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A generic Scheme For the time-dependent Navier-Stokes Equation Coupled With The Heat Equation

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Pith's one-line read A generic numerical scheme converges to a weak solution of the coupled Navier-Stokes/heat equations with temperature-dependent viscosity.

desk verdict A legitimate GDM extension for the time-dependent Navier-Stokes/heat system, but the main theorem has a load-bearing regularity gap and the numerics test the wrong boundary conditions. read the letter →

arxiv 2411.14650 v1 pith:DORK7Q3E submitted 2024-11-22 math.NA cs.NA

Yahya Alnashri This is my paper · ORCID
classification math.NAcs.NA MSC 65M0876D0535Q3080A19
keywords Navier-Stokesproblemheatequationtime-dependentgradientdiscretisationmethodschemesfinitevolumeschemeconvergenceanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper designs a generic space-time discretisation, in the gradient discretisation framework, for the time-dependent Navier-Stokes equations coupled with the heat equation through a temperature-dependent viscosity, and proves convergence without imposing non-physical conditions. The main theorem states that any sequence of space-time gradient discretisations that is coercive, consistent, limit-conforming, trilinear limit-conforming and compact produces discrete solutions whose velocity and temperature reconstructions converge strongly in $L^2$, whose gradients converge weakly in $L^2$, and whose pressure converges weakly in $L^2$, to a weak solution of the continuous coupled problem. The proof works by deriving uniform energy estimates for the discrete solutions and then using compactness and the abstract consistency properties to identify the limit. Numerical experiments with a hybrid finite volume scheme on triangular and distorted meshes, for both constant viscosity and the temperature-dependent viscosity $V(S)=\sqrt{S^2+1}+2$, show first-order convergence rates. If the theorem is right, a single checklist of five abstract properties certifies convergence for every concrete scheme that fits this format.

What carries the argument

The central object is the space-time gradient discretisation $D_T=(D,J_D,\tilde J_D,(t^{(n)})_{n=0,\dots,N})$, a tuple of finite-dimensional spaces and linear reconstruction operators that encodes how a concrete numerical method represents velocity, temperature, pressure, gradients and divergence. Its role is to separate the specific scheme (finite volume, virtual element, nonconforming finite element, and so on) from the mechanism of convergence: if the discretisation satisfies the abstract properties of coercivity, consistency, limit-conformity, trilinear limit-conformity and compactness, then the generic variational scheme (2.6) inherits uniform discrete energy estimates and can be passed to the limit. The positivity assumptions on the discrete convection forms $A_D$ and $B_D$ and the discrete inf-sup quantity $B_D$ in (2.5) are what make the energy estimates and the pressure limits possible.

What would settle it

Run the fully discrete scheme with a bounded discontinuous viscosity such as $V(S)=1$ for $S<0$ and $V(S)=2$ for $S\ge 0$ on a manufactured temperature field that crosses the discontinuity, and test whether the viscosity term converges to the expected limit as the mesh and time step shrink; if that limit is wrong, or if the discrete pressures grow without bound, Theorem 3.4 needs an additional hypothesis.

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Extended reading notes

Core claim

The paper claims that the fully implicit gradient scheme (2.6) for the coupled problem is convergent in this precise sense: for any sequence of space-time gradient discretisations satisfying the five abstract properties, any sequence of discrete solutions $(u_m,p_m,S_m)$ has a subsequence for which the reconstructed velocity and temperature converge strongly in $L^2(0,T;L^2(\Omega))$, the discrete gradients converge weakly in $L^2(\Omega\times(0,T))^d$, and the reconstructed pressures converge weakly in $L^2(\Omega\times(0,T))$, and the limit triple is a weak solution of the continuous coupled Navier-Stokes/heat system (1.1)-(2.4).

Load-bearing premise

The argument depends on an unstated regularity property of the viscosity $V$: the proof exchanges a limit in temperature with a limit in viscosity, which requires $V(S_m)$ to converge in a strong enough sense, while the stated assumptions only guarantee that $V$ is bounded between two positive constants.

Editorial extensions

If this is right

  • Any concrete scheme that can be written as a space-time gradient discretisation and that satisfies the five abstract properties converges to a weak solution of the coupled problem, so the theorem acts as a certification checklist for finite volume, virtual element and other polytopal methods.
  • The uniform energy bounds of Lemma 3.1 give a constructive numerical proof of existence of weak solutions, since the limit of the bounded discrete sequence is the desired continuous solution.
  • Because the scheme is implicit in time and no CFL-type condition appears in the abstract assumptions, convergence holds for arbitrary time-step/mesh-size ratios as long as the time step tends to zero.
  • In the numerical tests, relative $L^2$ errors for velocity, pressure and temperature decay at a rate close to one on both triangular and distorted meshes, for constant viscosity and for $V(S)=\sqrt{S^2+1}+2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the proof can be made fully rigorous by adding an explicit continuity or strong-composition assumption on $V$; the numerical examples use smooth $V$, so they do not probe the borderline case.
  • Editorial inference: a missing uniform $L^2$ bound on the discrete pressure means the claimed weak convergence of the pressure rests on an estimate that the paper does not supply; adding a discrete pressure-stability inequality would close the gap.
  • Editorial inference: the abstract framework should extend to other nonconforming discretisations, such as discontinuous Galerkin or virtual element methods, provided they satisfy discrete inf-sup stability; a natural test is to implement one and verify the five properties.
  • Editorial inference: the convergence statement is for weak solutions and does not address uniqueness or time regularity; if the coupled system has multiple weak solutions, the theorem only guarantees convergence to one of them.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper extends the gradient discretisation method (GDM) to the evolutionary incompressible Navier-Stokes system coupled with a heat equation through a temperature-dependent viscosity V(S). After defining a space-time gradient discretisation and a fully implicit scheme (2.6), the author states coercivity, consistency, limit-conformity, trilinear limit-conformity and compactness assumptions, proves discrete energy estimates (Lemmas 3.1 and 3.3), and claims in Theorem 3.4 that any sequence of discrete solutions associated with a suitable sequence of discretisations converges, up to a subsequence, to a weak solution of the continuous problem (2.4), with strong L2 convergence of reconstructed velocity and temperature and weak convergence of gradients and pressure. Numerical experiments with hybrid finite volume discretisations on triangular and distorted meshes report first-order convergence rates.

Significance. If the proof gaps are repaired, the result is a useful generic framework: it covers families of polytopal schemes for a non-isothermal Navier-Stokes model and avoids artificial smoothness assumptions beyond boundedness of V. The numerical section provides concrete tests, although no code is shipped, and the paper does not contain machine-checked proofs. The extension over the stationary analysis in [4] and the abstract GDM machinery in [11] is incremental but within the normal scope of a numerical analysis journal. However, the claimed generality of Theorem 3.4 currently rests on an unstated continuity/composition assumption on V and on an unproved pressure estimate.

major comments (3)
  1. [Theorem 3.4, Eq. (3.10)] The passage to the limit in the viscosity term is not justified. Assumption (2.3) only gives V in L∞(R) with a1 <= V <= a2. Strong L2 convergence of ePi_{D_m} S_m to S does not imply strong, or even almost-everywhere, convergence of V(ePi_{D_m} S_m) to V(S), and the discrete gradients ∇_{D_m} u_m are only weakly convergent, so the product cannot be controlled. The proof needs an additional assumption such as V continuous and bounded, or a monotonicity/composition condition ensuring V(ePi_{D_m} S_m) -> V(S) in some L^q space, before the dominated convergence argument in the proof of Theorem 3.4 can be applied. Without such an assumption, the central convergence claim is unsupported.
  2. [Theorem 3.4, third bullet and proof] The weak convergence χ_{D_m} p_m -> p̄ in L2(Ω×(0,T)) is asserted but never proved. No L2 bound on the reconstructed pressure is derived from the scheme; Lemmas 3.1 and 3.3 only give estimates for velocity and temperature. One needs to use the discrete inf-sup constant B_D from (2.5) together with the momentum equation (2.6a) to bound χ_{D_m} p_m in L2(0,T;L2(Ω)) before one can pass to the limit in the pressure term of (3.10). This is a load-bearing gap in Theorem 3.4.
  3. [Lemma 3.1, proof] The algebraic inequality (a−b)·a >= 1/2(|a|²+|b|²) is false; the correct inequality is (a−b)·a >= 1/2(|a|²−|b|²). The subsequent displayed estimate in the proof uses the correct form, so the energy estimate is repairable, but the written statement must be corrected for the proof to be valid.
minor comments (6)
  1. [Lemma 3.1] Lemma 3.1 states that (u,p,S) is a solution to the discrete problem (2.4); it should refer to the gradient scheme (2.6).
  2. [Definition 2.3] In Definition 2.3(2), the variable v is used twice ('for all v ∈ L²(Ω)^d and for all v ∈ L²(Ω)'); the second occurrence should be a scalar test function, for example r or φ.
  3. [Eq. (2.4a)] In the weak formulation (2.4a), the pressure term −∫_Ω (div v)(x) p̄(x,t) dx is missing the dt inside the time integral.
  4. [Theorem 3.4, proof] The proof claims that [11, Theorem 4.14] gives strong convergence in L∞(0,T;L2(Ω)); the usual GDM compactness result gives strong convergence in L2 and weak-* convergence in L∞. The statement of Theorem 3.4 only requires strong L2 convergence, so this should be corrected.
  5. [Section 4] The numerical test uses V(S)=√(S²+1)+2, which is not in L∞(R) as required by Assumption (2.3). The test should use a bounded truncation or explicitly state that the unbounded growth is never attained on the range of the discrete temperature.
  6. [Throughout] There are numerous typographical errors ('inforced', 'trail functions', 'm =∈ {0,...,N}', 'Defintion') that should be corrected in a revision.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the convergence proof rests on external GDM compactness results and independent manufactured-solution tests; the main gap is an unstated regularity assumption on V, not a circular reduction.

full rationale

The derivation chain in Theorem 3.4 is not circular by the standards of this review. The discrete scheme (2.6) is solved forward in time, and the claimed convergence is obtained by combining energy estimates (Lemmas 3.1 and 3.3) with external compactness and interpolation results from the GDM monograph [11], not from the target continuous solution. The author's own prior work [4] is used only to import definitions of coercivity, limit-conformity, compactness, and the stationary discrete elements, and those definitions do not contain the convergence conclusion. The numerical section uses exact manufactured solutions (4.2)-(4.4) and reports measured L2 errors, so no fitted parameter is renamed as a prediction. The only serious defect is a proof gap in the passage to the limit in the viscosity term: the text claims 'the assumptions inforced on V enable us to apply the dominated convergence theorem', but Assumptions (2.3) only give V in L∞ with a1≤V≤a2, and strong L2 convergence of eΠ_{D_m}S_m does not imply V(eΠ_{D_m}S_m)→V(S) without a continuity or composition property. This is a missing hypothesis in the proof of Theorem 3.4 and should be reported as a correctness risk, not as circularity, because the missing condition is not the target result and the claimed reduction does not equate an input with an output.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the model constants are prescribed. The proof leans on the abstract GDM framework of [11] and [12], on the stationary coupled GDM in [4] for the discrete structures, and on an unstated continuity or composition property of V needed to take the limit in the viscosity term. The latter is the main fragile premise.

assumptions (4)
  • domain assumption The spatial gradient discretisation satisfies Definition 2.1, including the discrete inf-sup condition BD > 0, and the space-time sequence satisfies the coercivity, consistency, limit-conformity, trilinear limit-conformity and compactness properties of Definition 2.3.
    Theorem 3.4 assumes these properties; the convergence proof depends on them for compactness and for identifying the limit.
  • standard math The external GDM results used in the proof, specifically [11, Lemma 4.8], [11, Theorem 4.14] and the discrete integration-by-parts formula [11, Eq. (D.15)], are applicable to the discrete operators defined in Definition 2.1.
    The proof invokes these lemmas without proving them; if they require stronger discrete Stokes or convection properties than stated, the argument has a gap.
  • ad hoc to paper V is continuous in its argument, or otherwise has a composition property such that strong L2 convergence of the reconstructed temperature implies a suitable limit for V(S_m).
    The proof of Theorem 3.4 uses dominated convergence on the viscosity term, but the stated assumption (2.3) only gives V in L^infinity; continuity is needed for the product limit.
  • domain assumption The data satisfy the regularity conditions in (2.3): V bounded above and below, g and h in L2, u0 and S0 in H^1_0.
    These are the standard assumptions for the weak formulation and are used in the energy estimates of Lemmas 3.1 and 3.3.

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Pith. "Pith review of A generic Scheme For the time-dependent Navier-Stokes Equation Coupled With The Heat Equation." pith.science (2026). https://pith.science/paper/DORK7Q3E

@misc{pith2026241114650,
  author       = {Pith},
  title        = {Pith review of: A generic Scheme For the time-dependent Navier-Stokes Equation Coupled With The Heat Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DORK7Q3E}},
  note         = {Machine review of arXiv:2411.14650}
}
read the original abstract

In this work, we study the gradient discretisation method (GDM) of the time-dependent Navier-Stokes equations coupled with the heat equation, where the viscosity depends on the temperature. We design the discrete method and prove its convergence without non-physical conditions. The paper is closed with numerical experiments that confirm the theoretical results.

Figures

Figures reproduced from arXiv: 2411.14650 by the authors.

Figure 4.1
Figure 4.1. Sample of the polygonal meshes. temperature (V (S) = 1). The exact solution is u¯(x, y) :=  u¯1 u¯2  =  sin((π + t)y) cos((π + t)x) cos((π + t)y) sin((π + t)x)  , (4.2) p¯(x, y) = sin((π + t)x) cos((π + t)y), (4.3) S¯(x, y) = tsin(x + y), (4.4) We post in [PITH_FULL_IMAGE:figures/full_fig_p010_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. The relative errors for the case V (S) = 1. References [1] R. Agroum, S. M. Aouadi, C. Bernardi, and J. Satouri, Spectral discretization of the navier-stokes problem coupled with the heat equation, ESAIM: M2AN, 49 (2013), pp. 621–639. [2] R. Agroum, C. Bernardi, and J. Satouri, Spectral discretization of the time-dependent navier–stokes problem coupled with the heat equation, Applied Mathematics and Computa￾tion, 26… view at source ↗
Figure 4.3
Figure 4.3. The relative errors for the case V (S) = 1. [8] F. Brezzi, J. Rappaz, and P.-A. Raviart, Finite dimensional approximation of nonlinear problems: Part i: branches of nonsingular solutions, Numerische Mathematik, 36 (1980), pp. 1–25. [9] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations i, Revue fran¸caise d’automatique informatique recherche… view at source ↗
Figures from the paper (2 more)
Figure 4.4
Figure 4.4. Figure 4.4: The relative errors for the case V (S) = √ S2 + 1 + 2 [PITH_FULL_IMAGE:figures/full_fig_p013_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: The relative errors for the case V (S) = √ S2 + 1 + 2 [PITH_FULL_IMAGE:figures/full_fig_p014_4_5.png]

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [4]

    Alnashri, The gradient discretisation method for the navier–stokes problem coupled with the heat equation , Results in Applied Mathematics, 11 (2021), p

    Y. Alnashri, The gradient discretisation method for the navier–stokes problem coupled with the heat equation , Results in Applied Mathematics, 11 (2021), p. 100176

    work page 2021

  2. [11]

    Droniou, R

    J. Droniou, R. Eymard, T. Gallou¨et, C. Guichard, and R. Herbin, The gradient dis- cretisation method, vol. 82, Springer, 2018

    work page 2018

  3. [1]

    Agroum, S

    R. Agroum, S. M. Aouadi, C. Bernardi, and J. Satouri, Spectral discretization of the navier-stokes problem coupled with the heat equation, ESAIM: M2AN, 49 (2013), pp. 621–639

    work page 2013

  4. [2]

    Agroum, C

    R. Agroum, C. Bernardi, and J. Satouri, Spectral discretization of the time-dependent navier–stokes problem coupled with the heat equation , Applied Mathematics and Computa- tion, 268 (2015), pp. 59–82

    work page 2015

  5. [3]

    Allali, F

    K. Allali, F. Bikany, A. Taik, and V. Volpert, Numerical simulations of heat explosion with convection in porous media , Combustion Science and Technology, 187 (2015), pp. 384– 395

    work page 2015

  6. [5]

    P. F. Antonietti, G. V acca, and M. Verani, Virtual element method for the navier–stokes equation coupled with the heat equation , IMA Journal of Numerical Analysis, 43 (2023), pp. 3396–3429

    work page 2023

  7. [6]

    Bercovier and O

    M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the stokes problem in the primitive variables , Numerische Mathematik, 33 (1979), pp. 211–224

    work page 1979

  8. [7]

    Bernardi, F

    C. Bernardi, F. Coquel, and P.-A. Raviart, Automatic coupling and finite element dis- cretization of the Navier-Stokes and heat equations, Internal Report 10001 , Laboratoire Jacques-Louis Lions, 2010. 12 YAHYA ALNASHRI Figure 4.3. The relative errors for the case V (S) = 1

    work page 2010

Show all 13 references
  1. [8]

    Brezzi, J

    F. Brezzi, J. Rappaz, and P.-A. Raviart, Finite dimensional approximation of nonlinear problems: Part i: branches of nonsingular solutions , Numerische Mathematik, 36 (1980), pp. 1–25

    1980

  2. [9]

    Crouzeix and P.-A

    M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations i , Revue fran¸ caise d’automatique informatique recherche op´ erationnelle. Math´ ematique, 7 (1973), pp. 33–75

    1973

  3. [10]

    Droniou, R

    J. Droniou, R. Eymard, and P. Feron, Gradient schemes for stokes problem , IMA Journal of Numerical Analysis, 36 (2016), pp. 1636–1669

    2016

  4. [12]

    Eymard, P

    R. Eymard, P. Feron, and C. Guichard, Family of convergent numerical schemes for the incompressible navier–stokes equations , Mathematics and Computers in Simulation, 144 (2018), pp. 196–218

    2018

  5. [13]

    McLean, Continuum Fluid Mechanics and the Navier-Stokes Equations , John Wiley and Sons, Ltd, 2012

    D. McLean, Continuum Fluid Mechanics and the Navier-Stokes Equations , John Wiley and Sons, Ltd, 2012. (Yahya Alnashri) Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Saudi Arabia Email address : yanashri@uqu.edu.sa ANALYSIS OF SCHEMES FOR T...

    2012

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