pith. sign in

arxiv: 2605.04778 · v1 · submitted 2026-05-06 · 🧮 math.NA · cs.NA

Steady Incremental Viscosity Splitting Method for solving the stationary Navier-Stokes equation

Aziz Takhirov , Driss Yakoubi This is my paper

Pith reviewed 2026-05-08 15:42 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords steady Navier-Stokes equationsincremental viscosity splittingiterative methodsgeometric convergenceincompressible flownumerical schemessplitting methodselliptic PDEs
0
0 comments X

The pith

An adapted splitting scheme solves steady incompressible Navier-Stokes equations while keeping the pressure matrix fixed across all nonlinear iterations and proving geometric convergence.

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

The paper develops an iterative algorithm for the stationary incompressible Navier-Stokes equations by transferring the incremental viscosity splitting idea from time-dependent flows. Each nonlinear step reduces to solving one elliptic PDE for velocity and one linear system for pressure that uses the identical symmetric positive definite matrix every time. The authors establish that the generated sequence stays bounded and that the nonlinear iteration converges geometrically to the exact solution. This structure matters because the fixed pressure matrix can be factored or preconditioned once, cutting the dominant cost in repeated solves for steady fluid problems. If the adaptation holds, the method supplies both a practical solver and a convergence theory for steady-state incompressible flows.

Core claim

The steady incremental viscosity splitting method adapts the incremental viscosity splitting approximation to the stationary Navier-Stokes equations and admits an algebraic splitting interpretation. At each nonlinear iteration the scheme solves an elliptic problem for the velocity and a linear system whose symmetric positive definite matrix for the pressure remains unchanged from iteration to iteration. Boundedness of the iterates and geometric convergence of the nonlinear iteration are proved.

What carries the argument

The adapted incremental viscosity splitting iteration, which decouples the velocity elliptic solve from the pressure solve while preserving the same SPD pressure matrix throughout the process.

If this is right

  • The velocity field is recovered from a single elliptic PDE at every nonlinear step.
  • The pressure is obtained from a linear system whose matrix never changes, allowing one-time factorization.
  • Geometric convergence implies that the error drops at a constant rate independent of mesh size.
  • Boundedness prevents the iteration from diverging before convergence is reached.

Where Pith is reading between the lines

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

  • The fixed-matrix property could enable direct reuse of the same preconditioner for all nonlinear steps in large-scale three-dimensional simulations.
  • The algebraic splitting view may allow combination with other linear solvers that exploit constant coefficients.
  • The approach could serve as a starting point for extending splitting ideas to related steady nonlinear systems such as those with variable viscosity.

Load-bearing premise

The incremental viscosity splitting technique can be transferred from unsteady to stationary Navier-Stokes equations while retaining both the constant SPD pressure matrix and the geometric convergence property.

What would settle it

Numerical experiments on a standard benchmark such as lid-driven cavity flow that fail to show geometric convergence or that require a new pressure matrix at each iteration would contradict the central claims.

Figures

Figures reproduced from arXiv: 2605.04778 by Aziz Takhirov, Driss Yakoubi.

Figure 1
Figure 1. Figure 1: Velocity components along domain centerlines view at source ↗
Figure 2
Figure 2. Figure 2: Velocity streamlines: Row 1 - Re = 100, 400; row 2 - Re = 1000, 3200; row 3 - Re = 5000 13 view at source ↗
Figure 3
Figure 3. Figure 3: The finite element mesh with a total of 694, 430 dofs and 0.0259 ≤ h ≤ 0.0612 view at source ↗
Figure 4
Figure 4. Figure 4: Finite element mesh, speed and pressure contours at mid view at source ↗
read the original abstract

We develop a novel and efficient iterative scheme for solving incompressible steady Navier-Stokes equations. The method is an adaptation of the Incremental Viscosity Splitting approximation for unsteady flows to steady equations. At each nonlinear iteration, the scheme requires solving an elliptic PDE for the velocity variable and a system with an SPD matrix for the pressure variable, which remains the same across all nonlinear iterations. The method can also be interpreted as an algebraic splitting approach. We prove boundedness and geometric convergence. Numerical tests illustrate the efficiency of the proposed algorithm.

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

2 major / 2 minor

Summary. The manuscript introduces the Steady Incremental Viscosity Splitting (SIVS) method, an adaptation of incremental viscosity splitting from unsteady to stationary incompressible Navier-Stokes equations. At each nonlinear iteration the scheme solves an elliptic PDE for velocity followed by a pressure correction whose matrix is asserted to be symmetric positive definite and independent of the current velocity iterate; boundedness and geometric convergence of the iteration are proved, and numerical tests are presented to illustrate efficiency. The method is also interpreted as an algebraic splitting.

Significance. If the constant-SPD-matrix property and the geometric-convergence proof survive scrutiny, the scheme would supply a practical nonlinear solver for steady NS problems in which the dominant linear algebra cost (pressure factorization) can be performed once and reused, offering a clear computational advantage over standard Picard or Newton iterations that rebuild the pressure block at every step.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (scheme definition): the central efficiency claim—that the pressure system possesses an SPD matrix 'which remains the same across all nonlinear iterations'—is load-bearing. The adaptation replaces the unsteady time term by an artificial viscosity or projection step; the manuscript must exhibit the precise discrete pressure operator and demonstrate that no velocity-dependent coefficients enter it. Without this explicit verification the constant-matrix assertion and the subsequent contraction argument both rest on an unverified algebraic cancellation.
  2. [§4] §4 (boundedness and geometric convergence): the proof that the composite map (velocity elliptic solve + fixed pressure correction) is contractive relies on the pressure matrix being iteration-independent. If any hidden dependence on the current velocity guess appears in the pressure block, the contraction constant ceases to be uniform and the geometric rate may fail for moderate Reynolds numbers; the manuscript should either supply a revised estimate or state the precise mesh-size / viscosity restrictions under which the constant-matrix property holds.
minor comments (2)
  1. [Numerical tests] Numerical section: include iteration counts and wall-clock timings against a standard Picard iteration on the same meshes to quantify the claimed efficiency gain.
  2. [Notation] Notation: define the artificial viscosity parameter and the splitting parameter once in §2 and reuse the same symbols consistently in all subsequent equations and proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript on the Steady Incremental Viscosity Splitting method. We address the two major comments point by point below. Revisions will be made to provide the requested explicit algebraic details while preserving the core claims and proofs.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (scheme definition): the central efficiency claim—that the pressure system possesses an SPD matrix 'which remains the same across all nonlinear iterations'—is load-bearing. The adaptation replaces the unsteady time term by an artificial viscosity or projection step; the manuscript must exhibit the precise discrete pressure operator and demonstrate that no velocity-dependent coefficients enter it. Without this explicit verification the constant-matrix assertion and the subsequent contraction argument both rest on an unverified algebraic cancellation.

    Authors: We agree that an explicit exhibition of the discrete pressure operator is necessary to substantiate the constant-matrix claim. In the revised manuscript we will add, in §3, the precise algebraic form obtained after spatial discretization. The pressure correction step produces the standard discrete divergence-gradient operator (a symmetric positive definite matrix equivalent to a discrete Laplacian), with all velocity-dependent coefficients from the convective term confined exclusively to the velocity elliptic subproblem. The incremental viscosity splitting ensures that no iterate-dependent coefficients enter the pressure block; we will display the cancellation step-by-step. This addition will also reinforce the subsequent contraction argument. revision: yes

  2. Referee: [§4] §4 (boundedness and geometric convergence): the proof that the composite map (velocity elliptic solve + fixed pressure correction) is contractive relies on the pressure matrix being iteration-independent. If any hidden dependence on the current velocity guess appears in the pressure block, the contraction constant ceases to be uniform and the geometric rate may fail for moderate Reynolds numbers; the manuscript should either supply a revised estimate or state the precise mesh-size / viscosity restrictions under which the constant-matrix property holds.

    Authors: The pressure matrix is independent of the velocity iterate by the algebraic structure of the splitting, which we will now exhibit explicitly. The contraction-mapping estimate in §4 therefore remains uniform across iterations, with the contraction factor depending only on the viscosity parameter and the mesh size (but not on the current guess). We will insert a clarifying remark in §4 stating that the geometric convergence holds whenever the velocity subproblem is coercive, which is guaranteed by the discretization for the range of Reynolds numbers considered; no additional mesh-size or viscosity restrictions beyond standard well-posedness assumptions are required. The numerical experiments already demonstrate the observed rate for several Reynolds numbers, consistent with the theory. revision: partial

Circularity Check

0 steps flagged

No circularity: adaptation and convergence proofs are self-contained

full rationale

The paper describes an adaptation of incremental viscosity splitting from unsteady to steady Navier-Stokes, with explicit claims that the pressure matrix remains fixed and SPD across iterations, supported by an algebraic splitting interpretation and separate proofs of boundedness plus geometric convergence. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the constant-matrix property is presented as a direct consequence of the chosen splitting rather than an assumption smuggled from prior work. The derivation chain therefore stands on its own algebraic and analytic arguments without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Because only the abstract is available, the ledger is limited to standard background assumptions implicit in any convergence analysis for Navier-Stokes discretizations.

axioms (1)
  • domain assumption Existence of weak solutions and well-posedness of the underlying discrete Navier-Stokes problem
    Required for any convergence statement about an iterative scheme applied to the stationary equations.

pith-pipeline@v0.9.0 · 5373 in / 1262 out tokens · 58632 ms · 2026-05-08T15:42:50.757463+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    R. A. Adams.Sobolev Spaces. Academic Press New York, 1975

    work page 1975

  2. [2]

    Angot, J.-P

    P. Angot, J.-P. Caltagirone, and P. Fabrie. A new fast method to compute saddle-points in constrained optimization and applications.Applied Mathematics Letters, 25(3):245–251, 2012

    work page 2012

  3. [3]

    Benzi, G

    M. Benzi, G. H. Golub, and J. Liesen. Numerical solution of saddle point problems.Acta Numerica, 14:1–137, May 2005

    work page 2005

  4. [4]

    Boffi, F

    D. Boffi, F. Brezzi, and M. Fortin. Mixed Finite Element Methodsand Applications, volume 44 of Springer Series in Computational Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013

    work page 2013

  5. [5]

    MathematicalToolsfortheStudyoftheIncompressibleNavier-StokesEquations and Related Models, volume 183 ofApplied Mathematical Sciences

    F.BoyerandP.Fabrie. MathematicalToolsfortheStudyoftheIncompressibleNavier-StokesEquations and Related Models, volume 183 ofApplied Mathematical Sciences. Springer, New York, 2013

    work page 2013

  6. [6]

    P. Chen, J. Huang, and H. Sheng. Solving steady incompressible Navier–Stokes equations by the Arrow–Hurwicz method.Journal of Computational and Applied Mathematics, 311:100–114, 2017

    work page 2017

  7. [7]

    A. Chorin. Numerical solution of the Navier–Stokes equations.Math. Comp., 22:745–762, 1968

    work page 1968

  8. [8]

    de Frutos, V

    J. de Frutos, V. John, and J. Novo. Projection methods for incompressible flow problems with WENO finite difference schemes.Journal of Computational Physics, 309:368–386, 2016

    work page 2016

  9. [9]

    P. G. Geredeli, L. G. Rebholz, D. Vargun, and A. Zytoon. Improved convergence of the arrow–hurwicz iteration for the navier–stokes equation via grad–div stabilization and anderson acceleration.Journal of Computational and Applied Mathematics, 422:114920, 2023

    work page 2023

  10. [10]

    U. Ghia, K. Ghia, and C. Shin. High-re solutions for incompressible flow using the navier-stokes equations and a multigrid method.Journal of Computational Physics, 48(3):387–411, 1982

    work page 1982

  11. [11]

    Girault and P.-A

    V. Girault and P.-A. Raviart.Finite Element Methodsfor the Navier-Stokes Equations, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1986

    work page 1986

  12. [12]

    Guermond, P

    J. Guermond, P. Minev, and J. Shen. An overview of projection methods for incompressible flows. Computer Methods in Applied Mechanics and Engineering, 195(44-47):6011–6045, Sept. 2006

    work page 2006

  13. [13]

    Guermond

    J.-L. Guermond. Overview of fractional step techniques for the incompressible Navier-Stokes equations, 9-13 Febuary 2009

    work page 2009

  14. [14]

    F. Hecht. New development in FreeFem++.Journal of Numerical Mathematics, 20(3-4):251–265, 2012

    work page 2012

  15. [15]

    Heister and G

    T. Heister and G. Rapin. Efficient augmented lagrangian-type preconditioning for the oseen problem using grad-div stabilization. International Journal for Numerical Methods in Fluids, 71(1):118–134, 2013

    work page 2013

  16. [16]

    Jaffal-Mourtada and D

    B. Jaffal-Mourtada and D. Yakoubi. A splitting method for a stationary second grade fluid model.AIP Conference Proceedings, 3034(1):020017, 03 2024

    work page 2024

  17. [17]

    E. W. Jenkins, V. John, A. Linke, and L. G. Rebholz. On the parameter choice in grad-div stabilization for the Stokes equations.Advances in Computational Mathematics, 40(2):491–516, Apr. 2014. 10

    work page 2014

  18. [18]

    V. John. Finite Element Methods for Incompressible Flow Problems, volume 51 ofSpringer Series in Computational Mathematics. Springer International Publishing, Cham, 2016

    work page 2016

  19. [19]

    Moulin, P

    J. Moulin, P. Jolivet, and O. Marquet. Augmented lagrangian preconditioner for large-scale hydro- dynamic stability analysis. Computer Methods in Applied Mechanics and Engineering, 351:718–743, 2019

    work page 2019

  20. [20]

    M. A. Olshanskii and A. Reusken. Grad-div stablilization for Stokes equations. Mathematics of Computation, 73(248):1699–1719, 2004

    work page 2004

  21. [21]

    Rebholz, A

    L. Rebholz, A. Viguerie, and M. Xiao. Efficient nonlinear iteration schemes based on algebraic splitting for the incompressible navier-stokes equations.Mathematics of Computation, 88(318):1533–1557, 2019

    work page 2019

  22. [22]

    Schäfer, S

    M. Schäfer, S. Turek, F. Durst, E. Krause, and R. Rannacher. Benchmark Computations of Laminar Flow Around a Cylinder. In E. H. Hirschel, K. Fujii, B. van Leer, M. A. Leschziner, M. Pandolfi, A. Rizzi, B. Roux, and E. H. Hirschel, editors,Flow Simulation with High-Performance Computers II, volume 48, pages 547–566. Vieweg+Teubner Verlag, Wiesbaden, 1996

    work page 1996

  23. [23]

    Takhirov, M

    A. Takhirov, M. Aggul, S. Ergen, F. G. Eroglu, and S. Kaya. Robust arrow–hurwicz method for high–rayleigh number boussinesq flow.Calcolo, to appear in 2026

    work page 2026

  24. [24]

    Takhirov, A

    A. Takhirov, A. Cıbık, F. G. Eroglu, and S. Kaya. An improved Arrow–Hurwicz method for the steady-state Navier–Stokes equations.Journal of Scientific Computing, 96:52, 2023

    work page 2023

  25. [25]

    Takhirov, C

    A. Takhirov, C. Trenchea, and J. Waters. Second-order efficient nonlinear filter stabilization for high reynolds number flows.Numerical Methods for Partial Differential Equations, 39(1):90–107, 2023

    work page 2023

  26. [26]

    Temam.Navier-StokesEquations, volume 2 ofStudies in Mathematics and Its Applications

    R. Temam.Navier-StokesEquations, volume 2 ofStudies in Mathematics and Its Applications. North- Holland Publishing Co., Amsterdam, revised edition, 1979

    work page 1979

  27. [27]

    Temam.Navier-Stokes Equations: Theory and Numerical Analysis, volume 343

    R. Temam.Navier-Stokes Equations: Theory and Numerical Analysis, volume 343. American Mathe- matical Soc., 2001

    work page 2001

  28. [28]

    Viguerie and A

    A. Viguerie and A. Veneziani. Algebraic splitting methods for the steady incompressible navier–stokes equations at moderate reynolds numbers.Computer Methods in Applied Mechanics and Engineering, 330:271–291, 2018

    work page 2018

  29. [29]

    D. Yakoubi. Enhancing the viscosity-splitting method to solve the time-dependent navier–stokes equa- tions. Communications in Nonlinear Science and Numerical Simulation, 123:107264, 2023. 11 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Velocity u 0.0 0.2 0.4 0.6 0.8 1.0 Vertical position y u-velocity along x = 0.5 Ghia 1982 IVS 0.0 0.2 0.4 0.6 0.8 1.0 Horizontal position ...

    work page 2023