulence branch): The reported decrease in Jacobi/multigrid iterations with Re is shown only up to moderate Re; the manuscript does not demonstrate that the trend persists once the dissipative range is fully resolved and small-scale gradient intermittency becomes prominent, which could alter the effective condition number and move the operating point across the phase-space boundary.

Authors: We acknowledge that the numerical demonstrations for 3D NS turbulence reach only moderate Re, where the dissipative range is not yet fully resolved and extreme intermittency is not dominant. The observed decrease in iteration count is consistent with the theoretical prediction for the relevant region of the phase space. Because the phase-space boundary is set by inertial-range spectral properties, localized intermittency at dissipative scales is not expected to shift the operating point across the boundary. In the revised manuscript we will add a short discussion of this point, clarifying the expected robustness of the trend at higher Re while noting that fully resolved ultra-high-Re simulations would provide further confirmation. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No circularity: scalings derived from independent physical arguments and numerical verification

full rationale

The paper derives power-law scalings for Poisson solver iterations versus Reynolds number by combining standard convergence theory for Jacobi and multigrid methods with physical arguments on the right-hand-side spectrum in turbulence. These are then tested numerically on Navier-Stokes and Burgers cases, producing a phase-space diagram whose boundaries are falsifiable by the simulations themselves. No equation reduces to a fitted parameter renamed as prediction, no load-bearing step collapses to a self-citation, and the central claim (decreasing complexity for 3-D NS turbulence) rests on externally checkable numerical evidence rather than definitional closure.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Since only abstract available, no specific free parameters or axioms identified.

pith-pipeline@v0.9.0 · 5442 in / 904 out tokens · 19059 ms · 2026-05-16T19:14:59.484962+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A theoretical convergence analysis for both Jacobi and multigrid solvers defines a two-dimensional phase space divided into two regions depending on whether the number of solver iterations tends to decrease or increase with the Reynolds number.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ˆr0_k ∝ Re^{-1} Δt^q k^β with β=11/6

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

1J. H. Ferziger, M. Peri´ c, and R. L. Street. Computational Methods for Fluid Dynamics . Springer, 4th edition, 2020. 2S. A. Orszag and G. S. Patterson. Numerical simulation of three-d imensional homogeneous isotropic turbulence. Physical Review Letters , 28:76–79, 1972. 3D. I. Pullin. Pressure spectra for vortex models of ﬁne-scale homo geneous turbulen...

work page 2020
[2]

8Yukio Kaneda and Mitsuo Yokokawa

Springer, Berlin, Heidelberg, 1988. 8Yukio Kaneda and Mitsuo Yokokawa. DNS of Canonical Turbulence wit h up to 4096 3 Grid Points. In Parallel Computational Fluid Dynamics , pages 23–32. Elsevier, May 2004. 9S. Hoyas and J. Jim´ enez. Scaling of velocity ﬂuctuations in turbulen t channels up to Reτ = 2003. Physics of Fluids , 18:011702, 2006. 10T. Ishihar...

work page 1988

[1] [1]

1J. H. Ferziger, M. Peri´ c, and R. L. Street. Computational Methods for Fluid Dynamics . Springer, 4th edition, 2020. 2S. A. Orszag and G. S. Patterson. Numerical simulation of three-d imensional homogeneous isotropic turbulence. Physical Review Letters , 28:76–79, 1972. 3D. I. Pullin. Pressure spectra for vortex models of ﬁne-scale homo geneous turbulen...

work page 2020

[2] [2]

8Yukio Kaneda and Mitsuo Yokokawa

Springer, Berlin, Heidelberg, 1988. 8Yukio Kaneda and Mitsuo Yokokawa. DNS of Canonical Turbulence wit h up to 4096 3 Grid Points. In Parallel Computational Fluid Dynamics , pages 23–32. Elsevier, May 2004. 9S. Hoyas and J. Jim´ enez. Scaling of velocity ﬂuctuations in turbulen t channels up to Reτ = 2003. Physics of Fluids , 18:011702, 2006. 10T. Ishihar...

work page 1988