pith. sign in

arxiv: 2606.10676 · v1 · pith:CXIUEWB3new · submitted 2026-06-09 · 🌌 astro-ph.EP · astro-ph.GA· astro-ph.IM· astro-ph.SR

Full one-fluid dusty gas with multiple grain species in SPH

Mark Hutchison , Guillaume Laibe , Giovanni Tedeschi-Prades , Timoth\'ee David-Cl\'eris , Alex Barret , Maxime Lombart , Daniel J. Price , Christine M. Koepferl This is my paper

Pith reviewed 2026-06-27 11:55 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.GAastro-ph.IMastro-ph.SR
keywords smoothed particle hydrodynamicsdusty gasmultiple grain speciesone-fluid formalismarbitrary drag regimesprotoplanetary disksstopping timeterminal velocity approximation
0
0 comments X

The pith

A new SPH implementation evolves multiple dust species under arbitrary drag while conserving mass, momentum, angular momentum and energy by construction.

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

The paper develops a smoothed particle hydrodynamics code that solves the full one-fluid dusty gas equations for any number of dust species and any stopping time. It extends earlier terminal-velocity approximations that break down for large grains. The method is verified on standard test problems including waves, shocks, settling and disks. Conservation properties hold automatically. The work shows that the terminal-velocity limiter used for stability can alter results for Stokes numbers above one and that the full equations avoid such artifacts.

Core claim

We present a Smoothed Particle Hydrodynamics implementation of the full one-fluid dusty gas algorithm for multiple dust species, generalising our previous terminal velocity approach to handle arbitrary drag regimes. By construction, mass, momentum, angular momentum, and energy are all conserved.

What carries the argument

The full one-fluid dusty gas algorithm for multiple dust species that evolves differential velocities explicitly and solves drag terms implicitly.

If this is right

  • The method recovers analytic solutions in drag regimes where the terminal velocity approximation fails.
  • Errors from the terminal velocity approximation can accumulate and affect other dust phases.
  • Disabling the stopping-time limiter changes outcomes for large grains but the discrepancy with the full solution remains comparable.
  • The formulation is required when coagulation or fragmentation must be included because large grains then dominate the dynamics.
  • Orbit-crossing trajectories cannot be captured and remain the main limitation of the approach.

Where Pith is reading between the lines

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

  • Adding an effective dust pressure term, as done for velocity dispersion in gases, could allow the one-fluid model to approximate orbit-crossing effects.
  • The method may enable more accurate long-term simulations of grain growth in protoplanetary disks where Stokes numbers exceed unity.
  • The five-to-ten times higher cost is acceptable only when the terminal velocity approximation demonstrably alters the science outcome.
  • Extension to three dimensions with self-gravity would test whether the conservation properties survive in more complex orbital dynamics.

Load-bearing premise

The one-fluid formalism remains valid and sufficient when differential velocities are evolved explicitly for multiple species without requiring explicit treatment of orbit-crossing trajectories.

What would settle it

A direct numerical comparison in a regime containing orbit-crossing dust trajectories that produces different macroscopic evolution from the one-fluid solution would show the formalism is insufficient.

Figures

Figures reproduced from arXiv: 2606.10676 by Alex Barret, Christine M. Koepferl, Daniel J. Price, Giovanni Tedeschi-Prades, Guillaume Laibe, Mark Hutchison, Maxime Lombart, Timoth\'ee David-Cl\'eris.

Figure 1
Figure 1. Figure 1: Evolution of the differential velocity Δ𝑣 𝑗 as a function of time 𝑡¯, both in code units, for the dustybox test assuming a dust-to-gas ratio of one. The dust is distributed across 10 species with grain sizes 𝑠𝑗 following a power-law distribution, as listed in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: compares the results from our SPH simulation (discrete points/circles) against a 1D spectral solution (solid/dashed lines) described in Section B. The 1D spectral solution was run with 1000 grid points in 𝑥 using 50 000 uniform time steps so as to be considered ‘exact’. The 𝐿2 errors are calculated according to ∥𝑒∥𝐿2 = " 1 𝑁 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Panels, from top to bottom, give the density, velocity, [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results from the dustysettle test in a vertical column of gas and dust at 𝑅 = 50, au from a solar-mass star. Left: Coloured points show the full one-fluid simulation; black and grey points cor￾respond to the terminal velocity approximation with a stopping-time limiter (TVA+lim) and without (TVA), respectively. Solid coloured curves with black dotted overlays show solutions from a high￾resolution 1D solver.… view at source ↗
Figure 5
Figure 5. Figure 5: Rendered cross-sections of the five dust species in the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: Rendered cross-sections of the two largest grains in the dustysettle test (𝑠4 for 𝑧 > 0 and 𝑠5 for 𝑧 < 0), shown at ∼ 35 yr intervals. Each timestep is divided into two columns, comparing the TVA simulation (left) with the full one-fluid simulation (right). The 𝑠4 grains in the upper atmosphere are poorly captured by the TVA for |𝑧| ≳ 5 au; however, the full one-fluid solution catches up by 𝑡 = 70 yr… view at source ↗
Figure 7
Figure 7. Figure 7: Average initial Stokes numbers for dust species along the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Averaged radial gas and dust velocities along the disc mid-plane in the [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The same TVA radial dust velocities shown in Figure [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Surface densities for all dust species. In each panel, the white dotted line separates the full one-fluid simulation (bottom left) [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the averaged volume densities at [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Left: Dust-to-gas ratios of particles in and around the disc mid-plane for the full one-fluid simulation (grey points), the TVA+lim (tan points), and TVA (orange points), with their respective averaged profiles shown as solid, dashed, and dotted lines. The broader spread of values in the full one-fluid simulation reflects its general settling and migration efficiency, but also its tendency towards dust cl… view at source ↗
Figure 14
Figure 14. Figure 14: Fractional differences in the gas and 𝑠1 dust density profiles between the different methods at 𝑡 = 2388 yr. The approximate migration-front locations of the four largest grains in the TVA and full one-fluid simulations are marked by vertical dashed and solid arrows, respectively. In the TVA simulation, the rapid migration of the 𝑠10 grains induces strong gas backreaction, which in turn propagates to the … view at source ↗
read the original abstract

We present a Smoothed Particle Hydrodynamics (SPH) implementation of the full one-fluid dusty gas algorithm for multiple dust species, generalising our previous terminal velocity approach to handle arbitrary drag regimes. By construction, mass, momentum, angular momentum, and energy are all conserved. We benchmark our method against a suite of tests -- DUSTYBOX, DUSTYWAVE, DUSTYSHOCK, DUSTYSETTLE, and DUSTYDISC -- each probing different aspects of the algorithm. Compared to the terminal velocity approximation, the full one-fluid approach incurs a computational cost increase of a factor of five to ten due to the added overhead of evolving the differential velocities and solving the drag terms implicitly. However, it accurately recovers analytic behaviour in regimes where the terminal velocity approximation fails. In such cases, errors from the terminal velocity approximation accumulate and propagate to other dust phases. We show that the stopping-time limiter commonly used in the terminal velocity approximation for numerical stability can substantially affect simulations containing large grains (Stokes numbers $\gtrsim 1$). While disabling the limiter leads to different outcomes, the discrepancy with the full one-fluid solution remains comparable, underscoring the importance of using a more general formulation for large grains. The full one-fluid formalism may be useful when including processes such as coagulation and fragmentation, where accurate treatment of large grains becomes essential. While the inability to model orbit-crossing dust trajectories remains a key limitation of the one-fluid formalism, this may eventually be addressed through the introduction of an effective dust pressure, mirroring how fluid models encapsulate microscopic velocity dispersion in gases.

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

0 major / 2 minor

Summary. The paper presents an SPH implementation of the full one-fluid dusty gas algorithm extended to multiple grain species. It generalizes the authors' prior terminal-velocity approximation to arbitrary drag regimes, asserting that mass, momentum, angular momentum, and energy are conserved by construction. The implementation is benchmarked on the standard DUSTYBOX, DUSTYWAVE, DUSTYSHOCK, DUSTYSETTLE, and DUSTYDISC tests, recovering analytic solutions in regimes where the terminal-velocity approximation fails, at a computational cost increase of a factor of five to ten due to evolving differential velocities and implicit drag solves. The work discusses the impact of the stopping-time limiter on large grains (St ≳ 1) and notes the one-fluid formalism's inability to model orbit-crossing trajectories as a key limitation.

Significance. If the implementation and conservation properties hold, the method supplies a more general tool for multi-species dusty gas simulations in astrophysical contexts, particularly where terminal-velocity assumptions break down or where coagulation/fragmentation processes require accurate treatment of large grains. The benchmarks are standard and independent, and the explicit discussion of the orbit-crossing limitation and limiter effects provides useful guidance for users.

minor comments (2)
  1. [Abstract] Abstract: the description of the implicit drag solver implementation is limited and no quantitative error metrics (e.g., L1 or L2 norms) from the benchmarks are supplied; adding these would improve clarity on the method's accuracy.
  2. The discussion of the stopping-time limiter's effect on large grains is noted, but the manuscript would benefit from explicit comparison of limiter-on vs. limiter-off results against the full one-fluid solution in at least one benchmark (e.g., DUSTYSETTLE or DUSTYDISC) to quantify the discrepancy.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

Minor self-citation to prior terminal-velocity work; central implementation claims remain independent

full rationale

The paper describes a new SPH implementation of the full one-fluid dusty-gas scheme for multiple grain species, generalizing the authors' earlier terminal-velocity approximation. Conservation of mass/momentum/angular-momentum/energy is stated to hold by construction of the numerical scheme (standard for conservative SPH discretizations) rather than derived from data or prior results. Benchmarks use standard test problems (DUSTYBOX, DUSTYWAVE, etc.) whose analytic solutions are external to the paper. Self-citations appear only to motivate the generalization and are not load-bearing for the new algorithm's validity or the reported accuracy gains in non-terminal regimes. No self-definitional loops, fitted inputs renamed as predictions, or uniqueness theorems imported from the same authors are present.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard conservation laws and the applicability of the one-fluid formalism to the tested drag regimes; no new physical entities or fitted parameters are introduced.

axioms (2)
  • standard math Standard conservation laws of mass, momentum, angular momentum and energy apply to the coupled gas-dust system.
    Invoked when stating that these quantities are conserved by construction.
  • domain assumption The one-fluid description remains adequate for the benchmark regimes even when differential velocities are non-zero.
    Underlying the claim that the method recovers analytic behaviour where the terminal velocity approximation fails.

pith-pipeline@v0.9.1-grok · 5864 in / 1429 out tokens · 35188 ms · 2026-06-27T11:55:21.373711+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

79 extracted references · 51 canonical work pages · 7 internal anchors

  1. [1]

    arXiv , arxivurl =:1111.3083 , journal =

    Pith/arXiv arXiv

  2. [2]

    Safronov, V. S. , title =

  3. [3]

    arXiv , author =:2508.19127 , journal =

    doi:10.1051/0004-6361/202554662 , eid =. arXiv , author =:2508.19127 , journal =

    work page doi:10.1051/0004-6361/202554662

  4. [4]

    arXiv , author =:2601.07088 , journal =

    doi:10.1051/0004-6361/202557277 , eid =. arXiv , author =:2601.07088 , journal =

    work page doi:10.1051/0004-6361/202557277

  5. [5]

    arXiv , author =:2509.25347 , journal =

    doi:10.1051/0004-6361/202556530 , eid =. arXiv , author =:2509.25347 , journal =

    work page doi:10.1051/0004-6361/202556530

  6. [6]

    doi:10.1093/mnras/stae722 , eprint =

    , keywords =. doi:10.1093/mnras/stae722 , eprint =

    work page doi:10.1093/mnras/stae722

  7. [7]

    arXiv , author =:2512.06409 , journal =

    doi:10.1093/mnras/stag078 , eid =. arXiv , author =:2512.06409 , journal =

    work page doi:10.1093/mnras/stag078

  8. [8]

    arXiv , arxivurl =:1005.4982 , journal =

    Pith/arXiv arXiv

  9. [9]

    arXiv , author =:2511.00631 , journal =

    doi:10.48550/arXiv.2511.00631 , eid =. arXiv , author =:2511.00631 , journal =

    work page doi:10.48550/arxiv.2511.00631

  10. [10]

    arXiv , author =:2310.04435 , journal =

    doi:10.3847/1538-4365/ad14f9 , eid =. arXiv , author =:2310.04435 , journal =

    work page doi:10.3847/1538-4365/ad14f9

  11. [11]

    arXiv , author =:2206.01023 , journal =

    doi:10.3847/1538-4365/ac76cb , eid =. arXiv , author =:2206.01023 , journal =

    work page doi:10.3847/1538-4365/ac76cb

  12. [12]

    doi:10.3847/2515-5172/abc7be , eid =

    Research Notes of the American Astronomical Society , keywords =. doi:10.3847/2515-5172/abc7be , eid =

    work page doi:10.3847/2515-5172/abc7be

  13. [13]

    doi:10.1093/mnras/staa3162 , eprint =

    , keywords =. doi:10.1093/mnras/staa3162 , eprint =

    work page doi:10.1093/mnras/staa3162

  14. [14]

    Streaming Instability for Particle-Size Distributions

    doi:10.3847/2041-8213/ab2596 , eid =. arXiv , author =:1905.13139 , journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.3847/2041-8213/ab2596 2041

  15. [15]

    doi:10.1093/mnras/staf444 , eprint =

    , keywords =. doi:10.1093/mnras/staf444 , eprint =

    work page doi:10.1093/mnras/staf444

  16. [16]

    doi:10.1093/mnras/stad2471 , eprint =

    , keywords =. doi:10.1093/mnras/stad2471 , eprint =

    work page doi:10.1093/mnras/stad2471

  17. [17]

    arXiv , author =:2011.07849 , journal =

    doi:10.3847/1538-4357/abca9b , eid =. arXiv , author =:2011.07849 , journal =

    work page doi:10.3847/1538-4357/abca9b 2011

  18. [18]

    doi:10.1093/mnrasl/slae011 , eprint =

    , keywords =. doi:10.1093/mnrasl/slae011 , eprint =

    work page doi:10.1093/mnrasl/slae011

  19. [19]

    doi:10.1146/annurev-astro-071221-052705 , eprint =

    , keywords =. doi:10.1146/annurev-astro-071221-052705 , eprint =

    work page doi:10.1146/annurev-astro-071221-052705

  20. [20]

    Grabowski , keywords =

    Wojciech W. Grabowski , keywords =. Chapter 3 - Rainfall modeling , editor =. Rainfall: Modeling, Measurement and Applications , publisher =. 2022 , isbn =. doi:https://doi.org/10.1016/B978-0-12-822544-8.00010-X , url =

    work page doi:10.1016/b978-0-12-822544-8.00010-x 2022

  21. [21]

    doi:10.1093/mnras/sty1701 , eprint =

    , keywords =. doi:10.1093/mnras/sty1701 , eprint =

    work page doi:10.1093/mnras/sty1701

  22. [22]

    doi:10.1093/mnras/stae2039 , eprint =

    , keywords =. doi:10.1093/mnras/stae2039 , eprint =

    work page doi:10.1093/mnras/stae2039

  23. [23]

    doi:10.1093/mnras/staa3682 , eprint =

    , keywords =. doi:10.1093/mnras/staa3682 , eprint =

    work page doi:10.1093/mnras/staa3682

  24. [24]

    doi:10.1093/mnras/stac2232 , eprint =

    , keywords =. doi:10.1093/mnras/stac2232 , eprint =

    work page doi:10.1093/mnras/stac2232

  25. [25]

    M., Huang, J., P´ erez, L

    doi:10.3847/2041-8213/aaf741 , eid =. arXiv , author =:1812.04040 , journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.3847/2041-8213/aaf741 2041

  26. [26]

    doi:10.1093/mnras/stv246 , eprint =

    , keywords =. doi:10.1093/mnras/stv246 , eprint =

    work page doi:10.1093/mnras/stv246

  27. [27]

    arXiv , author =:2407.09343 , journal =

    doi:10.3847/1538-4357/ad8087 , eid =. arXiv , author =:2407.09343 , journal =

    work page doi:10.3847/1538-4357/ad8087

  28. [28]

    arXiv , author =:2310.09132 , journal =

    doi:10.1051/0004-6361/202346767 , eid =. arXiv , author =:2310.09132 , journal =

    work page doi:10.1051/0004-6361/202346767

  29. [29]

    arXiv , author =:2303.13172 , journal =

    doi:10.1051/0004-6361/202346167 , eid =. arXiv , author =:2303.13172 , journal =

    work page doi:10.1051/0004-6361/202346167

  30. [30]

    Dust-obscured star formation and the contribution of galaxies escaping UV/optical color selections at z~2

    doi:10.1051/0004-6361/201116950 , eid =. arXiv , author =:1106.0498 , journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004-6361/201116950

  31. [31]

    2024, A&A, 684, A162, doi: 10.1051/0004-6361/202349015 Planck Collaboration, Aghanim, N., Akrami, Y., et al

    doi:10.1051/0004-6361/202349015 , eid =. arXiv , author =:2404.10821 , journal =

    work page doi:10.1051/0004-6361/202349015

  32. [32]

    arXiv , author =:2009.02321 , journal =

    doi:10.1088/1674-4527/20/10/164 , eid =. arXiv , author =:2009.02321 , journal =

    work page doi:10.1088/1674-4527/20/10/164 2009

  33. [33]

    arXiv , author =:2005.14097 , journal =

    doi:10.1051/0004-6361/201936576 , eid =. arXiv , author =:2005.14097 , journal =

    work page doi:10.1051/0004-6361/201936576 2005

  34. [34]

    Protostars and Planets VII , date-added =

  35. [35]

    doi:10.1093/mnras/stad2460 , eprint =

    , keywords =. doi:10.1093/mnras/stad2460 , eprint =

    work page doi:10.1093/mnras/stad2460

  36. [36]

    Dust as interstellar catalyst I. Quantifying the chemical desorption process

    doi:10.1051/0004-6361/201525981 , eid =. arXiv , author =:1510.03218 , journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004-6361/201525981

  37. [37]

    doi:10.1038/s41550-024-02454-x , eprint =

    Nature Astronomy , keywords =. doi:10.1038/s41550-024-02454-x , eprint =

    work page doi:10.1038/s41550-024-02454-x

  38. [38]

    doi:10.1093/mnras/staf2028 , eprint =

    , keywords =. doi:10.1093/mnras/staf2028 , eprint =

    work page doi:10.1093/mnras/staf2028

  39. [39]

    doi:10.1093/mnras/stac3044 , eprint =

    , keywords =. doi:10.1093/mnras/stac3044 , eprint =

    work page doi:10.1093/mnras/stac3044

  40. [40]

    doi:10.1016/j.icarus.2024.116452 , eid =

    , keywords =. doi:10.1016/j.icarus.2024.116452 , eid =

    work page doi:10.1016/j.icarus.2024.116452 2024

  41. [41]

    arXiv , author =:2412.09426 , journal =

    doi:10.48550/arXiv.2412.09426 , eid =. arXiv , author =:2412.09426 , journal =

    work page doi:10.48550/arxiv.2412.09426

  42. [42]

    doi:10.1016/j.pss.2021.105357 , eid =

    , keywords =. doi:10.1016/j.pss.2021.105357 , eid =

    work page doi:10.1016/j.pss.2021.105357 2021

  43. [43]

    doi:10.1175/1520-0469(1972)029<0400:TEODOT>2.0.CO;2 , journal =

    work page doi:10.1175/1520-0469(1972)029 1972

  44. [44]

    Insights from Synthetic Star-forming Regions: I. Reliable Mock Observations from SPH Simulations

    doi:10.3847/1538-4365/233/1/1 , eid =. arXiv , author =:1603.02270 , journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.3847/1538-4365/233/1/1

  45. [45]

    Royal Society of London Proceedings Series A , keywords =

  46. [46]

    Journal of Computational Physics , keywords =

  47. [47]

    astro-ph/0310790 , journal =

    Pith/arXiv arXiv

  48. [48]

    arXiv , arxivurl =:1111.3090 , journal =

    Pith/arXiv arXiv

  49. [49]

    arXiv , arxivurl =:1106.1736 , journal =

    Pith/arXiv arXiv

  50. [50]

    arXiv , arxivurl =:0709.2772 , journal =

    Pith/arXiv arXiv

  51. [51]

    arXiv , arxivurl =:1006.1524 , journal =

    Pith/arXiv arXiv

  52. [52]

    Journal of Computational Physics , month = sep, pages =

  53. [53]

    doi:10.1093/mnras/sty642 , eprint =

    , keywords =. doi:10.1093/mnras/sty642 , eprint =

    work page doi:10.1093/mnras/sty642

  54. [54]

    arXiv , arxivurl =:1702.03930 , journal =

    Pith/arXiv arXiv

  55. [55]

    arXiv , arxivurl =:1002.0335 , journal =

    Pith/arXiv arXiv

  56. [56]

    doi:10.1093/mnras/stac765 , eprint =

    , keywords =. doi:10.1093/mnras/stac765 , eprint =

    work page doi:10.1093/mnras/stac765

  57. [57]

    doi:10.1093/mnras/staa1366 , eprint =

    , keywords =. doi:10.1093/mnras/staa1366 , eprint =

    work page doi:10.1093/mnras/staa1366

  58. [58]

    doi:10.1093/mnras/staa3171 , eprint =

    , keywords =. doi:10.1093/mnras/staa3171 , eprint =

    work page doi:10.1093/mnras/staa3171

  59. [59]

    2016 , bdsk-file-1 =

    On dust entrainment in photoevaporative winds , volume =. 2016 , bdsk-file-1 =. http://mnras.oxfordjournals.org/content/461/1/742.full.pdf+html , journal =

    2016

  60. [60]

    Asymptotically stable numerical method for multispecies momentum transfer: gas and multifluid-dust test suite and implementation in FARGO3D

    doi:10.3847/1538-4365/ab0a0e , eid =. arXiv , author =:1811.07925 , journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.3847/1538-4365/ab0a0e

  61. [61]

    arXiv , author =:2104.02356 , journal =

    doi:10.1016/j.jcp.2020.110035 , eid =. arXiv , author =:2104.02356 , journal =

    work page doi:10.1016/j.jcp.2020.110035 2020

  62. [62]

    doi:10.1093/mnras/stac2625 , eprint =

    , keywords =. doi:10.1093/mnras/stac2625 , eprint =

    work page doi:10.1093/mnras/stac2625

  63. [63]

    doi:10.1093/mnras/stz2035 , eprint =

    , keywords =. doi:10.1093/mnras/stz2035 , eprint =

    work page doi:10.1093/mnras/stz2035

  64. [64]

    Small dust grain dynamics on adaptive mesh-refinement grids. I. Methods

    doi:10.1051/0004-6361/201834147 , eid =. arXiv , author =:1905.01948 , journal =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004-6361/201834147 1905

  65. [65]

    astro-ph/0409263 , journal =

    Pith/arXiv arXiv

  66. [66]

    arXiv , arxivurl =:1505.00969 , journal =

    Pith/arXiv arXiv

  67. [67]

    arXiv , arxivurl =:1407.3569 , journal =

    Pith/arXiv arXiv

  68. [68]

    arXiv , arxivurl =:1402.5248 , journal =

    Pith/arXiv arXiv

  69. [69]

    arXiv , arxivurl =:1402.5249 , journal =

    Pith/arXiv arXiv

  70. [70]

    doi:10.1088/0305-4470/23/7/028 , journal =

    work page doi:10.1088/0305-4470/23/7/028

  71. [71]

    doi:10.1088/0305-4470/33/6/309 , journal =

    work page doi:10.1088/0305-4470/33/6/309

  72. [72]

    Zeitschrift fur Physik , month = jan, pages =

  73. [73]

    arXiv , arxivurl =:1706.05107 , journal =

    Pith/arXiv arXiv

  74. [74]

    Transactions of the Cambridge Philosophical Society , month = jan, pages =

  75. [75]

    Physical Review , month = jun, pages =

  76. [76]

    arXiv , arxivurl =:1111.3089 , journal =

    Pith/arXiv arXiv

  77. [77]

    arXiv , arxivurl =:1802.03213 , journal =

    Pith/arXiv arXiv

  78. [78]

    arXiv , arxivurl =:1012.1885 , journal =

    Pith/arXiv arXiv

  79. [79]

    astro-ph/0610872 , journal =

    Pith/arXiv arXiv