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A numerical method for low Mach number compressible flows by simultaneous relaxation of dependent variables

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Pith's one-line read A simultaneous-relaxation finite difference method solves low Mach number compressible flows without low Mach number approximations, conserving momentum and total energy at roundoff level.

desk verdict A competent incremental advance in pressure-based low-Mach solvers, with an unverified iteration count and an entropy-conservation overclaim that need fixing before it is publishable. read the letter →

arxiv 2502.08116 v1 pith:ZTFZQZAE submitted 2025-02-12 physics.flu-dyn physics.comp-ph

Hideki Yanaoka , Yuji Sato This is my paper · ORCID
classification physics.flu-dynphysics.comp-ph MSC 65M0676M2076N15 PACS 47.11.-j47.11.Bc47.40.-x
keywords lowMachnumbercompressibleflowconservativefinitedifferencesimultaneousrelaxationtotalenergyconservationpressure-basedmethodinternalequationdensityvariation
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 proposes a numerical method for low Mach number compressible flows that avoids low Mach number approximations and still achieves discrete conservation of momentum, total energy, and entropy. The method relaxes velocity, pressure, density, and internal energy simultaneously inside each Newton step, using a fully conservative finite difference scheme on a staggered grid in space and time. Its central finding is that when kinetic and internal energies are stored at the same time level $n+1$, total energy is conserved at roundoff level (about $10^{-15}$) and momentum to about $10^{-17}$ in a three-dimensional periodic inviscid test. A sympathetic reader would care because this could give long-time low Mach number simulations without the drift and instability that often accompany approximate low Mach treatments.

What carries the argument

The central object is the simultaneous relaxation loop in Eqs. (3.23a)–(3.23e): inside each Newton iteration, a Helmholtz equation for the pressure correction replaces the usual Poisson equation (adding the density derivative $\partial\rho/\partial p|_e$), and then velocity, pressure, internal energy, and density are corrected together, with density updated through the equation of state $\rho = (\kappa Ma^2 p + 1)/e$. Around this loop sits a fully conservative finite difference scheme with square-root density weighted interpolation of velocity and internal energy, a spatiotemporal staggered grid, and the implicit midpoint rule. The identity that carries the argument is the discrete placement of kinetic and internal energy at the same temporal level $n+1$, which makes the total energy constant at roundoff level; placing internal energy at $n+3/2$ instead degrades the total energy error to about $10^{-7}$.

What would settle it

Rerun the three-dimensional periodic inviscid test of Section 4.2 with the same grids and time steps but with the simultaneous relaxation limited to 1, 2, 5, 10, and 100 iterations, and record $|\varepsilon_{\rho E}|$ at $t/(L/U_0)=10$. If roundoff-level conservation appears only beyond some iteration count, or fails to appear at any finite count, then the conservation claim depends on the loop being run to convergence rather than being an intrinsic property of the discretization. A second check is the entropy error, which the paper reports at $10^{-6}$ rather than roundoff; grid-refinement and iteration-count studies of $|\varepsilon_{\rho s}|$ would show whether entropy conservation improves or saturates.

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Extended reading notes

Core claim

On its own terms, the paper claims that a pressure-based, fully conservative finite difference scheme can solve the compressible Navier–Stokes equations at low Mach numbers without invoking low Mach number, Boussinesq, or incompressibility approximations, and that this scheme discretely conserves the total amounts of momentum, total energy, and entropy. The key numerical finding is that placing kinetic and internal energies at the same temporal level $n+1$ makes the discrete total energy constant at roundoff level ($|\varepsilon_{\rho E}| \approx 10^{-15}$), whereas the earlier approach of Wall et al. with internal energy at $n+3/2$ gives $|\varepsilon_{\rho E}| \sim 10^{-7}$ on the same test. The scheme also reproduces sound wave amplitude and frequency, turbulence statistics, Taylor–Green vortex decay, natural convection in a cavity, and three-dimensional Taylor vortex decay, supporting its accuracy across inviscid, viscous, compressible, and incompressible regimes.

Load-bearing premise

The load-bearing premise is that the simultaneous relaxation loop is iterated enough times that the linearizations used in the Newton loop (replacing $\rho^{n+1,m+1}$ by $\rho^{n+1,m}$ and $\hat{u}^{n+1,m+1}$ by $\hat{u}^{n+1,m}$) become negligible, but the paper does not report iteration counts, tolerances, or a study of how conservation depends on them.

Editorial extensions

If this is right

  • Low Mach number flows with large temperature differences, where density varies and incompressibility fails, can be computed with the same solver as compressible and incompressible regimes, without switching models.
  • Long-time simulations of periodic inviscid flows will not accumulate momentum or total-energy drift, since those quantities are conserved at roundoff level on both uniform and nonuniform grids.
  • The scheme remains stable at Courant numbers above 1 (up to CFL 8 for sound waves) because the pressure is treated implicitly and the pressure correction takes Helmholtz form.
  • The method extends to viscous compressible turbulence and natural convection, matching benchmark dissipation rates and Nusselt numbers while using fewer grid points than some earlier work.

Reading between the lines

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

  • Because the scheme avoids low Mach number approximation, the same discretization could in principle handle flows that transition between compressible and incompressible behavior, such as combustion or high-temperature heat sources, without model switching; the paper does not test this directly.
  • The reported entropy conservation at $10^{-6}$ is orders of magnitude weaker than momentum and total energy conservation, suggesting that entropy is only approximately conserved; entropy-based diagnostics should be used with caution in long integrations.
  • The missing iteration-count reporting makes a reproducibility check natural: fixing a small, fixed number of relaxation iterations would determine whether roundoff-level energy conservation is robust to under-relaxation, which matters for production codes.
  • The same simultaneous relaxation idea could be applied to other tightly coupled systems, such as magnetohydrodynamic or reacting flows, where pressure, density, and energy updates interact strongly; the paper does not address these.
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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

1 major / 5 minor

Summary. The paper proposes a pressure-based finite-difference method for low Mach number compressible flows, built on the fully conservative scheme of Morinishi and the semi-implicit acoustic-treatment ideas of Wall, Pierce, and Moin. The key algorithmic novelty is a simultaneous relaxation loop in which velocity, pressure, internal energy, and density are corrected together, with density obtained from the equation of state. The method is tested on sound-wave propagation, three-dimensional periodic inviscid compressible flow, decaying compressible isotropic turbulence, Taylor-Green vortex flows, natural convection in a cavity, and a three-dimensional Taylor decaying vortex. The central claims are second-order accuracy and discrete conservation of momentum and total energy at roundoff level when kinetic and internal energies are stored at the same time level n+1, with a weaker entropy conservation result.

Significance. If the central conservation claims hold, the method would be a practically useful pressure-based low Mach number solver with excellent long-time conservation properties, validated against several external references including the exact three-dimensional Navier-Stokes solution of Antuono, the DNS of Honein and Moin, the wall-bounded natural-convection benchmark of Vierendeels et al., and the Taylor-Green reference data. The broad benchmark coverage is a genuine strength, as is the use of externally defined reference solutions rather than self-consistency checks. The technical novelty is incremental, however: the roundoff-level total-energy conservation at time level n+1 is essentially a property of the underlying Morinishi fully conservative scheme, and the paper's own Fig. 10 confirms that Morinishi's method already achieves this behavior. The contribution of the simultaneous relaxation loop is stability and practical iteration behavior, but that contribution is not quantified anywhere in the manuscript.

major comments (1)
  1. [§3.2, Eqs. (3.15), (3.23), and Steps 2–7; Figs. 9–10] The text in Section 3.2 says that the Poisson equation for pressure should be solved after correcting the velocity, and that the present method 'does not require the Poisson equation for the pressure to be solved.' This wording is confusing because the Helmholtz equation for Δp is derived from the mass conservation constraint and serves as the pressure-correction equation. Please clarify the distinction between a pure Poisson solve and the Helmholtz form used here.
minor comments (5)
  1. [§3.4, Eq. (3.30)] The third definition in Eq. (3.30) repeats ερu instead of defining ερw; this is a typo that should be corrected.
  2. [§3.2] The sentence 'The present numerical method does not require the Poisson equation for the pressure to be solved' is misleading because Eqs. (3.19) and (3.23a) solve a Helmholtz equation for the pressure correction. Please rephrase to say that the pressure-correction Helmholtz equation is solved directly.
  3. [§3.2] There is a typo in 'hihgly SMAC method'; it should read 'highly SMAC method.'
  4. [§4.2] The conclusion that the present method and the Wall et al. variant show 'no difference in characteristics' in Fig. 8 is contradicted by the large difference in total-energy conservation error in Fig. 9; the sentence should be qualified.
  5. [§2.1] The Eckert number Ec is used in Eq. (2.11) but defined only later in the text; the definition should be moved before first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central conservation and accuracy claims are scheme identities plus external-benchmark validation; self-citations are contextual only.

full rationale

The paper's central claims are not circular. The discrete conservation of momentum and total energy (Section 4.2, Figs. 9-10) is presented as a property of the constructed fully conservative finite difference scheme, in which kinetic and internal energies are stored at the same time level; this follows from the discretized equations (3.15a-d, 3.23a-e) rather than from any fitted parameter or from the paper's own prior results. The accuracy and validity claims are checked against independent external benchmarks: the acoustic dispersion relation (Eq. 4.2, from Wall et al. 2002), the exact tri-periodic Navier-Stokes solution of Antuono (2020), DNS data of Honein and Moin (2004), the cavity benchmark of Vierendeels et al. (2003), and reference Taylor-Green results from Wang et al. (2013), Orszag (1974), Shirokov and Elizarova (2014), and Kulikov and Son (2018). No parameter is fitted to force agreement. The only self-citations (Yanaoka and Inafune, 2023; Yanaoka, 2023) are used contextually to state that a previously used simultaneous velocity-pressure relaxation idea is here extended to compressible flows; they are not load-bearing for the conservation identities or the benchmark comparisons. The unstated iteration count and tolerance for the simultaneous relaxation loop raise reproducibility and verification concerns, but an unreported convergence parameter is not a circular definition and does not reduce the reported conservation results to their inputs by construction. Therefore no circular step is present.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The paper's contribution is algorithmic: a new way to couple the correction steps, not a new conservation law or new material law. The method relies on ideal gas thermodynamics, Newtonian viscosity, and the conservation properties of Morinishi's scheme.

free parameters (2)
  • epsilon (stabilization parameter) = 0 (basically)
    Introduced in Eq. (3.14) for double-time interpolation of pressure to prevent high-wavenumber acoustic oscillations; set to 0 in this study, so not fitted but chosen by hand.
  • simultaneous relaxation iteration count l = unspecified ('predetermined')
    Inner iterations in Eqs. (3.23a-e); the number is never reported, and the method's conservation and accuracy properties may depend on it.
assumptions (5)
  • domain assumption Ideal gas equation of state: κMa^2 p + 1 = ρe
    Eq. (2.8); all low Mach number derivations and the density update (Eq. 3.13) rely on this.
  • domain assumption Fluid is Newtonian with constant specific heat ratio κ and Sutherland's law for viscosity and conductivity in the cavity test
    Section 2.1 and Section 4.5; needed for the viscous stress tensor and heat flux.
  • standard math Morinishi's fully conservative finite difference scheme preserves kinetic and internal energy when both are at the same time level
    The paper inherits this property from Morinishi (2009, 2010); the central conservation claim in Section 4.2 depends on it.
  • domain assumption The Helmholtz equation (3.19) with the ∂ρ/∂p|_e term removes the acoustic CFL restriction without artificial dissipation
    Adopted from Wall et al. (2002); the large-CFL tests in Sections 4.1 and 4.6 rely on this.
  • standard math Implicit midpoint rule and SMAC-type splitting are second-order accurate
    Used throughout Section 3; convergence plots with slope -2 in Figures 5, 10, 24, and 28 support this.

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Cite this review

Pith. "Pith review of A numerical method for low Mach number compressible flows by simultaneous relaxation of dependent variables." pith.science (2026). https://pith.science/paper/ZTFZQZAE

@misc{pith2026250208116,
  author       = {Pith},
  title        = {Pith review of: A numerical method for low Mach number compressible flows by simultaneous relaxation of dependent variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZTFZQZAE}},
  note         = {Machine review of arXiv:2502.08116}
}
read the original abstract

Density varies spatiotemporally in low Mach number flows. Hence, incompressibility cannot be assumed, and the density must be accurately solved. Various methods have been proposed to analyze low Mach number flows, but their energy conservation properties have not been investigated in detail. This study proposes a new method for simultaneously relaxing velocity, pressure, density, and internal energy using a conservative finite difference scheme with excellent energy conservation properties to analyze low Mach number flows. In the analysis for sound wave propagation in an inviscid compressible flow, the amplitude amplification ratio and frequency of sound wave obtained by this numerical method agree well with the theoretical values. In the analysis for a three-dimensional periodic inviscid compressible flow, each total amount for the momentum, total energy, and entropy are discretely conserved. When no approximation, such as low Mach number approximations, is applied to the fundamental equations, the excellent conservation properties of momentum, total energy, and entropy are achieved. In decaying compressible isotropic turbulence, this computational method can capture turbulence fluctuations. Analyzing the Taylor-Green decaying vortex, we confirmed the validity of this computational scheme for compressible viscous flows. In the calculation for the natural convection in a cavity, the validity of this numerical method was presented even in incompressible flows considering density variation. In a three-dimensional Taylor decaying vortex problem, it was shown that this numerical method can accurately calculate incompressible flows. We clarified the accuracy and validity of the present numerical method by analyzing various flow models and demonstrated the possibility of applying this method to complex flow fields.

Figures

Figures reproduced from arXiv: 2502.08116 by the authors.

Figure 1
Figure 1. Velocity distribution: N = 21 The initial values for the density and temperature are uniform and defined as ρ0 and T0, respectively. The initial value of sound speed is given as c0 = p κ(κ − 1)cvT0. As in the existing research (Wall et al., 2002), the following velocity distribution is imposed as an initial disturbance, and the sound wave is propagated: u(x) = ∆umax cos  2πx λ  , (4.1) where λ and ∆umax are the wa… view at source ↗
Figure 2
Figure 2. Comparison of acoustic wave amplitude: CFL = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Comparison of acoustic wave frequency calculated by Wall et al. (Wall et al., 2002) is shown below: ωn ω = 1 (CFL)k∆x tan−1  2(CFL)(k ′∆x) 2 − 1 2 [(CFL)(k ′∆x)]2  , (4.2) where k ′ represents the modified wavenumber. Using the second-order central difference method, the relationship between k∆x and k ′∆x is given as k ′∆x = p 2 − 2 cos(k∆x). (4.3) Regardless of the CFL, the present calculation result agrees well … view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: Influences of space and time discretizations on acoustic wave frequency [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Errors of amplitude and frequency of acoustic wave for CFL [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Relative errors of momentum and total energy, and absolute error of entropy: [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Relative errors of momentum and total energy, and absolute error of entropy for CFL [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Total amounts of momentum, total energy, and entropy: [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Relative errors of momentum and total energy, and absolute error of entropy: [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Relative error of total energy: t/(L/U0) = 10; No approximation 0 2 4 6 8 10 10-20 10-16 10-12 10-8 10-4 100 |< ρ u>|, |< ρ v>|, |< ρ w>| t/(L/U0 ) |<ρu>| |<ρv>| |<ρw>| (a) Momentum 0 2 4 6 8 10 10-18 10-14 10-10 10-6 10-2 102 < ρ E>, |< ρ s>| t/(L/U0 ) <ρE> |<ρs>| (b…
Figure 11
Figure 11. Figure 11: Total amounts of momentum, total energy, and entropy for nonuniform grid: [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Relative errors of momentum and total energy, and absolute error of entropy for nonuniform grid: [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Total amounts of momentum, total energy, and entropy: [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Relative errors of momentum and total energy, and absolute error of entropy: [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: Time variations of turbulence kinetic energy and fluctuation intensities of pressure, specific volume, and [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Total amounts of momentum, total energy, and entropy for [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: Relative errors of momentum and total energy, and absolute error of entropy for [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 18
Figure 18. Figure 18: Time variations of turbulence kinetic energy and fluctuation intensities of pressure, specific volume, and [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]
Figure 19
Figure 19. Figure 19: Total amounts of momentum, kinetic energy, and total energy for [PITH_FULL_IMAGE:figures/full_fig_p022_19.png]
Figure 20
Figure 20. Figure 20: Errors of momentum and total energy for Re = ∞: N = 41 0 2 4 6 8 10 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 ε k/( ρ 0 V 3 0 /L ) t/(L/V0 ) Present Orszag (1974) Shirokov and Elizarova (2014) Kulikov and Son (2018) (a) Re = 100 0 2 4 6 8 10 12 0.000 0.002…
Figure 21
Figure 21. Figure 21: Comparisons of dissipation rate of kinetic energy for [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]
Figure 22
Figure 22. Figure 22: Isosurfaces of z-direction vorticity for Re = 1600: N = 256 0 5 10 15 20 0.000 0.004 0.008 0.012 0.016 0.020 ε k/( ρ 0 V 3 0 /L ) t/(L/V0 ) Present (N=256) Wang et al. (2013) Shirokov and Elizarov (2014) Kulikov and Son (2018) (a) Comparison with existing value 0 5 10…
Figure 23
Figure 23. Figure 23: Dissipation rate of kinetic energy for Re = 1600: N = 256 10 100 10-9 10-7 10-5 10-3 10-1 101 |ε εk|, | ε ρ E | grid number N |ε εk | |ε ρE | slope-2 [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 24
Figure 24. Figure 24: Relative errors of dissipation rate of kinetic energy and total amount of total energy [PITH_FULL_IMAGE:figures/full_fig_p024_24.png]
Figure 25
Figure 25. Figure 25: Velocity, temperature, density, and pressure fields for [PITH_FULL_IMAGE:figures/full_fig_p025_25.png]
Figure 26
Figure 26. Figure 26: Average pressure for ∆T = 720◦C 101 102 103 104 105 106 107 0 4 8 12 16 20 24 Numax Ra No approximation Boussinesq approximation Low Mach number approximation Vierendeels et al. (a) Maximum Nusselt number 101 102 103 104 105 106 107 0 2 4 6 8 10 Nu Ra No approximation…
Figure 27
Figure 27. Figure 27: Nusselt number distributions without approximation and with Boussinesq approximation for [PITH_FULL_IMAGE:figures/full_fig_p026_27.png]
Figure 28
Figure 28. Figure 28: Relative errors of maximamu and average Nusselt numbers for [PITH_FULL_IMAGE:figures/full_fig_p027_28.png]
Figure 29
Figure 29. Figure 29: Total amount of kinetic energy Similarly, for the Taylor–Green decaying vortex problem, the momentum and total energy are discretely conserved in an inviscid flow. The time variation in the dissipation rate of kinetic energy agrees well with the previous results. We c…

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Works this paper leans on

40 extracted references · 24 canonical work pages

  1. [8]

    A collocated finite volume method for predicting flows at all speeds. Int. J. Numer. Methods Fluids 16, 1029–1050. doi:doi:https://doi.org/10.1002/fld.1650161202. Gray, D.D., Giorgini, A.,

    work page doi:10.1002/fld.1650161202

  2. [20]

    Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 143, 90–124. doi:doi:https://doi.org/10.1006/jcph.1998.5962. Morinishi, Y .,

    arXiv 1998

  3. [21]

    JSME Ser

    Fully conservative finite difference scheme for low–Mach number unsteady compressible flow simulations. JSME Ser. B 75, 2153–2162. doi:doi:https://doi.org/10.1299/kikaib.75.759_2153. (in Japanese). Morinishi, Y .,

    work page doi:10.1299/kikaib.75.759_2153

  4. [27]

    Pressure-based Navier–Stokes solver using the multigrid method. AIAA J. 27, 1017–1018. doi:doi:https://doi.org/10.2514/3.10213. Ross Ethier, C., Steinman, D.A.,

    work page doi:10.2514/3.10213

  5. [34]

    Preconditioned methods for solving the incompressible and low speed compressible equations. J. Comput. Phys. 72, 277–298. doi:doi:http://dx.doi.org/10.1016/0021-9991(87)90084-2. Van der V orst, H.A.,

    doi:10.1016/0021-9991(87)90084-2

  6. [35]

    Bi-CGSTAB: A fast and smoothly converging variant of Bi–CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644. doi:doi:https://doi.org/10.1137/0913035. Vierendeels, J., Merci, B., Dick, E.,

    doi:10.1137/0913035

  7. [37]

    A semi-implicit method for resolution of acoustic waves in low Mach number flows. J. Comput. Phys. 181, 545–563. doi:doi:https://doi.org/10.1006/jcph.2002.7141. Wang, Z., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H., Kroll, N., May, G., Persson, P.O., van Leer, B., , Visbal, M.,

    arXiv 2002

  8. [40]

    Frequency response of three-dimensional natural convection of nanoflu- ids under microgravity environments with gravity modulation. Numer. Heat Tr. A-Appl. 83, 745–769. doi:doi:https://doi.org/10.1080/10407782.2022.2161437. 31

    work page arXiv 2022

Show all 40 references
  1. [1937]

    Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A 158, 499–521. doi:doi:https://doi.org/10.1098/rspa.1937.0036. Turkel, E.,

    1937

  2. [1960]

    Astrophys

    On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442–447. doi:doi:https://doi.org/10.1086/146849. Takemitsu, N.,

    doi

  3. [1967]

    A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26. doi:doi:https://doi.org/10.1016/0021-9991(67)90037-X. Demirdˇzi´c, I., ˇZ. Lilek, Peri´c, M.,

    doi

  4. [1970]

    A simplified MAC technique for incompressible fluid flow calculations. J. Comput. Phys. 6, 322–325. doi:doi:https://doi.org/10.1016/0021-9991(70)90029-X. Antuono, M.,

    doi

  5. [1974]

    (Eds.), Computing Methods in Applied Sciences and Engineering Part 2, Springer Berlin Heidelberg, Berlin, Heidelberg

    Numerical simulation of the Taylor–Green vortex, in: Glowinski, R., Lions, J.L. (Eds.), Computing Methods in Applied Sciences and Engineering Part 2, Springer Berlin Heidelberg, Berlin, Heidelberg. pp. 50–64. doi:doi:https://doi.org/10.1007/3-540-06769-8_4. 30 A numerical meth...

    doi

  6. [1975]

    Technical Report LA-5852

    SOLA: A numerical solution algorithm for transient fluid flows. Technical Report LA-5852. Los Alamos Scientific Lab., N. Mex.(USA). doi:doi:https://doi.org/10.2172/4205348. Honein, A.E., Moin, P.,

    doi

  7. [1976]

    The validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Transf. 19, 545–551. doi:doi:https://doi.org/10.1016/0017-9310(76)90168-X. Ham, F.E., Lien, F.S., Strong, A.B.,

    doi

  8. [1978]

    The equations of motion for thermally driven, buoyant flows. J. Res. Natl. Bur. Stand. 83, 297–308. doi:doi:http://dx.doi.org/10.6028/jres.083.019. Rhie, C.M.,

    doi

  9. [1983]

    On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49, 151–164. doi:doi:https://doi.org/10.1016/0021-9991(83)90118-3. Hennink, A., Tiberga, M., Lathouwers, D.,

    doi

  10. [1985]

    Finite difference method to solve incompressible fluid flow. J. Comput. Phys. 61, 499–518. doi:doi:https://doi.org/10.1016/0021-9991(85)90077-4. Taylor, G.I., Green, A.E.,

    doi

  11. [1987]

    A barely implicit correction for flux-corrected transport. J. Comput. Phys. 71, 1–20. doi:doi:https://doi.org/10.1016/0021-9991(87)90016-7. Qu´er´e, P.L., Masson, R., Perrot, P.,

    doi

  12. [1989]

    Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J. 27, 1167–1174. doi:doi:https://doi.org/10.2514/3.10242. Kulikov, Y .M., Son, E.E.,

    doi

  13. [1991]

    Primitive variable, strongly implicit calculation procedure for viscous flows at all speeds. AIAA J. 29, 1241–1249. doi:doi:http://dx.doi.org/10.2514/3.10728. Choi, Y .H., Merkle, C.I.,

    doi

  14. [1992]

    A Chebyshev collocation algorithm for 2D non-Boussinesq convection. J. Comput. Phys. 103, 320–335. doi:doi:https://doi.org/10.1016/0021-9991(92)90404-M. Rehm, R.G., Baum, H.R.,

    doi

  15. [1993]

    The application of preconditioning in viscous flows. J. Comput. Phys. 105, 207–223. doi:doi:https://doi.org/10.1006/jcph.1993.1069. Chorin, A.J.,

    1993

  16. [1994]

    Exact fully 3D Navier-–Stokes solutions for benchmarking. Int. J. Numer. Methods Fluids 19, 369–375. doi:doi:https://doi.org/10.1002/fld.1650190502. Samtaney, R., Pullin, D.I., Kosovi´c, B.,

    doi

  17. [1996]

    JSME Ser

    Conservative properties of finite difference schemes for incompressible flow (1st report, an- alytical requirements, discrete operators and schemes in a regular grid system. JSME Ser. B 62, 4090–4097. doi:doi:https://doi.org/10.1299/kikaib.62.4090. (in Japanese). Morinishi, Y .,

    doi

  18. [1998]

    A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comput. Phys. 141, 153–173. doi:doi:https://doi.org/10.1006/jcph.1998.5914. Boscheri, W., Tavelli, M.,

    1998

  19. [2001]

    Direct numerical simulation of decaying compressible turbulence and shocklet statistics. Phys. Fluids 13, 1415–1430. doi:doi:https://doi.org/10.1063/1.1355682. Shirokov, I.A., Elizarova, T.G.,

    doi

  20. [2002]

    A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids. J. Comput. Phys. 177, 117–133. doi:doi:https://doi.org/10.1006/jcph.2002.7006. Harten, A.,

    2002

  21. [2003]

    Benchmark solutions for the natural convective heat transfer problem in a square cavity with large horizontal temperature differences. Int. J.Numer. Method Heat Fluid Flow 13, 1057–1078. doi:doi:https://doi.org/10.1108/09615530310501957. Wall, C., Pierce, C.D., Moin, P.,

    doi

  22. [2004]

    Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comput. Phys. 201, 531–545. doi:doi:https://doi.org/10.1016/j.jcp.2004.06.006. Hou, Y ., Mahesh, K.,

    2004 doi

  23. [2005]

    A robust, colocated, implicit algorithm for directnumerical simulation of compressible, turbulent flows. J. Comput. Phys. 205, 205–221. doi:doi:https://doi.org/10.1016/j.jcp.2004.10.039. Karki, K.C., Patankar, S.V .,

    2004 doi

  24. [2009]

    A method for avoiding the acoustic time step restriction in compressible flow. J. Comput. Phys. 228, 4146–4161. doi:doi:https://doi.org/10.1016/j.jcp.2009.02.027. Morinishi, Y .,

    2009 doi

  25. [2010]

    Skew-symmetric form of convective terms and fully conservative finite dif- ference schemes for variable density low-mach number flows. J. Comput. Phys. 229, 276–300. doi:doi:https://doi.org/10.1016/j.jcp.2009.09.021. Orszag, S.,

    2009 doi

  26. [2013]

    High-order CFD methods: current status and perspective. Int. J. Numer. Methods Fluids 72, 811–845. doi:doi:https://doi.org/10.1002/fld.3767. Yanaoka, H.,

    doi

  27. [2014]

    Simulation of laminar–turbulent transition in compress- ible Taylor—Green flow basing on quasi-gas dynamic equations. J. Turbul. 15, 707–730. doi:doi:https://doi.org/10.1080/14685248.2014.927581. Spiegel, E.A., Veronis, G.,

    2014

  28. [2018]

    Taylor–Green vortex simulation using CABARET scheme in a weakly compressible formulation. Eur. Phys. J. E 41, 12 pages. doi:doi:https://doi.org/10.1140/epje/i2018-11645-4. Kwatra, N., Su, J., Gr ´etarsson, J.T., Fedkiw, R.,

    doi

  29. [2020]

    Tri-periodic fully three-dimensional analytic solutions for the Navier-–Stokes equations. J. Fluid Mech. 890, A23. doi:doi:https://doi.org/10.1017/jfm.2020.126. Bijl, H., Wesseling, P.,

    2020 doi

  30. [2021]

    A pressure-based solver for low-Mach number flow using a discontinuous galerkin method. J. Comput. Phys. 425, 109877. doi:doi:https://doi.org/10.1016/j.jcp.2020.109877. Hirt, C.W., Nichols, B.D., Romero, N.C.,

    2020

  31. [2022]

    High order semi-implicit schemes for viscous compressible flows in 3D. Appl. Math. Comput. 434, 127457. doi:doi:https://doi.org/10.1016/j.amc.2022.127457. Chen, K.H., Pletcher, R.H.,

    2022

  32. [2023]

    Influences of conservative and non-conservative Lorentz forces on energy conserva- tion properties for incompressible magnetohydrodynamic flows. J. Comput. Phys. 491, 112372 (36 pages). doi:doi:https://doi.org/10.1016/j.jcp.2023.112372. Yanaoka, H., Inafune, R.,

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.