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arxiv: 2512.05919 · v2 · submitted 2025-12-05 · 🧮 math.NA · cs.MS· cs.NA

A Discontinuous Galerkin Consistent Splitting Method for the Incompressible Navier-Stokes Equations

Dominik Still , Natalia Nebulishvili , Richard Schussnig , Katharina Kormann , Martin Kronbichler This is my paper

Pith reviewed 2026-05-17 00:30 UTC · model grok-4.3

classification 🧮 math.NA cs.MScs.NA
keywords discontinuous Galerkinconsistent splitting schemeincompressible Navier-Stokespressure Poisson equationLeray projectionfinite element methodsnumerical fluid dynamics
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The pith

Discontinuous Galerkin discretization of Liu's consistent splitting scheme for incompressible Navier-Stokes equations yields a decoupled solver with no splitting error.

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

The paper presents a discontinuous Galerkin finite element method applied to the consistent splitting scheme for the incompressible Navier-Stokes equations. This discretization implicitly enforces the divergence-free condition on the velocity, which removes velocity-pressure compatibility requirements and eliminates artificial pressure boundary layers. Appropriate numerical fluxes for the divergence operators and a Leray projection with penalty terms allow the consistent pressure boundary condition to simplify to contributions from only the acceleration and viscous terms. The resulting scheme decouples into one pressure Poisson equation and one vector convection-diffusion-reaction equation per time step, with semi-implicit treatment of convection. The method supports higher-order time integrators and shows optimal convergence rates on standard flow benchmarks.

Core claim

When the consistent splitting scheme is discretized with symmetric interior penalty Galerkin methods for second derivatives, chosen fluxes for the divergence of velocity and the convective operator, and Leray projection combined with divergence and normal continuity penalties, the pressure boundary condition reduces to terms arising solely from the acceleration and viscous contributions in the L2 discretization. This produces a fully decoupled time-stepping procedure consisting of a single pressure Poisson solve followed by a single vector-valued convection-diffusion-reaction equation, with no splitting error that would otherwise limit temporal accuracy.

What carries the argument

The consistent splitting scheme discretized via symmetric interior penalty discontinuous Galerkin fluxes together with Leray projection and penalty terms for divergence and convective operators.

If this is right

  • Optimal spatial and temporal convergence rates are attained for both velocity and pressure fields.
  • The scheme remains accurate with higher-order time integration methods without degradation from splitting.
  • Consistent boundary conditions apply directly to open and traction boundaries.
  • Mass conservation improves through the combination of Leray projection and penalty terms.
  • The method applies to benchmark flows such as cylinder flow and Taylor-Green vortex without special compatibility fixes.

Where Pith is reading between the lines

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

  • The single Poisson plus single CDR structure per step may simplify the design of efficient preconditioners or domain-decomposition solvers for large-scale problems.
  • The approach could be tested on moving-boundary or fluid-structure interaction problems where implicit divergence enforcement reduces remeshing costs.
  • Extension to variable-density or non-Newtonian flows would require checking whether the same flux choices preserve the reduction of the pressure boundary condition.
  • The semi-implicit convection treatment might combine with explicit subgrid-scale models for under-resolved turbulent simulations.

Load-bearing premise

Liu's consistent splitting scheme remains consistent and free of splitting error after discretization with the chosen DG fluxes, interior penalty terms, and Leray projection for the divergence and convective operators.

What would settle it

Numerical tests on the Taylor-Green vortex showing that the observed temporal convergence order falls below the order of the chosen time integrator or that spurious pressure boundary layers appear near walls would falsify the absence of splitting error.

Figures

Figures reproduced from arXiv: 2512.05919 by Dominik Still, Katharina Kormann, Martin Kronbichler, Natalia Nebulishvili, Richard Schussnig.

Figure 1
Figure 1. Figure 1: Temporal convergence of velocity (left) and pressure (right) in the Taylor–Green vortex bench [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spatial convergence of the consistent splitting scheme with different polynomial degrees. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The inflow boundary condition at x1 = 0 is given as gu (x1 = 0, x2, t) =  4Um x2(H−x2) H2 sin(πt/T) 0  . (48) The average velocity is given as Um = 1.5 and the viscosity as ν = 10−3 such that the resulting maximal Reynolds number is Remax = 100. The time interval is chosen as 0 ≤ t ≤ T = 8. At x1 = L, an open 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Flow around a cylinder benchmark: computational grid after 2 uniform refinements (top) and [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Kinetic energy, molecular dissipation and numerical dissipation of the consistent splitting [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between the solutions of the coupled solution approach, the dual splitting scheme [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Kinetic energy, molecular dissipation and numerical dissipation of the consistent splitting [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
read the original abstract

This work presents the discontinuous Galerkin discretization of the consistent splitting scheme proposed by Liu [J. Liu, J. Comp. Phys., 228(19), 2009]. The method enforces the divergence-free constraint implicitly, removing velocity--pressure compatibility conditions and eliminating pressure boundary layers. Consistent boundary conditions are imposed, also for settings with open and traction boundaries. Hence, accuracy in time is no longer limited by a splitting error. The symmetric interior penalty Galerkin method is used for second spatial derivatives. The convective term is treated in a semi-implicit manner, which relaxes the CFL restriction of explicit schemes while avoiding the need to solve nonlinear systems required by fully implicit formulations. For improved mass conservation, Leray projection is combined with divergence and normal continuity penalty terms. By selecting appropriate fluxes for both the divergence of the velocity field and the divergence of the convective operator, the consistent pressure boundary condition can be shown to reduce to contributions arising solely from the acceleration and the viscous term for the $L^2$ discretization. Per time step, the decoupled nature of the scheme with respect to the velocity and pressure fields leads to a single pressure Poisson equation followed by a single vector-valued convection-diffusion-reaction equation. We verify optimal convergence rates of the method in both space and time and demonstrate compatibility with higher-order time integration schemes. A series of numerical experiments, including the two-dimensional flow around a cylinder benchmark and the three-dimensional Taylor--Green vortex problem, verify the applicability to practically relevant flow problems.

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

1 major / 2 minor

Summary. The paper presents a discontinuous Galerkin discretization of Liu's 2009 consistent splitting scheme for the incompressible Navier-Stokes equations. It employs the symmetric interior penalty Galerkin method for viscous terms, a semi-implicit treatment of convection, and combines Leray projection with divergence and normal-continuity penalty terms to improve mass conservation. The central claim is that suitably chosen numerical fluxes for the divergence of velocity and of the convective operator allow the consistent pressure boundary condition to reduce exactly to acceleration and viscous contributions in the L2 discretization, even for open and traction boundaries; this yields a decoupled scheme consisting of one pressure Poisson solve followed by one vector convection-diffusion-reaction solve per time step, with no splitting error and optimal convergence rates in space and time.

Significance. If the algebraic cancellation of extraneous interface and boundary terms survives the DG weak formulation and stabilization, the method would provide a high-order, splitting-error-free DG scheme for incompressible flows that handles open/traction boundaries without pressure boundary layers or velocity-pressure compatibility constraints. The combination of semi-implicit convection and higher-order time integrators, together with the reported verification on the cylinder benchmark and Taylor-Green vortex, would represent a useful contribution to the DG literature for incompressible flows.

major comments (1)
  1. [§3] §3 (or the section deriving the weak form of the pressure Poisson equation): The headline claim that the consistent pressure boundary condition reduces to acceleration plus viscous terms only rests on the assertion that chosen DG fluxes for div(u) and div(convective operator), together with symmetric interior penalty terms and the Leray projection, produce exact cancellation of all pressure-dependent and projection-dependent residuals. The manuscript must supply the explicit integration-by-parts identities and interface term cancellations (including the precise penalty-parameter choices) that establish this reduction for the L2 discretization; without them the temporal accuracy claim cannot be verified.
minor comments (2)
  1. [Abstract] Abstract and §1: Replace the phrase 'can be shown' with an explicit forward reference to the theorem or lemma that proves the cancellation.
  2. [Numerical experiments] Numerical experiments section: The reported convergence tables should include the precise penalty-parameter values used for divergence and normal continuity; these parameters appear in the free-parameter list and affect the claimed cancellation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the single major comment below and will revise the paper to incorporate the requested details.

read point-by-point responses

  1. Referee: [§3] §3 (or the section deriving the weak form of the pressure Poisson equation): The headline claim that the consistent pressure boundary condition reduces to acceleration plus viscous terms only rests on the assertion that chosen DG fluxes for div(u) and div(convective operator), together with symmetric interior penalty terms and the Leray projection, produce exact cancellation of all pressure-dependent and projection-dependent residuals. The manuscript must supply the explicit integration-by-parts identities and interface term cancellations (including the precise penalty-parameter choices) that establish this reduction for the L2 discretization; without them the temporal accuracy claim cannot be verified.

    Authors: We agree that the explicit derivations strengthen the presentation. In the revised manuscript we will expand the weak-form derivation in Section 3 to include the full integration-by-parts identities for the pressure Poisson equation. These will detail: the precise DG numerical fluxes selected for both div(u) and the divergence of the convective operator; the resulting interface jump and average terms; the contributions of the symmetric interior penalty stabilization and the Leray projection; and the concrete penalty-parameter choices (standard SIPG scaling proportional to (k+1)^2/h) that produce exact cancellation of all pressure-dependent and projection-dependent residuals. The expanded identities will confirm that the consistent pressure boundary condition reduces exactly to acceleration and viscous terms in the L2 sense, thereby rigorously supporting the claimed absence of splitting error. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no reduction to inputs by construction

full rationale

The paper discretizes Liu's consistent splitting scheme using DG methods with specific fluxes, interior penalties, and Leray projection. The central claim—that appropriate flux choices make the consistent pressure BC reduce exactly to acceleration and viscous contributions—is presented as a result to be shown via the weak-form integration and stabilization terms, not as a definitional identity or fitted parameter. The scheme's decoupling into one Poisson solve and one convection-diffusion-reaction solve follows directly from the splitting structure without self-referential closure. Self-citation to Liu is load-bearing only for the continuous scheme; the DG-specific cancellation is an independent algebraic verification within the present discretization. No self-definitional loops, renamed empirical patterns, or uniqueness theorems imported from the authors' prior work appear in the provided derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the properties of Liu's 2009 splitting scheme transferring to the DG setting and on the existence of appropriate numerical fluxes that simplify the pressure boundary condition.

free parameters (1)
  • penalty parameters for divergence and normal continuity
    Introduced to improve mass conservation; their specific values are chosen as part of the method formulation.
axioms (1)
  • domain assumption Liu's consistent splitting scheme remains consistent and eliminates splitting error when discretized with the chosen DG interior penalty and flux terms.
    Invoked to justify that time accuracy is no longer limited by splitting error.

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