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An implicit $ P $-multigrid flux reconstruction method for simulation of locally preconditioned unsteady Navier-Stokes equations at low Mach numbers

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A P-multigrid solver for high-order flux reconstruction reaches its fastest convergence for low-Mach unsteady flows when coarse levels use polynomial degrees about half the fine degree, and it preserves high-order accuracy.

desk verdict Solid engineering paper worth a serious referee: packages known components into a working low-Mach P-multigrid solver and gives an empirically useful hierarchy rule, though the accuracy study has warts and the design rule lacks sensitivity analysis. read the letter →

arxiv 1908.03972 v1 pith:IERUCZFV submitted 2019-08-11 physics.comp-ph

Lai Wang , Meilin Yu This is my paper · ORCID
classification physics.comp-ph
keywords P-multigridfluxreconstructionFR/CPRlowMachnumberlocalpreconditioningESDIRKunsteadyNavier-Stokesunder-resolvedsimulation
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 develops an implicit solver for low-Mach unsteady compressible flows using high-order flux reconstruction with ESDIRK time stepping and local preconditioning confined to pseudo-time. It claims that P-multigrid acceleration works best when the polynomial degree roughly halves between adjacent levels, and that this rule holds for two- and three-level V-cycles across P3, P4, and P5 discretizations and across 2D and 3D test cases. The payoff if true is a simple, practical recipe for reducing the cost of high-order low-Mach simulations, with demonstrated application to under-resolved transitional flow over a wing.

What carries the argument

The central object is the P-multigrid V-cycle on a hierarchy of polynomial spaces, with an element Jacobi smoother performing a few relaxed updates per level. The smoother update is $q^{m+1}=q^m+\alpha_r D^{-1}F$, where $D$ is the block diagonal of the linearized pseudo-time operator and $\alpha_r=1$ in this study. The load-bearing idea is the degree-halving hierarchy: adjacent levels should differ by roughly half the finer polynomial degree, so coarse-grid corrections stay effective instead of introducing new errors. This rule is what carries the convergence acceleration.

What would settle it

Rerun the NACA0012 convergence comparisons with a different smoother configuration, for example $\alpha_r=0.5$, smoothing counts $\{10-20-40\}$, or CFL growth ratio 2.0, and check whether the fastest hierarchy shifts away from $\{P_0-P_0/2-P_0\}$ and $\{P_0-P_0/2-P_0/4-P_0/2-P_0\}$. If the ranking changes, the claimed design rule is not robust.

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Extended reading notes

Core claim

The paper establishes that an implicit P-multigrid V-cycle can be combined with locally preconditioned flux reconstruction and ESDIRK time integration without destroying high-order accuracy. Its central finding is empirical: the fastest convergence for both two-level and three-level cycles occurs when coarse polynomial degrees are near half the fine degree, specifically hierarchies close to $\{P_0-P_0/2-P_0\}$ and $\{P_0-P_0/2-P_0/4-P_0/2-P_0\}$. In the tested cases, three-level cycles beat two-level cycles, and excessive degree gaps such as $\{P_5-P_1\}$ either stall or contaminate convergence. Local preconditioning appears only in the pseudo-transient-continuation loop, so the unsteady physical time integration retains the design order of the ESDIRK method.

Load-bearing premise

The load-bearing premise is that the hierarchy ranking observed with the paper's fixed solver settings, including Jacobi relaxation $\alpha_r=1$, smoothing counts $\{5-10-20\}$, SER CFL growth ratio 1.5, smoother update intervals, and cutoff parameter $\kappa$, remains valid when those settings change; if the ranking is an artifact of the tuning, the recommended degree-halving rule collapses.

Editorial extensions

If this is right

  • For a chosen fine polynomial degree, setting the coarse level near $P_0/2$ and the next near $P_0/4$ for three levels should cut both CPU time and V-cycle count relative to single-level iteration.
  • Three-level cycles are preferable to two-level cycles in the tested low-Mach regime.
  • Hierarchies with large adjacent degree gaps should be avoided, since they can stall convergence or cause residual oscillation.
  • The combination of pseudo-time local preconditioning and P-multigrid preserves nominal high-order spatiotemporal accuracy at Ma = 0.005.
  • The solver supports under-resolved transitional low-Mach wing simulations, with lift, drag, and separation/reattachment predictions consistent with prior studies.

Reading between the lines

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

  • The degree-halving rule likely transfers to other compact high-order discretizations such as discontinuous Galerkin or spectral difference, because the mechanism is coarse-level correction effectiveness rather than the flux-reconstruction formulation; this is an inference, not tested in the paper.
  • A two-grid local-mode analysis of the element Jacobi smoother with preconditioned eigenvalues could turn the empirical rule into a predictive criterion and show how the optimal ratio shifts with cell aspect ratio or Reynolds number.
  • The paper's comparison of Ma = 0.1 and Ma = 0.01 wing flows leaves the cause of the difference in lift and separation bubble ambiguous; a controlled study with matched cutoff parameter and matched numerical dissipation could separate weak-compressibility effects from solver-settings effects.
  • Extending the solver to Newton-Krylov with P-multigrid as a preconditioner, which the paper names as future work, is the natural test of whether the same hierarchy rule accelerates very-low-Mach nonlinear solves.
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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 manuscript develops an implicit P-multigrid solver for high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) discretizations of the locally preconditioned unsteady compressible Navier–Stokes equations at low Mach numbers. The solver combines a FAS-type V-cycle with an element Jacobi smoother and ESDIRK temporal integration, with local preconditioning applied only in the pseudo-time continuation to preserve time accuracy. Accuracy is verified on an isentropic vortex at Ma=0.005, the influence of polynomial-degree hierarchies is studied on 2D inviscid and viscous NACA0012 flows and a 3D sphere flow at Ma=0.001, and the solver is applied to an under-resolved transitional SD7003 wing at Re=60000 and Ma=0.1/0.01. The authors recommend hierarchies close to {P0-P0/2-P0} for two-level and {P0-P0/2-P0/4-P0/2-P0} for three-level V-cycles as giving the best convergence acceleration.

Significance. If the empirical hierarchy rule is robust, the paper offers a practical acceleration strategy for high-order low-Mach FR/CPR solvers, addressing a known stiffness problem in a way that is directly useful for coarse-resolution turbulence simulation. The manuscript is clearly organized, provides a complete algorithmic description, and includes a challenging 3D application with quantitative comparisons to previous experimental and numerical results. The central claims are conditional, however: the order-of-accuracy verification is incomplete for density and pressure at the finest resolutions, and the hierarchy recommendation is obtained with a fixed set of solver parameters and without work-per-cycle normalization, so its generality is not yet established.

major comments (3)
  1. [§4.1, Tables 1 and 2] The order-of-accuracy verification does not fully support the claim of verified high-order spatiotemporal accuracy. In Table 1, ESDIRK4 shows order drops to 0.79 for density and 1.39 for pressure at the finest time step, while velocity retains order 4.00. In Table 2, the P3 and P4 FR schemes similarly show density and pressure order collapse at the finest grid (e.g., P3 density order 0.49, P4 density order 0.02), while velocity continues to converge. The authors attribute these reductions to "the accuracy limit of the solver," but do not identify this limit or explain why velocity is unaffected. Because the abstract and conclusions state that high-order spatiotemporal accuracy is verified, this inconsistency must be addressed, either by tightening the pseudo-time tolerance, analyzing the effect of the 1e-4 tolerance on each variable, or rephrasing the accuracy claim.
  2. [§4.2, Figures 4–6, 8–10, 12–13] The central hierarchy recommendation is based on a fixed set of solver parameters (I{5-10-20}, alpha_r=1, SER growth ratio 1.5, CFLinit=1e2, CFLmax=1e5, kappa=1) with no sensitivity study. It is therefore unclear whether the observed ranking of hierarchies, e.g., P{4-2} over P{4-1} or P{5-3-1} over P{5-4-1}, persists for other smoothing schedules, relaxation parameters, or CFL update strategies. A sensitivity study or a two-grid analysis is needed to show that the recommended rule "polynomial degree difference close to half of the finer degree" is a property of the multigrid correction rather than an artifact of the chosen parameters.
  3. [§4.2, CPU-time and V-cycle comparisons] The comparison across hierarchies is not normalized for the cost of smoothing sweeps at different polynomial degrees. All two-level cases use I{5-10} and all three-level cases use I{5-10-20}, so a V-cycle for P{5-4-1} performs the same number of sweeps as one for P{5-2-1}, but the sweeps at P4 are much more expensive than those at P2 or P1. The residual-versus-V-cycle plots partially mitigate this, but the CPU-time comparisons would be more interpretable if the paper reported the cost per smoothing sweep at each degree and the total work per V-cycle for each hierarchy. Without this information, the reader cannot separate the benefit of coarse-grid correction from the effect of concentrating smoothing effort on cheaper lower-degree spaces.
minor comments (6)
  1. [§2.3, Eq. (15)] There is a typo: "The ensure conservation" should read "To ensure conservation."
  2. [§4.2.1] The statement that P{3-2} and P{3-1} have "almost the same computational cost" is vague, since P{3-1} uses a cheaper coarse level; please clarify whether the similar cost arises because P{3-1} requires more V-cycles, which is not apparent from the figures.
  3. [§4.3] The choice of the global cutoff parameter kappa=2.5 for Ma=0.01, while kappa=1.0 for Ma=0.1, is stated without justification; a brief explanation of how this cutoff value was selected would improve reproducibility.
  4. [Figure 18] The axis labels contain typos: "Peudo iteration m" and "resisual" should be "Pseudo iteration m" and "residual."
  5. [§3] The restriction and prolongation operators (PP0_P1 and IP1_P0) are not defined; specifying whether they are L2 projections or interpolation operators would make the algorithm fully reproducible.
  6. [§4.2.1] The convergence histories in Figures 4–6 are shown down to residual 1e-9, but the stopping criterion or target tolerance for the steady-state cases is not stated; please include it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are supported by in-paper numerical experiments and external benchmarks, not by self-citation or by fitted inputs renamed as predictions.

full rationale

The paper's central claims are: (1) the P-multigrid solver preserves high-order spatiotemporal accuracy; (2) polynomial hierarchies near {P0-P0/2-P0} or {P0-P0/2-P0/4-P0/2-P0} give the best convergence acceleration; and (3) coarse SD7003 simulations agree with prior experimental and numerical data. Claim (1) is validated against the analytic isentropic vortex in Section 4.1, Tables 1 and 2, so it does not reduce to an input or fitted parameter. Claim (2) is an empirical comparison made in Sections 4.2.1-4.2.3 using convergence histories for many fixed hierarchies; the paper explicitly says 'our numerical experiments consistently suggest' the hierarchy rule, and the evidence is the in-paper residual and CPU-time histories. The conference paper [52] is cited only as prior preliminary work, not as the load-bearing justification for the hierarchy recommendation. Claim (3) is checked against experimental results (Selig et al.) and independent numerical studies (Vermeire et al., Beck et al., Galbraith and Visbal), so it is externally falsifiable. The self-citations ([8], [16], [26], [52]) are contextual or preliminary and do not carry the central derivation. The fixed smoothing schedules and the cutoff parameter kappa are hand-tuned, and no sensitivity or work-normalization study is provided, but that is a robustness and cost-accounting concern rather than evidence that any prediction reduces to its own inputs by construction. No self-definitional step, fitted quantity renamed as prediction, or author-supplied uniqueness theorem was found.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central claims rest on established numerical components plus several hand-chosen algorithmic constants. The freshest content is empirical: the polynomial-hierarchy rule and its demonstration. No free parameter is fitted to the validation targets (vortex, SD7003 force coefficients), reducing circularity, but the convergence ranking is sensitive to the stated solver settings.

free parameters (6)
  • global cutoff parameter kappa = 1.0 for most cases; 2.5 for SD7003 at Ma=0.01
    Appears in epsilon (Eq. 11); sets the floor for the preconditioning reference velocity. The authors state a larger kappa accelerates convergence for the Ma=0.01 case and acknowledge it may affect the physical dissipation in the reported results.
  • CFLinit and CFLmax = 1e2 and 1e5 for airfoil/sphere studies; for the vortex, Delta_tau_init=0.01 and Delta_tau_max=10; for SD7003…
    Pseudo-time step limits in the SER schedule (Eq. 31-32); chosen for stability and computational efficiency. They directly affect measured convergence speed but not the steady solution.
  • Smoothing iteration counts I{n0-n1-n2} = I{5-10-20}, I{5-10}, I{10}, I{20-20-40}, I{5-5-10}, I{10-10-16}
    Number of pre and post smoothing sweeps per level; chosen by hand for each experiment. These counts affect the CPU-time convergence comparisons and the ranking of hierarchies.
  • SER growth ratio = 1.5
    Fixed in Eq. (31); the authors note a larger value would be more aggressive. It shapes the pseudo-time step evolution and thus the convergence histories.
  • Smoother update interval = every 10 pseudo iterations; every 20 for Ma=0.01 SD7003
    The element Jacobi smoother is not updated every iteration; the interval is chosen by hand and affects both cost and convergence.
  • Relaxation parameter alpha_r = 1.0
    Used in the element Jacobi smoother update, Eq. (30); set to one in this study with no sensitivity analysis. Under-relaxation could stabilize the smoother and change convergence behavior.
assumptions (7)
  • domain assumption The Weiss-Smith local preconditioning matrix Gamma (Eq. 9) with reference velocity Ur = epsilon c balances characteristic speeds and preserves the steady state of the unpreconditioned equations.
    Taken from Ref. [7]; the eigenvalues in Eq. (13) are stated without derivation. If the preconditioner is not uniformly robust at very low Mach, the convergence claims weaken.
  • standard math The FR/CPR differential form (Eq. 20) with correction field delta_e is equivalent to the weak form, and tensor-product correction polynomials are valid on quadrilaterals and hexahedra.
    Invoked in Section 2.3; equivalence is cited to Ref. [53]. This is accepted background for FR/CPR.
  • domain assumption The approximate Riemann solver of Ref. [7] (Eq. 21) provides a stable common inviscid flux at low Mach numbers.
    Used to evaluate f_com_n_inv; no numerical analysis is provided for this specific flux at Ma < 0.01.
  • domain assumption The BR2 viscous flux treatment from Ref. [55] yields stable and consistent viscous common fluxes.
    Used in Section 2.3; standard in DG/FR but not proven here.
  • domain assumption The dual-time stepping fixed point of Eq. (27) equals the ESDIRK stage solution F(q)=0, so preconditioning only in pseudo-time preserves physical-time accuracy.
    Section 2.4; plausible because preconditioning multiplies only the pseudo-time derivative, but the authors rely on numerical verification (Table 1) rather than proof. Note the observed order reduction for ESDIRK4.
  • ad hoc to paper The element Jacobi smoother (Eq. 30) with block-diagonal D is a convergent smoother for the P-multigrid V-cycle on the tested meshes.
    No smoothing analysis is given; the paper relies on residual histories. For P{5-1} in the viscous NACA0012 run, the smoother stalls below residual 1e-4 (Section 4.2.2), showing it is not universally effective.
  • ad hoc to paper The FAS P-multigrid correction algorithm (Section 3) converges for low-Mach preconditioned systems when the hierarchy satisfies the recommended rule.
    The rule is inferred from empirical experiments without a two-grid or Fourier analysis; it may depend on the specific smoother and CFL settings.

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Pith. "Pith review of An implicit $ P $-multigrid flux reconstruction method for simulation of locally preconditioned unsteady Navier-Stokes equations at low Mach numbers." pith.science (2026). https://pith.science/paper/IERUCZFV

@misc{pith2026190803972,
  author       = {Pith},
  title        = {Pith review of: An implicit $ P $-multigrid flux reconstruction method for simulation of locally preconditioned unsteady Navier-Stokes equations at low Mach numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IERUCZFV}},
  note         = {Machine review of arXiv:1908.03972}
}
abstract

We develop a $ P $-multigrid solver to simulate locally preconditioned unsteady compressible Navier-Stokes equations at low Mach numbers with implicit high-order methods. Specifically, the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) method is employed for spatial discretization and the high-order time integration is conducted by means of the explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) method. Local preconditioning is used to alleviate the stiffness of the compressible Navier-Stokes equations at low Mach numbers and is only conducted in pseudo transient continuation to ensure the high-order accuracy of ESDIRK methods. We employ the element Jacobi smoother to update the solutions at different $ P $-levels in the $ P $-multigrid solver. High-order spatiotemporal accuracy of the new solver for low-Mach-number flow simulation is verified with the isentropic vortex propagation when the Mach (Ma) number of the free stream is 0.005. The impact of the hierarchy of polynomial degrees on the convergence speed of the $ P $-multigrid method is studied via several numerical experiments, including two dimensional (2D) inviscid and viscous flows over a NACA0012 airfoil at $\text{Ma} = 0.001$, and a three dimensional (3D) inviscid flow over a sphere at $\text{Ma} = 0.001$. The $ P $-multigrid solver is then applied to coarse resolution simulation of the transitional flows over an SD7003 wing at $ 8^\circ $ angle of attack when the Reynolds number is 60000 and the Mach number is 0.1 or 0.01.

Figures

Figures reproduced from arXiv: 1908.03972 by the authors.

Figure 1
Figure 1. Illustration of a typical three-level V-cycle of the [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Unstructured meshes around a NACA0012 airfoil. (a) A global [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. (a) Normalized pressure field and (b) Ma number field of the [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Convergence histories of different P-multigrid solvers for the P 3 FR discretization when solving the inviscid flow over a NACA0012 airfoil at Ma = 0.001. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Convergence histories of different P-multigrid solvers for the P 4 FR discretization when solving the inviscid flow over a NACA0012 airfoil at Ma = 0.001. Number of V-cycles Residual 0 200 400 600 800 1000 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 P{5} P{5-3} P{5-2} P{5-…
Figure 6
Figure 6. Figure 6: Convergence histories of different P-multigrid solvers for the P 5 FR discretization when solving the inviscid flow over a NACA0012 airfoil at Ma = 0.001. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: (a) Normalized pressure field and (b) Ma number field of the [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 11
Figure 11. Figure 11: The residual histories of the P 3 and P 4 FR discretization with different hierarchies of polynomial degrees are presented in [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 8
Figure 8. Figure 8: Convergence histories of different P-multigrid solvers for the P 3 FR discretization when solving viscous flow over a NACA0012 airfoil at Ma = 0.001 and Re = 5000. Number of V-cycles Residual 0 200 400 600 800 1000 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 P{4} P{4-3} P{…
Figure 9
Figure 9. Figure 9: Convergence histories of different P-multigrid solvers for the P 4 FR discretization when solving the viscous flow over a NACA0012 airfoil at Ma = 0.001 and Re = 5000. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Convergence histories of different P-multigrid solvers for the P 5 FR discretization when solving the viscous flow over a NACA0012 airfoil at Ma = 0.001 and Re = 5000. X Y Z X Y Z Pnorm: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X Y Z Ma: 0.0001 0.0003 0.0005 0.0007 0.0…
Figure 11
Figure 11. Figure 11: Inviscid flow over a sphere at Ma = 0.001. (a) Meshes in the near￾wall region, (b) contour of the normalized pressure pnorm and (c) contour of the Mach number. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Convergence histories of different P-multigrid solvers for the P 3 FR discretization when solving the inviscid flow over a sphere at Ma = 0.001. Number of V-cycles Residual 0 50 100 150 10-9 10-8 10-7 10-6 10-5 10-4 10-3 P{4} P{4-3} P{4-2} P{4-1} P{4-3-2} P{4-3-1} P{4…
Figure 13
Figure 13. Figure 13: Convergence histories of different P-multigrid solvers for the P 4 FR discretization when solving the inviscid flow over a sphere at Ma = 0.001. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: Meshes around an SD7003 wing for under-resolved transitional [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: Iso-surfaces of the Q-criterion colored by the instantaneous [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: Mean flow fields of the transitional flow over an SD7003 wing at [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: (a) Time-averaged surface pressure coefficient [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: Typical convergence histories of the relative residual for the [PITH_FULL_IMAGE:figures/full_fig_p026_18.png]

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Pith tools

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