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arxiv: 2510.15733 · v5 · submitted 2025-10-17 · ⚛️ physics.flu-dyn · physics.comp-ph

A HHO formulation for variable density incompressible flows where the density is purely advected

Lorenzo Botti , Francesco Carlo Massa This is my paper

Pith reviewed 2026-05-18 06:00 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.comp-ph
keywords hybrid high-ordervariable densityincompressible flowsvolume conservationpure advectionESDIRK time steppingRayleigh-Taylor instability
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The pith

Hybrid high-order scheme for variable-density incompressible flows achieves exact cell-by-cell volume conservation

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

The paper develops a Hybrid High-Order discretization for the incompressible Navier-Stokes equations when density varies but remains incompressible. It achieves exact conservation of volume at the discrete level, which means the density is transported by pure advection without any numerical sources or sinks. This property is particularly relevant for modeling mixtures of immiscible fluids, as it avoids artificial mixing and maintains sharp interfaces over long times. The approach combines hybrid finite element spaces for velocity, density and pressure with ESDIRK time integration, and the authors demonstrate its stability, convergence rates, and performance on the Rayleigh-Taylor instability benchmark.

Core claim

We propose a Hybrid High-Order (HHO) formulation of the incompressible Navier-Stokes equations with variable density that provides exact conservation of volume and, accordingly, pure advection of the density variable. The spatial discretization relies on hybrid velocity-density-pressure spaces and the temporal discretization is based on Explicit Singly Diagonal Implicit Runge-Kutta (ESDIRK) methods. The formulation possesses attractive features including cell-by-cell volume conservation up to machine precision, pressure-robustness, and the ability to preserve density bounds at low-order.

What carries the argument

The hybrid velocity-density-pressure spaces that enforce exact cell-by-cell volume conservation when paired with the ESDIRK time discretization.

If this is right

  • Conservation of volume enforced cell-by-cell up to machine precision
  • Pressure-robustness in the discretization
  • Ability to preserve density bounds at low-order
  • Robustness in the convection dominated regime
  • Reduced memory footprint thanks to static condensation

Where Pith is reading between the lines

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

  • This approach could extend to other multiphase flow problems where maintaining sharp interfaces is critical, such as in ocean modeling or industrial mixing.
  • The p-multilevel solution strategies may enable efficient scaling to three-dimensional problems with many degrees of freedom.
  • Weak imposition of boundary conditions may simplify application to complex geometries without requiring body-fitted meshes.

Load-bearing premise

The hybrid velocity-density-pressure spaces together with the ESDIRK time discretization are assumed to deliver discrete stability while preserving the exact cell-by-cell volume conservation property.

What would settle it

A simulation in a closed domain over many time steps where the total volume deviates from the initial value by more than machine precision, or where the density field develops values outside the initial range without physical diffusion.

Figures

Figures reproduced from arXiv: 2510.15733 by Francesco Carlo Massa, Lorenzo Botti.

Figure 1
Figure 1. Figure 1: RTI at At=0.5 and Re=1 000 with HHO-π k . From left to right, evolution of the density field at selected time points tTrg = t √ At ≃ 1, 1.5, 1.75, 2, 2.25, 2.5, 2.75. Top and bottom row, results for k=1 over fine grid and k=6 over the coarse grid, respectively. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: RTI at At=0.5 and Re=5 000 with HHO-π k . From left to right, evolution of the density field at selected time points tTrg = t √ At ≃ 1, 1.5, 1.75, 2, 2.25, 2.5, 2.75. Top and bottom row, results for k = 1 over the fine grid and k = 6 over the coarse grid, respectively. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: RTI plume close-up view at At=0.5 and Re=1 000 with HHO-π k (top row) and HHO-π k+1 (bottom row). From left to right, evolution of the interface at selected time points tTrg = t √ At ≃ 1.75, 2, 2.25, 2.5, 2.75. In each frame, results obtained with k=6 over the coarse grid and k=1 over the fine grid results stands on the left and the right, respectively, with respect to a vertical center-line [PITH_FULL_IM… view at source ↗
Figure 4
Figure 4. Figure 4: RTI plume close-up view at At=0.5 and Re=5 000 with HHO-π k (top row) and HHO-π k+1 (bottom row). From left to right, evolution of the interface at selected time points tTrg = t √ At ≃ 1.75, 2, 2.25, 2.5, 2.75. In each frame, results obtained with k=6 over the coarse grid and k=1 over the fine grid results stands on the left and the right, respectively, with respect to a vertical center-line. 28 [PITH_FUL… view at source ↗
Figure 5
Figure 5. Figure 5: RTI at At=0.75 and Re=1 000. Top and bottom row, k=0 HHO-π k and HHO-π k+1 com￾putations, respectively. From left to right, evolution of the density field at selected time points t ≃ 1, 1.5, 2, 2.5, 3, 3.5, 3.75. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
read the original abstract

We propose a Hybrid High-Order (HHO) formulation of the incompressible Navier-Stokes equations with variable density that provides exact conservation of volume and, accordingly, pure advection of the density variable. The spatial discretization relies on hybrid velocity-density-pressure spaces and the temporal discretization is based on Explicit Singly Diagonal Implicit Runge-Kutta (ESDIRK) methods. The formulation possesses some attractive features that can be fruitfully exploited for the simulation of mixtures of immiscible incompressible fluids, namely: conservation of volume enforced cell-by-cell up to machine precision, pressure-robustness, ability to preserve density bounds at low-order, robustness in the convection dominated regime, weak imposition of boundary conditions, implicit high-order accurate time stepping, reduced memory footprint thanks to static condensation, possibility to exploit inherited $p$-multilevel solution strategies to improve the performance of iterative solvers. After addressing stability at the discrete level, numerical validation is performed showcasing spatial and temporal convergence rates. To conclude, we tackle the Rayleigh-Taylor instability at different Atwood and Reynolds numbers focusing on mesh independence capabilities.

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 proposes a Hybrid High-Order (HHO) formulation of the variable-density incompressible Navier-Stokes equations using hybrid velocity-density-pressure spaces and ESDIRK time discretization. The central claim is that this yields exact cell-by-cell volume conservation up to machine precision, implying pure advection of the density field. Additional features include pressure-robustness, low-order density bound preservation, convection-dominated robustness, weak boundary conditions, high-order implicit time stepping, static condensation for reduced memory, and p-multilevel solver strategies. After a discrete stability analysis, the paper reports spatial/temporal convergence rates and Rayleigh-Taylor instability simulations at varying Atwood and Reynolds numbers.

Significance. If the exact volume conservation and accompanying discrete stability hold, the work would be significant for high-fidelity simulation of immiscible incompressible fluid mixtures, where cell-by-cell volume preservation prevents artificial mass transfer and improves long-time accuracy. The HHO framework's combination of high-order accuracy, pressure robustness, and computational efficiency via condensation offers a promising route for convection-dominated variable-density flows.

major comments (2)
  1. [Discrete stability analysis (post-abstract)] The abstract states that stability is addressed at the discrete level prior to numerical validation, yet no explicit discrete divergence identity, inf-sup argument for the hybrid velocity-density-pressure triple, or proof that exact cell-by-cell volume conservation survives hybridization and static condensation is supplied. This construction is load-bearing for the headline claim of machine-precision volume conservation and pure density advection.
  2. [Numerical results, Rayleigh-Taylor tests] In the Rayleigh-Taylor section, mesh independence is demonstrated at different Atwood and Reynolds numbers, but no quantitative error tables (e.g., interface displacement or kinetic energy norms versus reference solutions) or direct comparison to existing variable-density schemes are provided; this weakens the ability to attribute observed improvements specifically to the conservation property.
minor comments (2)
  1. [Abstract] The abstract lists attractive features including 'inherited p-multilevel solution strategies'; a short clarifying sentence or citation would aid readers new to HHO methods.
  2. [Introduction] Notation for the hybrid spaces (velocity, density, pressure) is introduced late; moving a compact definition to the introduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address the major comments point by point below, and we believe these revisions will strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Discrete stability analysis (post-abstract)] The abstract states that stability is addressed at the discrete level prior to numerical validation, yet no explicit discrete divergence identity, inf-sup argument for the hybrid velocity-density-pressure triple, or proof that exact cell-by-cell volume conservation survives hybridization and static condensation is supplied. This construction is load-bearing for the headline claim of machine-precision volume conservation and pure density advection.

    Authors: We thank the referee for highlighting this point. The stability analysis in Section 3 of the manuscript establishes the discrete properties through the design of the hybrid spaces, where the velocity reconstruction ensures a cell-wise divergence-free condition that is preserved under the hybridization and static condensation procedures. This leads to the exact volume conservation up to machine precision. However, we agree that an explicit step-by-step derivation of the divergence identity and a brief inf-sup discussion for the hybrid triple would make the argument clearer. We will add a dedicated paragraph or subsection in the revised version to provide this explicit construction without altering the core results. revision: yes

  2. Referee: [Numerical results, Rayleigh-Taylor tests] In the Rayleigh-Taylor section, mesh independence is demonstrated at different Atwood and Reynolds numbers, but no quantitative error tables (e.g., interface displacement or kinetic energy norms versus reference solutions) or direct comparison to existing variable-density schemes are provided; this weakens the ability to attribute observed improvements specifically to the conservation property.

    Authors: We appreciate this suggestion for enhancing the numerical section. The current Rayleigh-Taylor results focus on demonstrating mesh independence and the method's robustness across parameter ranges through high-resolution visualizations and qualitative behavior matching expected physics. To address the concern, we will include quantitative data in the revised manuscript, such as tables reporting the interface displacement at specific times and L2 norms of the velocity field for different mesh sizes, compared against available reference values from the literature where possible. Regarding direct comparisons to other schemes, while we can reference existing methods, performing side-by-side simulations would require substantial additional computational effort; we believe the conservation property's impact is evident from the exact preservation and the observed convergence, but we can expand the discussion to better highlight this. revision: partial

Circularity Check

0 steps flagged

No circularity: direct discretization of standard equations using established HHO and ESDIRK components

full rationale

The paper proposes an HHO spatial discretization combined with ESDIRK time stepping for the variable-density incompressible Navier-Stokes system. The core properties (exact cell-by-cell volume conservation and pure density advection) are presented as direct consequences of the hybrid velocity-density-pressure spaces and the chosen discrete divergence operator, without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. Stability is stated to be addressed separately prior to numerical tests, and the formulation draws on known HHO ingredients rather than deriving its key identities from the target results themselves. No equations or claims in the abstract or described structure collapse by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard incompressible Navier-Stokes equations with variable density and on the algebraic properties of hybrid polynomial spaces that allow exact integration-by-parts identities for volume conservation; no free parameters or new physical entities are introduced in the abstract.

axioms (2)
  • domain assumption Incompressible Navier-Stokes equations with variable density
    The paper starts from the standard variable-density incompressible flow model.
  • standard math Hybrid High-Order spaces admit exact cell-wise volume conservation
    The formulation relies on known algebraic properties of HHO spaces for conservation.

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Reference graph

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