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REVIEW 3 major objections 5 minor 14 references

On the Removal of Solver-Induced Dependencies in Momentum-Weighted Interpolation for Primal and Continuous-Adjoint Flow Solvers

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Momentum-weighted interpolation can be made independent of solver relaxation and time-step parameters while preserving the original discretization.

desk verdict A clean, parameter-free fix for solver-induced MWI dependence; the derivation holds up and the numerics support it, but the paper needs a data-availability correction and a softer physical-consistency claim. read the letter →

arxiv 2607.21736 v1 pith:RBCEMFKC submitted 2026-07-23 physics.comp-ph

Niklas Kühl This is my paper · ORCID
classification physics.comp-ph MSC 76M1276D05 PACS 47.11.-j
keywords Momentum-WeightedInterpolationRhie–ChowPressure–velocitycouplingContinuousadjointsensitivityUnder-relaxationdependenceTime-stepCollocatedfinite-volumemethodSolverparameterindependence
topics Navier-Stokes Regularity
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

Momentum-Weighted Interpolation (MWI) is the standard way collocated finite-volume solvers reconstruct face fluxes for pressure–velocity coupling. This paper shows that in typical implementations the diagonal momentum coefficient entering MWI includes under-relaxation and time-discretization contributions, so the computed face flux — and with it forces, wake quantities, and adjoint sensitivities — changes when the user changes relaxation or time-step, even though the spatial discretization is identical. It proposes a reformulation that removes those solver-induced terms from the mobility coefficient, replacing them with stored recursive correction terms from the previous iteration and time level. The result is a face flux whose pressure-correction part is unaffected by relaxation and time-step, applicable identically to primal and continuous-adjoint systems. Numerical tests on a 2D cylinder and a 3D turbulent ship hull show that the uncorrected form varies by about 1% for forces and up to about 20% for wake fields, with adjoint wake sensitivities even changing sign; the corrected form stays constant over wide parameter ranges.

What carries the argument

Momentum-Weighted Interpolation (MWI) itself — the face-flux reconstruction scheme that combines an interpolated cell-centered predictor with a face-based pressure-gradient correction weighted by a mobility coefficient b_F proportional to Omega/A_P. The key move is to divide the relaxed, time-discretized momentum equation by the spatial diagonal coefficient A_P only, so the mobility entering the pressure correction no longer contains 1/omega or 1/Delta-t; the previously absorbed terms become recursive multipliers acting on stored flux-correction differences, as in Eq. (16).

What would settle it

Run the corrected MWI on a sequence of progressively more skewed or polyhedral meshes for the same cylinder or ship case at a fixed relaxation factor far from 1, and check whether the computed drag or wake quantity stays constant as mesh quality degrades; if the flux varies with mesh skewness while solver parameters are fixed, the cancellation underlying the corrected formulation is not exact at the discrete level.

Watch

Extended reading notes

Core claim

The paper establishes that the classical MWI face flux inherits a spurious dependence on solver parameters because the mobility coefficient b_F is proportional to the inverse of the interpolated diagonal momentum coefficient A_P, which in practical solvers contains 1/omega and 1/Delta-t contributions. By writing the relaxed, time-integrated momentum balance in a divided form and applying the same elimination of the non-pressure term as in the classical derivation, the author derives a modified flux expression in which the pressure-correction mobility is scaled only by the spatial coefficient, while the temporal and relaxation effects reappear as coefficients multiplying stored flux-correctio

Load-bearing premise

The derivation requires that the non-pressure part of the momentum balance, when interpolated from cell centers to a face, equals the same term evaluated directly at the face; on a real collocated mesh these are different discrete reconstructions, so the cancellation is exact only in a formal continuous sense.

Editorial extensions

If this is right

  • CFD results obtained with the corrected MWI are reproducible across different relaxation factors and time steps, removing a hidden source of variance between supposedly identical simulations.
  • Adjoint shape sensitivities become trustworthy when the adjoint solver requires small relaxation or pseudo-time steps, since the corrected form eliminates the parameter-induced sign changes observed in wake sensitivities.
  • The correction is a minimal change to existing MWI implementations: the pressure-correction term is scaled differently and previously stored flux corrections are reused, so it can be dropped into current segregated solvers without altering the spatial discretization.
  • Force-based quantities stabilize at the one-percent level, while field-based quantities such as wake homogeneity stabilize at the twenty-percent level, indicating that sensitivity studies benefit most from the correction.
  • Because the formulation is derived on unstructured polyhedral collocated grids and applies equally to primal and adjoint systems, it covers the class of industrial finite-volume solvers.

Reading between the lines

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

  • If the cancellation of the non-pressure term is exact only in the continuous limit, the residual dependence on relaxation might not vanish on highly skewed or otherwise low-quality meshes; a dedicated grid-quality sensitivity study would test this.
  • The recursive form of the corrected flux resembles a time filter or discrete integrator for the flux correction, suggesting a possible connection to deferred-correction and dual-time-stepping strategies where stored corrections are re-weighted.
  • A direct extension to higher-order time schemes (e.g., BDF2 or Crank–Nicolson) would require consistent recursive weights; the practical stability of the recursive term under strongly varying time steps has not been analyzed and is a natural next test.
  • The observed sign change in adjoint wake sensitivity implies that optimization loops using the classical formulation may converge to the wrong descent direction, making the corrected formulation necessary for reliable gradient-based design.
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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 / 5 minor

Summary. The paper proposes a correction to Momentum-Weighted Interpolation (MWI) that removes contributions of under-relaxation and time discretization from the diagonal momentum coefficient appearing in the pressure-driven face-flux correction. The derivation is presented in a recipe-like way: classical MWI (Eq. 6), relaxation-only (Appendix A), time-only (Appendix B), and the combined case (Eq. 16). The combined formulation introduces stored MWI corrections from the previous time level and outer iteration, so that the converged fixed point gives a face flux whose pressure-correction term scales with the spatial mobility only. The numerical experiments on a 2D laminar cylinder and a 3D turbulent ship hull (JBC) compare the classical and corrected formulations under variations of relaxation factor and pseudo-time-step size. The corrected formulation yields nearly flat integral and field-based quantities, whereas the classical formulation exhibits deviations of order 1% in forces, up to 25% in wake-related quantities, and sign changes in adjoint wake sensitivities.

Significance. If the central claim holds, the paper removes a real practical artifact: solver-parameter dependence of converged MWI fluxes in both primal and continuous-adjoint solvers. The derivation is analytic and parameter-free, with no fitted coefficients; the fixed-point cancellation of the recursive terms is a structural property, not a calibration result. The numerical study is carefully designed, especially in fixing the primal state for all adjoint evaluations, which isolates the MWI effect on sensitivities. The work is an incremental but useful extension of Kühl & Rung (2022), and the recipe-like presentation should make the correction easy to implement in other finite-volume codes. The main limitations are the absence of external code/data for reproducibility and a few inconsistencies in the printed equations, but these do not undermine the analytical core.

major comments (3)
  1. [Section 3, Eq. (14)] The first recursive term after the pressure-correction term is printed as \omega d_F^n(\dot{V}^{n,k-1}_F - \tilde{V}^{n,k-1}_F). Subtracting Eq. (12) from Eq. (13) gives \omega d_F^n(\dot{V}^{n-1}_F - \tilde{V}^{n-1}_F), not an n,k-1 term. Since Eq. (16) correctly uses the previous-time-level correction \Delta\dot{V}^{n-1}_F, this is likely a typographical error in a central equation, but it must be corrected.
  2. [Section 3, Eq. (8)] The under-relaxation term on the right-hand side is printed with v^{n,k}_{P,i}, while Eq. (9), obtained by dividing Eq. (8), has v^{n,k-1}_{P,i}. These two equations are inconsistent; the correct under-relaxation history term is v^{n,k-1}, as used in Eqs. (9) and (11). Please fix Eq. (8).
  3. [Section 3, after Eq. (16)] The sentence 'the pressure-correction scaling becomes independent of both the chosen time-step size and the relaxation parameter' is not immediate from Eq. (16) alone, because the explicit pressure-correction coefficient is \omega b_F/(1+d_F). The independence holds at the converged fixed point, where the recursive terms combine to produce \dot{V} = \tilde{V} - b_F[\nabla p term]. This fixed-point reduction should be shown explicitly; as written, the claim appears too strong at the level of the transient equation.
minor comments (5)
  1. [Section 9, Data Availability] The statement 'No new data were generated or analyzed in this study' contradicts Section 4, which reports numerical results generated for this study. Please rephrase, e.g., 'No experimental data were generated; numerical data are available from the author upon request.'
  2. [Section 2, Eqs. (3)-(5) and Section 3, Eqs. (12)-(13)] The H/A terms are written without the overbar/non-overbar distinction defined in the text. For clarity, explicitly mark the interpolated H/A in the predictor equations and state that the face-based form uses the same interpolated H/A by construction, so that the cancellation in Eqs. (6) and (16) is unambiguous.
  3. [Abstract and Section 4.2] The abstract and conclusions state wake deviations 'up to 20%', but Sec. 4.2 reports deviations up to approximately +25% for W and beyond -100% for S_W. Please harmonize these numbers.
  4. [Section 4.1 and 4.2, experiment labels] The labels E1.1/E2.1 are reused with different meanings in the cylinder and ship sections. This is not fatal, but adding a prefix (e.g., C1.1, J1.1) would improve readability.
  5. [Section 3, reference to Eq. 22] The text says 'In complete analogy to Eqn. 22' before Eq. (13), but Eq. (22) appears only later in Appendix A. Please adjust the cross-reference or renumber.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: parameter-independence is a derived fixed-point property, not a fitted or self-referential input.

full rationale

The derivation chain is a self-contained algebraic reformulation. Starting from the semi-discrete momentum equation with under-relaxation and implicit Euler time integration (Eq. 8), the paper rearranges to Eq. 11, constructs the interpolated predictor flux (Eq. 12) and the formal face-based momentum balance (Eq. 13), and cancels the shared non-pressure term H_P,i/A_P to obtain the recursive MWI correction (Eqs. 14–16). No parameter is fitted and no empirical target is used to determine the formula. The claimed independence from ω and Δt is a genuine fixed-point consequence: at convergence the recursive terms ΔV^{n,k−1} and ΔV^{n−1} cancel, leaving Ẍ_F = ẽ_F − b_F [∂p/∂x_i − f_i]_F^face ΔΓ, in which the mobility b_F is built from the spatial-only diagonal coefficient. This is a derivation, not a construction whose conclusion is identical to its assumption. The numerical sections compare the classical and corrected formulations in the same in-house solver; they demonstrate the effect but do not fit the correction. Concerns that are real but non-circular: the cancellation of interpolated versus face-based H/A is exact only in the continuous/formal sense (the paper itself notes the discrete reconstructions differ in Sec. 2), so on skewed grids the independence may be approximate; the numerical evidence relies on self-cited FreSCo+ validation; and the Data Availability statement (Sec. 9, "No new data were generated or analyzed in this study") contradicts the presented simulations. These affect robustness and reproducibility, not the circularity of the analytical claim.

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

No free parameters are fitted in the derivation; the correction is an algebraic rearrangement of the semi-discrete momentum equation. The main implicit input is the formal face evaluation of the cell-centered balance and the equivalence of interpolated and face-based non-pressure terms; the numerical portion relies on the author's own FreSCo+ solver and on the stated convergence criteria. No new physical entities are introduced.

assumptions (4)
  • domain assumption The cell-centered semi-discrete momentum balance (Eq. 1) can be evaluated 'formally' at face centers with the same discrete operators.
    Sec. 2, around Eqs. 2-5; the cancellation of the non-pressure term requires a face-based reading of the cell-centered equation. On collocated grids this is an operational assumption, not a derived identity.
  • domain assumption The face-interpolated non-pressure contribution (H_P,i/A_P)^bar_F equals the face-evaluated H_P,i/A_P in the face-based balance (Eqs. 12/13).
    Sec. 3; the elimination between predictor and face-based flux is exact only if interpolation of H/A and the face evaluation commute. On non-uniform/polyhedral grids they differ by reconstruction error, leaving a residual that could reintroduce parameter dependence.
  • domain assumption Baseline time integration is implicit Euler and under-relaxation enters as in Eq. 8; extensions to other schemes require 'a consistent modification' but are not derived.
    Sec. 3 and App. B; the universality of the recipe for arbitrary time integrators and relaxation variants is asserted, not demonstrated.
  • domain assumption The iterative/recursive process reaches a fixed point satisfying the stored-correction relations (Eq. 15) with the stated residual tolerances.
    Sec. 4; the parameter-independence claim is shown at converged states ('residuals below 10^-10', 'no further variations'), and the pseudo-unsteady convergence criterion is not precisely defined.

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Cite this review

Pith. "Pith review of On the Removal of Solver-Induced Dependencies in Momentum-Weighted Interpolation for Primal and Continuous-Adjoint Flow Solvers." pith.science (2026). https://pith.science/paper/RBCEMFKC

@misc{pith2026260721736,
  author       = {Pith},
  title        = {Pith review of: On the Removal of Solver-Induced Dependencies in Momentum-Weighted Interpolation for Primal and Continuous-Adjoint Flow Solvers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RBCEMFKC}},
  note         = {Machine review of arXiv:2607.21736}
}
read the original abstract

Momentum-Weighted Interpolation (MWI) is a key component in pressure--velocity coupling schemes on collocated cell-centered finite-volume methods for both primal and continuous adjoint formulations. In many practical implementations, MWI relies on diagonal momentum coefficients that include contributions from under-relaxation and time discretization. As a result, both primal quantities of interest and adjoint sensitivities may exhibit a non-physical dependence on solver parameters such as relaxation factors and time-step size, and no well-defined limit is obtained as these parameters approach zero. In this work, building on previous developments in discrete-consistent MWI formulations, a simple correction is proposed that removes solver-induced contributions from the diagonal momentum coefficients in the pressure-driven term. The resulting formulation preserves the original discretization while eliminating artificial dependencies on relaxation and time-stepping parameters and is applied consistently to both primal and adjoint systems. To facilitate its application, the derivation is presented in a structured, recipe-like manner that can be readily followed and transferred to different finite volume-based solver configurations. The proposed modification is assessed for a two-dimensional laminar cylinder flow and a three-dimensional turbulent ship hull flow configuration. In both cases, the uncorrected formulation leads to significant variations in forces, wake-related quantities, and shape sensitivities when solver parameters are altered, despite all simulations being iterated to converged residual levels and stable integral quantities. In contrast, the corrected formulation yields consistent results across a wide range of relaxation factors and time-step sizes.

Figures

Figures reproduced from arXiv: 2607.21736 by the authors.

Figure 1
Figure 1. Schematic representation of a finite-volume arrangement (a) in the field and (b) at a boundary. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Submerged cylinder case (ReD = 10): Overview of the computational grid (a) and near-wall resolution (b). • E1.2: Steady simulations under identical conditions using the proposed formulation, i.e., Eqn. 25. • E2.1: Pseudo-unsteady simulations with implicit time integration using the classical MWI formulation from Eqn. 6, in which neither temporal nor relaxation contributions are excluded in the mobility coefficient. … view at source ↗
Figure 3
Figure 3. Submerged cylinder case (ReD = 10): Influence of the relaxation factor ω on the primal drag coefficient (left) and the corresponding adjoint sensitivity (center) for Experiments E1.1 and E1.2. The right-hand side shows the relative deviation of both quantities. Exemplary normalized primal pressure fields corresponding to the steady simulations are shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Submerged cylinder case (ReD = 10): Exemplary primal pressure fields. Left to right: classical MWI without relaxation contributions for ω = 0.8 and ω = 0.06, and proposed MWI with relaxation-consistent formulation for ω = 0.8 and ω = 0.06. to the primal pressure fields…
Figure 5
Figure 5. Figure 5: Submerged cylinder case (ReD = 10): Exemplary adjoint pressure fields. Left to right: classical MWI without relaxation contributions for ω = 0.8 and ω = 0.06, and proposed MWI with relaxation-consistent formulation for ω = 0.8 and ω = 0.06. The corresponding results fo…
Figure 6
Figure 6. Figure 6: Submerged cylinder case (ReD = 10): Influence of the normalized pseudo-time step ∆t/(V /D) on the primal drag coefficient (left) and the corresponding adjoint sensitivity (center) for Experiments E2.1–E2.4. The right-hand side shows the relative deviation between the f…
Figure 7
Figure 7. Figure 7: Japan Bulk Carrier case (ReL = 107 ): Unstructured computational grid. Far-field (a) and near-hull view (b) on the utilized numerical grid. • E1.1: Steady simulations are carried out on a half-domain configuration exploiting symmetry with respect to the center plane. T…
Figure 8
Figure 8. Figure 8: Japan Bulk Carrier case (Re = 107 ): Influence of the relaxation factor ω on the normalized total resistance (left) and the corresponding normalized drag sensitivity (center) for Experiments E1.1 and E1.2. The right-hand side shows the relative deviation between both f…
Figure 9
Figure 9. Figure 9: Japan Bulk Carrier case (Re = 107 ): Influence of the relaxation factor ω on the wake homogeneity W (left) and the corresponding shape sensitivity S ω (center) for Experiments E1.1 and E1.2. The right-hand side shows the relative deviation between both formulations. Co…
Figure 10
Figure 10. Figure 10: Japan Bulk Carrier case (Re = 107 ): Exemplary nominal wake fields. (a)–(b) Classical MWI formulation without correction for ω = 0.8 and ω = 0.06, respectively; (c)–(d) proposed MWI formulation for the same relaxation factors. overall behavior is consistent with the p…
Figure 11
Figure 11. Figure 11: Japan Bulk Carrier case: Exemplary adjoint axial wake fields. (a)–(b) Classical MWI formulation [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: Japan Bulk Carrier case (Re = 107 ): Influence of the normalized time-step size on the propeller thrust T (left) and the corresponding sensitivity S T (center) for Experiments E2.1 and E2.2. The right-hand side shows the relative deviation between both formulations. 1…
Figure 13
Figure 13. Figure 13: Japan Bulk Carrier case (Re = 107 ): Influence of the normalized time-step size on the effective wake homogeneity E (left) and the corresponding sensitivity S E (center) for Experiments E2.1 and E2.2. The right-hand side shows the relative deviation between both formu…

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