Pith. sign in

REVIEW 4 major objections 8 minor 40 references

Gas mixing through a Smoothed Particle Hydrodynamics approach

T0 review · 4 major / 8 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single exponential relaxation step lets SPH gas-mixing simulations advance on the hydrodynamic timestep instead of the collision timescale.

desk verdict A useful and honest SPH method paper for binary gas mixing, with a real efficiency advance from operator splitting, but the transient accuracy of the splitting in the stiff regime is not yet demonstrated. read the letter →

arxiv 2509.06590 v1 pith:WP3TEOXJ submitted 2025-09-08 astro-ph.EP astro-ph.IM

Luca Maggioni , Matteo Teodori , Gianfranco Magni , Michelangelo Formisano , Maria Cristina De Sanctis , Francesca Altieri This is my paper · ORCID
classification astro-ph.EPastro-ph.IM
keywords smoothedparticlehydrodynamicsbinarygasmixturescollisionalrelaxationLie-TrottersplittingkinetictheoryplanetaryatmospheresChapman-Enskogdiffusionmultiphaseflow
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 an SPH scheme for simulating the mixing of two monatomic gases, each evolved as its own inviscid Euler fluid, with interspecies momentum and energy exchange supplied by a kinetic relaxation model built on the Boltzmann equation. Its central move is to separate the fast collisional exchange from the slower hydrodynamics through a first-order Lie–Trotter splitting, so the collisional step is applied as an exact exponential relaxation rather than a stiff explicit update. This lets the simulation advance on the hydrodynamic timestep instead of the much shorter molecular-collision timestep. The paper validates the scheme against a fully coupled SPH implementation and against analytic equilibrium states, reporting agreement within about 6 percent in the splitting comparison, roughly 8–9 percent in final mean density, and about 2 percent in equilibrium temperature for argon–krypton and neon–xenon mixtures, with total energy conserved to order $10^{-5}$ and total entropy increasing. A sympathetic reader would take the contribution to be a computationally efficient, thermodynamically consistent baseline for binary gas mixing in confined planetary settings.

What carries the argument

The load-bearing object is the exponential relaxation update for the interspecies velocity difference and energy. The paper reduces the coupled collisional system to $d\Delta\mathbf{v}/dt=-(\rho_{\alpha}+\rho_{\beta})K_{\mathrm{mix}}\Delta\mathbf{v}$, freezes $(\rho_{\alpha}+\rho_{\beta})K_{\mathrm{mix}}$ over the hydrodynamic timestep, and applies the exact solution $\Delta\mathbf{v}^{n+1}=\Delta\mathbf{v}^{*}e^{-(\rho_{\alpha}+\rho_{\beta})K_{\mathrm{mix}}\Delta t}$, then reconstructs each species' velocity from the conserved center-of-mass velocity. Energies relax analogously, $\epsilon_{\mathrm{tk}}^{n+1}=\hat{\epsilon}+(\epsilon_{\mathrm{tk}}^{*}-\hat{\epsilon})e^{-f\Delta t}$, toward collision-target energies $\hat{\epsilon}$. Here $K_{\mathrm{mix}}$ is a mixture coefficient built from Chapman–Cowling collision integrals for a Lennard-Jones potential; in the strong-collision limit the exponential factors drive the gases to a common velocity and temperature, and in the no-collision limit the hydrodynamic values are untouched. Together with the first-order Lie–Trotter split, this object is what lets the scheme advance on the hydrodynamic timestep while still reaching the correct equilibrium.

What would settle it

Run the argon–krypton test at increasing resolution with timesteps chosen so that the ratio of the collisional timescale to the hydrodynamic timestep varies from 0.01 to 1; if the final density and temperature errors grow steeply as that ratio approaches 1, or if a fully coupled small-timestep reference solution disagrees with the splitting result by more than the reported 8–9 percent in density and 2 percent in temperature, the central accuracy claim fails.

Watch

Extended reading notes

Core claim

The central claim is that binary monatomic gas mixing can be modeled in SPH by giving each species its own continuity, momentum, and energy equations and coupling them through the collision terms of a Gross–Krook relaxation model, and that the resulting stiff coupling can be handled without an implicit solver. The paper claims that after the hydrodynamic step, the velocity difference between the two gases relaxes according to $\Delta\mathbf{v}^{n+1}=\Delta\mathbf{v}^{*}\exp[-(\rho_{\alpha}+\rho_{\beta})K_{\mathrm{mix}}\Delta t]$, with individual velocities reconstructed from the center-of-mass velocity, and that the thermokinetic energies relax with the analogous exponential factors toward the collision-target energies. With this splitting, the timestep is set by hydrodynamics rather than by the collisional timescale, and the method reproduces the equilibration of density and temperature over a range of molecular mass ratios, including the high-ratio neon–xenon case, while conserving total momentum and energy and increasing entropy.

Load-bearing premise

The load-bearing premise is that during one hydrodynamic timestep the total density and the mixing coefficient stay nearly constant while the gases relax exponentially; if densities or collision rates change substantially within that timestep, the correction is only first-order accurate and the reported speed-versus-accuracy tradeoff may not hold.

Editorial extensions

If this is right

  • The validation runs reproduce equilibrium densities to within about 8–9 percent and equilibrium temperatures to within about 2 percent for argon–krypton and neon–xenon, so the scheme is accurate enough for order-of-magnitude mixing studies in similar confined geometries.
  • Because the collisional correction is an exponential closure rather than an explicit stiff step, the simulation timestep is set by the hydrodynamic CFL condition, which in the reported comparison reduced wall-clock time from about six hours to about forty minutes at the same particle count.
  • Each species keeps its own density, velocity, and energy fields, so the model can represent transient non-equilibrium states in which the lighter gas is compressed before redistributing, rather than forcing instantaneous mixing.
  • The modular two-fluid structure gives a direct path to adding polyatomic degrees of freedom and gas–solid drag with dust or ice, which the paper identifies as the next step toward Mars-relevant volatile-release scenarios.
  • The scheme conserves total momentum and energy and produces positive entropy production during mixing, so its equilibrium states are thermodynamically consistent.

Reading between the lines

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

  • Inference: the same exponential-closure machinery should extend directly to polyatomic mixtures by replacing the monatomic target energies with values that include rotational and vibrational contributions; the paper names this extension but does not test it, and the argon–krypton mass-ratio test gives a ready proxy for a water-vapor/carbon-dioxide system.
  • Inference: the persistent 8–9 percent density underestimate, which the paper attributes to boundary mirroring and kernel truncation, can be tested by repeating the same tests in a periodic domain; if the underestimate persists there, the collisional closure itself would be implicated.
  • Inference: the paper's use of Taylor–Aris dispersion to estimate effective diffusion suggests a testable corollary: in the drill-hole geometry, measured species concentration profiles over time should be consistent with the model's effective diffusion coefficient, which the paper currently validates only through equilibrium values and relaxation times.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 8 minor

Summary. The manuscript proposes a two-fluid Smoothed Particle Hydrodynamics (SPH) model for binary monatomic gas mixtures. Each gas is evolved with its own Euler equations, and interspecies momentum and energy exchange are modeled by collisional terms derived from a Boltzmann-based kinetic relaxation model (Gross–Krook) with Lennard-Jones collision integrals. To avoid the small collisional timestep, the authors use first-order Lie–Trotter operator splitting: a pure hydrodynamic SPH step is followed by an exponential relaxation step for the velocity difference and for the energies. The model is tested on (i) a Ne–Xe comparison between the formal SPH drag-like implementation and the splitting scheme, (ii) Ar–Kr and Ne–Xe closed-cylinder mixing tests with temperature and density equilibration, and (iii) DUSTYBOX-style tests in the non-stiff and stiff regimes. The paper claims that the splitting approach allows a hydrodynamic timestep while accurately reproducing density and temperature equilibration, and it outlines extensions to polyatomic gases and gas–dust interactions for ExoMars contexts.

Significance. If the central claims hold, this is a useful and modular SPH method for simulating binary gas mixing in confined geometries, with a clear computational advantage over explicit integration of the stiff collisional terms. The physical modeling is grounded in standard kinetic theory (Chapman–Cowling, Neufeld correlations, Zahmatkesh et al.), and the paper includes several strengths: validation against analytic equilibrium densities and temperatures, DUSTYBOX solutions in both non-stiff and stiff regimes, total-energy conservation to about 1e-5, a thermodynamic-consistency check through entropy production, and a kernel-insensitivity test. The comparison with an independent formal SPH implementation, although limited, is valuable. However, the paper's main efficiency-accuracy claim is not yet fully supported because the splitting error in the stiff regime is not quantified, and the number-weighted SPH interpolation used for the exchange terms is introduced empirically without a precise definition or convergence study.

major comments (4)
  1. [§2.5 and Appendix A] The central efficiency-accuracy claim rests on the validity of first-order Lie–Trotter splitting in the stiff regime, but no error bound or convergence study is provided. During the uncoupled hydrodynamic substep, each species evolves alone, so the exponential collision correction in Eq. (25) can only damp the relative velocity; it cannot undo the density redistribution produced by that substep. The stiff DUSTYBOX test with K=10^8 explicitly does not temporally resolve the intermediate decay, so it validates only the terminal state. The Ne–Xe comparison in §3.1 reports a 6% difference in mean densities but no dt-convergence test and no comparison of local fields. Please add a systematic dt-convergence study for the Trotter scheme in a stiff case and compare transient density and velocity fields against the formal SPH implementation or an analytic reference.
  2. [§2.3, Eq. (17)] The total-energy conservation identity in Eq. (17) is asserted with "it can be shown" and is not derived. This identity is the basis for the total-energy conservation claim in Fig. 5 and for the closed-system validation. The identity is not immediate because the target energies in Eq. (16) involve temperature and velocity differences with different mass weightings. Please prove Eq. (17) from Eqs. (6), (8), (15), and (16), or give a specific reference that contains this derivation for the per-mass formulation used here.
  3. [§2.5, Eqs. (31)-(32) and §3.3] The number-weighted SPH interpolation introduced for the exchange terms is not defined. The text states that employing number-weighted interpolation "improves the accuracy of thermal relaxation and energy conservation in high mass ratio mixtures," but it does not specify how this differs from the kernel-weighted velocity estimate in Eq. (32), how it is normalized, or whether it preserves the conservation properties of the continuum exchange terms. Please define the number-weighted estimates explicitly and test their consistency (conservation of total momentum and energy, and convergence with particle number) rather than introducing it as an empirical fix.
  4. [§3.2 and Appendices B-C] The persistent 8–9% underestimate of equilibrium mean densities is attributed to boundary effects, but the evidence is indirect. Appendix C shows a similar bias in a no-collision two-gas simulation, and Appendix B shows insensitivity to the kernel, but there is no quantitative demonstration that the boundary treatment is the dominant source of the offset. A single-species hydrostatic SPH test in the same cylindrical geometry, or the same mixing test with an improved boundary treatment, would directly support the attribution. The statement that the error stabilizes at 7–9% even when increasing the particle number from 2×10^5 to 3×10^5 also needs an explanation.
minor comments (8)
  1. [§3.1] The 'maximum discrepancy of only 6%' refers to mean densities; please state this explicitly and, in addition, report a local error measure such as the L2 norm of the density field.
  2. [§2.5, Eq. (31)] The displayed formula for the local density of the other species appears garbled: the expression multiplies the SPH sum by N_neigh. Please write the intended SPH estimate unambiguously and define all symbols (N_neigh, h_a, h_i).
  3. [§2.5, Eq. (23)] The regularization parameter is written as η=0.001h^2, which makes the denominator |r|^2 + η^2 dimensionally inconsistent; presumably η^2 = 0.001 h^2 is intended.
  4. [§2.5 and Conclusions] The phrase 'two-step Euler integrator' (later 'explicit two-steps Euler') is never defined. Please specify the time-integration scheme explicitly, for example whether it is a predictor-corrector or a two-stage Runge–Kutta method.
  5. [Abstract and §4] The wording 'good accuracy in reproducing the equilibration of density' overstates the reported 8–9% density offset; please temper the wording or state the offset explicitly in the abstract.
  6. [Table 1] The column header 'Ar and kr' should use proper capitalization: 'Ar and Kr'.
  7. [Appendix A, Fig. A2] The two panels are described in the text, but the figure should include axis labels and legends or panel titles so that the reader can identify which panel corresponds to gas g1 and which to gas g2.
  8. [Data availability] The statement 'No data was used for this article' is misleading for a numerical study; please state that simulation data are available on request or provide a link to a code/data repository.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: collisional coefficients and validation targets come from independent kinetic theory and analytic equilibrium relations.

full rationale

The paper's derivation chain is self-contained with respect to its inputs. The collisional momentum and energy exchange terms are taken from an external kinetic relaxation model (Gross & Krook 1956; Chapman & Cowling 1970), with collision integrals computed using Neufeld et al. (1972) empirical fits and Lennard-Jones parameters from Oh (2013). The target equilibrium densities follow from mass conservation plus the doubling of the volume available to each gas in the closed domain, and the equilibrium temperature follows from the independent energy-weighted average in Eq. (33); neither is derived from a parameter fitted to the simulation output. The comparison between the Trotter-split scheme and the formal SPH collisional discretization in Eq. (23) is an implementation consistency check, not a claim that one of them predicts the other from first principles. The DUSTYBOX tests compare against analytic solutions from Laibe & Price (2011); the stiff-regime test explicitly states that the intermediate velocity decay is not temporally resolved and only the asymptotic terminal state is reproduced, so it does not overclaim a transient prediction. The use of the simulated peak velocity difference U to estimate an effective diffusion timescale is explicitly labeled 'an order of magnitude reference rather than a precise prediction,' so it is not dressed up as an independent validation. The number-weighted SPH interpolation choice is a discretization decision, not a fitted parameter whose value is then rediscovered as a result. No load-bearing step reduces to a self-citation, a renamed known result, or an ansatz smuggled in from the authors' prior work. The central claims are therefore not circular.

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

All model assumptions are kinetic-theory or numerical approximations. The paper draws Lennard-Jones parameters and collision-integral fits from prior literature, and the main unproven burden is the splitting/constancy approximation plus the number-weighted interpolation choice.

assumptions (6)
  • domain assumption Each gas is an inviscid ideal gas with zero thermal conductivity and obeys the ideal EOS P_i = c_s,i^2 rho_i / gamma_i (Eq. 4).
    The Euler equations in Eq. (3) omit viscous stresses and heat conduction; this is standard ideal-gas modeling but is an assumption for the planetary scenarios.
  • domain assumption Self-collisions and inelastic collisions are neglected; only elastic cross-collisions couple the species.
    Section 2.1 states that the kinetic model retains only cross-collision relaxation, appropriate for monatomic gases but an idealization that omits species self-diffusion transport.
  • domain assumption Cross-collisions are approximated by the Gross-Krook relaxation with a Gaussian equilibrium distribution (Eqs. 5-6).
    The collision integral is not solved directly; the relaxation form fixes the target temperature and velocity per Vega Reyes et al. 2007.
  • domain assumption Lennard-Jones (12,6) potential with mixing rules epsilon_alpha_beta = sqrt(epsilon_alpha epsilon_beta) and sigma_alpha_beta = (sigma_alpha + sigma_beta)/2 describes the intermolecular interaction.
    Eqs. (1)-(2), standard combining rules from Chapman-Cowling; Lennard-Jones parameters for Ar, Kr, Ne, Xe are taken from Oh 2013.
  • ad hoc to paper In the operator splitting, (rho_alpha + rho_beta) K_mix in Eq. (25) and f_alpha_beta, hat-epsilon in Eq. (29) are constant over the timestep.
    This constancy enables the exact exponential updates; it is a numerical approximation without a stated error bound, introduced in Section 2.5.
  • ad hoc to paper The collisional exchange terms use number-weighted SPH interpolation for the other-species velocity and target energies.
    End of Section 2.5: chosen because it improves accuracy for Neon-Xenon; no derivation or independent justification is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gas mixing through a Smoothed Particle Hydrodynamics approach." pith.science (2026). https://pith.science/paper/WP3TEOXJ

@misc{pith2026250906590,
  author       = {Pith},
  title        = {Pith review of: Gas mixing through a Smoothed Particle Hydrodynamics approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WP3TEOXJ}},
  note         = {Machine review of arXiv:2509.06590}
}
read the original abstract

Transport and mixing of gas species are of particular interest in planetary environments, where interactions among multiple species can occur within confined or porous media. In this work, we present a novel Smoothed Particle Hydrodynamics (SPH) approach for modeling the mixing of binary gas species. The model treats each gas as a separate fluid governed by its own set of Euler equations, coupled through collisional momentum and energy exchange terms derived from a kinetic relaxation model based on the Boltzmann equation. The numerical scheme employs a first-order operator splitting approach combined with a two-step Euler integrator. In this setup, the hydrodynamic evolution is first computed using standard SPH techniques to handle pressure forces. This is followed by a separate correction step that accounts for interspecies collisional exchanges. Such a decoupled treatment enables the use of a larger timestep dictated by hydrodynamics rather than the typically much smaller collisional timescale, enhancing computational efficiency. The model achieves good accuracy in reproducing the equilibration of density and temperature in a range of molecular mass ratios. Its modular structure supports natural extensions to polyatomic mixtures and enables the inclusion of additional physics, such as gas-solid interactions with dust and ice. These features make the method particularly well-suited for applications involving confined, multi-component gas systems, such as those expected during the ESA ExoMars mission.

Figures

Figures reproduced from arXiv: 2509.06590 by the authors.

Figure 1
Figure 1. Illustration of the initial system setup. Xenon occupies the upper half of the cylinder, while Neon occupies the lower half. xenon (𝑚𝑋𝑒 = 131.29 u). Both gases are initially set at a pressure of 𝑃 = 610 Pa and a fixed temperature of 𝑇 = 300 K. They are treated as ideal gases with an adiabatic index of 𝛾 = 1.66. Xenon is initially placed in the upper half of the cylindrical domain, while neon occupies the lower half.… view at source ↗
Figure 3
Figure 3. Time evolution of the mean densities of argon (blue) and krypton (red) during the mixing process. underestimated by approximately 8 − 9% relative to the theoretical prediction. This underestimation error decreases as the number of SPH particles increases. As shown in Fig.2, with 5×104 the discrep￾ancy exceeds 25%. Increasing the resolution leads to a progressive reduction in the error, which stabilizes around 7 − 9%… view at source ↗
Figure 2
Figure 2. Time evolution of the mean densities for neon (top) and xenon (bottom). Solid lines represent the trend obtained using the formal SPH collisional term from Eq. (23), while dotted lines correspond to the Trotter splitting approach. The discrepancy between the two methods is at most 6% and they show the same asymptotic behavior. same pressure (610 Pa), but with different temperatures: 400 K for argon and 300 K for kry… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Time evolution of the mean temperature of argon (blue) and krypton (red) during the mixing process [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Change in total energy over time, normalized by the initial energy. throughout the simulation, except for a brief initial transient phase. In addition, the model satisfies the second law of thermodynamics. Indeed, as demonstrated in Appendix D, the entropy of the syste…
Figure 7
Figure 7. Figure 7: Time evolution of the mean temperatures of neon (blue) and xenon (red) during the mixing process. 4 CONCLUSIONS In this work, we present a novel SPH framework for simulating bi￾nary gas mixing, with explicit treatment of interspecies collisional momentum and energy exc…
Figure 6
Figure 6. Figure 6: Time evolution of the mean densities of neon (blue) and xenon (red) during the mixing process. and the initial temperature difference between the two gases is also larger [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 21 canonical work pages

  1. [1]

    Springer Singapore, Singapore, pp 114--123

    Ai M., Zheng A., Li F., 2018, in Wang Y., Jiang Z., Peng Y., eds, Image and Graphics Technologies and Applications. Springer Singapore, Singapore, pp 114--123

    work page 2018

  2. [2]

    I., 1956, @doi [Proceedings of the Royal Society of London

    Aris R., Taylor G. I., 1956, @doi [Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences] 10.1098/rspa.1956.0065 , 235, 67

    arXiv 1956

  3. [3]

    C., Phan-Thien N., 2019, @doi [Physics of Fluids] 10.1063/1.5122671 , https://ui.adsabs.harvard.edu/abs/2019PhFl...31j3303B 31, 103303

    Bertevas E., Tran-Duc T., Le-Cao K., Khoo B. C., Phan-Thien N., 2019, @doi [Physics of Fluids] 10.1063/1.5122671 , https://ui.adsabs.harvard.edu/abs/2019PhFl...31j3303B 31, 103303

    work page doi:10.1063/1.5122671 2019

  4. [4]

    G., 1970, The mathematical theory of non-uniform gases

    Chapman S., Cowling T. G., 1970, The mathematical theory of non-uniform gases. an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases

    work page 1970

  5. [5]

    Colagrossi A., Antuono M., Le Touz \'e D., 2009, @doi [ ] 10.1103/PhysRevE.79.056701 , https://ui.adsabs.harvard.edu/abs/2009PhRvE..79e6701C 79, 056701

    doi:10.1103/physreve.79.056701 2009

  6. [6]

    Coradini A., et al., 2001, @doi [] 10.1016/S0273-1177(01)00283-6

    work page doi:10.1016/s0273-1177(01)00283-6 2001

  7. [7]

    C., et al., 2017, @doi [Astrobiology] 10.1089/ast.2016.1541 , 17, 612

    De Sanctis M. C., et al., 2017, @doi [Astrobiology] 10.1089/ast.2016.1541 , 17, 612

    work page arXiv 2017

  8. [8]

    pp 477--518, @doi 10.1007/978-3-662-04886-3_15

    Emch G., Liu C., 2002, The Logic of Thermostatistical Physics. pp 477--518, @doi 10.1007/978-3-662-04886-3_15

    work page doi:10.1007/978-3-662-04886-3_15 2002

Show all 40 references
  1. [9]

    A., Monaghan J

    Gingold R. A., Monaghan J. J., 1977, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/181.3.375 , 181, 375

    1977 doi

  2. [10]

    H., Glover S

    Greif T. H., Glover S. C. O., Bromm V., Klessen R. S., 2009, @doi [Monthly Notices of the Royal Astronomical Society] 10.1111/j.1365-2966.2008.14169.x , 392, 1381

    2009

  3. [11]

    P., Krook M., 1956, @doi [Phys

    Gross E. P., Krook M., 1956, @doi [Phys. Rev.] 10.1103/PhysRev.102.593 , 102, 593

    1956 doi

  4. [12]

    L., Aharonson O., Schorghofer N., Farmer C

    Hudson T. L., Aharonson O., Schorghofer N., Farmer C. B., Hecht M. H., Bridges N. T., 2007, @doi [Journal of Geophysical Research: Planets] https://doi.org/10.1029/2006JE002815 , 112

    2007 doi

  5. [13]

    Kwon J., 2019, @doi [Journal of Computational Physics] https://doi.org/10.1016/j.jcp.2018.12.007 , 384, 114

    2019 doi

  6. [16]

    J., 2012b, @doi [Monthly Notices of the Royal Astronomical Society] 10.1111/j.1365-2966.2011.20201.x , 420, 2365

    Laibe G., Price D. J., 2012b, @doi [Monthly Notices of the Royal Astronomical Society] 10.1111/j.1365-2966.2011.20201.x , 420, 2365

    2011

  7. [17]

    J., 2014, @doi [MNRAS] 10.1093/mnras/stu355 , https://ui.adsabs.harvard.edu/abs/2014MNRAS.440.2136L 440, 2136

    Laibe G., Price D. J., 2014, @doi [MNRAS] 10.1093/mnras/stu355 , https://ui.adsabs.harvard.edu/abs/2014MNRAS.440.2136L 440, 2136

    2014 doi

  8. [18]

    R., 2014, @doi [ ] 10.1093/mnras/stu1173 , https://ui.adsabs.harvard.edu/abs/2014MNRAS.443..927L 443, 927

    Lor \'e n-Aguilar P., Bate M. R., 2014, @doi [ ] 10.1093/mnras/stu1173 , https://ui.adsabs.harvard.edu/abs/2014MNRAS.443..927L 443, 927

    2014 doi

  9. [19]

    B., 1977, @doi [ ] 10.1086/112164 , https://ui.adsabs.harvard.edu/abs/1977AJ.....82.1013L 82, 1013

    Lucy L. B., 1977, @doi [ ] 10.1086/112164 , https://ui.adsabs.harvard.edu/abs/1977AJ.....82.1013L 82, 1013

    1977 doi

  10. [20]

    Monaghan J., 1997, @doi [Journal of Computational Physics] https://doi.org/10.1006/jcph.1997.5846 , 138, 801

    1997

  11. [21]

    J., 2005, @doi [Reports on Progress in Physics] 10.1088/0034-4885/68/8/R01 , https://ui.adsabs.harvard.edu/abs/2005RPPh...68.1703M 68, 1703

    Monaghan J. J., 2005, @doi [Reports on Progress in Physics] 10.1088/0034-4885/68/8/R01 , https://ui.adsabs.harvard.edu/abs/2005RPPh...68.1703M 68, 1703

    2005 doi

  12. [22]

    Monaghan J., 2020, @doi [European Journal of Mechanics - B/Fluids] https://doi.org/10.1016/j.euromechflu.2019.10.006 , 79, 454

    2020 doi

  13. [23]

    Monaghan J., Kocharyan A., 1995, @doi [Computer Physics Communications] https://doi.org/10.1016/0010-4655(94)00174-Z , 87, 225

    1995 doi

  14. [24]

    D., Janzen A

    Neufeld P. D., Janzen A. R., Aziz R. A., 1972, @doi [ ] 10.1063/1.1678363 , https://ui.adsabs.harvard.edu/abs/1972JChPh..57.1100N 57, 1100

    1972 doi

  15. [25]

    Oh S.-K., 2013, @doi [Journal of Thermodynamics] 10.1155/2013/828620 , 2013

    2013 doi

  16. [26]

    V., 1980, Numerical Heat Transfer and Fluid Flow, 1st edn

    Patankar S. V., 1980, Numerical Heat Transfer and Fluid Flow, 1st edn. CRC Press, @doi 10.1201/9781482234213 , https://doi.org/10.1201/9781482234213

    1980 doi

  17. [27]

    Prakash M., Cleary P., Ha J., Mehidi M., Blackburn H., Brooks G., 2007, @doi [Progress in Computational Fluid Dynamics - PROG COMPUT FLUID DYN] 10.1504/PCFD.2007.013001 , 7

    2007 arXiv

  18. [28]

    Ramachandran P., et al., 2021, @doi [ACM Trans. Math. Softw.] 10.1145/3460773 , 47

    2021 doi

  19. [29]

    D., 2002, @doi [American Journal of Physics] 10.1119/1.1463737 , https://ui.adsabs.harvard.edu/abs/2002AmJPh..70..508R 70, 508

    Ramshaw J. D., 2002, @doi [American Journal of Physics] 10.1119/1.1463737 , https://ui.adsabs.harvard.edu/abs/2002AmJPh..70..508R 70, 508

    2002 doi

  20. [30]

    I., Hayfield T., Agertz O., 2010, @doi [Monthly Notices of the Royal Astronomical Society] 10.1111/j.1365-2966.2010.16577.x , 405, 1513

    Read J. I., Hayfield T., Agertz O., 2010, @doi [Monthly Notices of the Royal Astronomical Society] 10.1111/j.1365-2966.2010.16577.x , 405, 1513

    2010

  21. [31]

    C., Bonet J., Hu S.-M., 2014, @doi [ACM Trans

    Ren B., Li C., Yan X., Lin M. C., Bonet J., Hu S.-M., 2014, @doi [ACM Trans. Graph.] 10.1145/2645703 , 33

    2014 doi

  22. [32]

    Sandnes T., Eke V., Kegerreis J., Massey R., Ruiz-Bonilla S., Schaller M., Teodoro L., 2025, @doi [Journal of Computational Physics] https://doi.org/10.1016/j.jcp.2025.113907 , 532, 113907

    2025

  23. [33]

    Stoyanovskaya O., Glushko T., Snytnikov N., Snytnikov V., 2018, @doi [Astronomy and Computing] https://doi.org/10.1016/j.ascom.2018.08.004 , 25, 25

    2018 doi

  24. [34]

    Strang G., 1968, @doi [SIAM Journal on Numerical Analysis] 10.1137/0705041 , 5, 506

    1968 doi

  25. [35]

    I., 1953, @doi [Proceedings of the Royal Society of London

    Taylor G. I., 1953, @doi [Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences] 10.1098/rspa.1953.0139 , 219, 186

    1953

  26. [36]

    F., 1959, Proceedings of the American Mathematical Society, 10, 545

    Trotter H. F., 1959, Proceedings of the American Mathematical Society, 10, 545

    1959

  27. [37]

    L., et al., 2017, @doi [Astrobiology] 10.1089/ast.2016.1533 , https://ui.adsabs.harvard.edu/abs/2017AsBio..17..471V 17, 471

    Vago J. L., et al., 2017, @doi [Astrobiology] 10.1089/ast.2016.1533 , https://ui.adsabs.harvard.edu/abs/2017AsBio..17..471V 17, 471

    2017

  28. [38]

    Vega Reyes F., Garz\'o V., Santos A., 2007, @doi [Phys. Rev. E] 10.1103/PhysRevE.75.061306 , 75, 061306

    2007 doi

  29. [39]

    M., Martínez-Sykora J., Hansteen V

    Wargnier Q. M., Martínez-Sykora J., Hansteen V. H., De Pontieu B., 2022, @doi [The Astrophysical Journal] 10.3847/1538-4357/ac6e62 , 933, 205

    2022 doi

  30. [40]

    Wendland H., 1995, @doi [Advances in Computational Mathematics] 10.1007/BF02123482 , 4, 389

    1995 doi

  31. [41]

    Zahmatkesh I., Emdad H., Alishahi M., 2013, @doi [Scientia Iranica] https://doi.org/10.1016/j.scient.2012.12.017 , 20, 162

    2013 doi

  32. [42]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.