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Minimising the numerical viscosity in Smoothed Particle Hydrodynamics simulations of discs

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For flat circular discs, the default quadratic artificial viscosity in SPH disc simulations is higher than needed, and the quadratic coefficient should be switched along with the linear one.

desk verdict Solid, useful demonstration that phantom's default beta=2 is over-viscous in smooth discs, but the shock regime is untested—the switched-beta recommendation is reasonable but extrapolated. read the letter →

arxiv 2505.24343 v1 pith:NXYKSM3K submitted 2025-05-30 astro-ph.SR astro-ph.EPastro-ph.HEastro-ph.IM

Cheng Chen , C. J. Nixon This is my paper
classification astro-ph.SRastro-ph.EPastro-ph.HEastro-ph.IM
keywords smoothedparticlehydrodynamicsnumericalviscosityartificialaccretiondiscsswitchessteady-stateShakura-Sunyaev
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

This paper asks whether the artificial viscosity that SPH codes add to handle shocks and keep particles ordered is stronger than it needs to be for ordinary disc simulations. It answers by running steady-state accretion discs in the phantom code, feeding mass at a fixed rate and letting it flow out through inner and outer boundaries, then comparing the resulting surface density to the semi-analytic solution of the thin-disc diffusion equation. The lower the numerical viscosity, the closer the simulated disc stays to the semi-analytic density, so the coefficient values that maximise the steady-state surface density are the ones that minimise spurious dissipation. The finding is that for flat circular discs the default quadratic coefficient $\beta_{\rm SPH} = 2$ is too high, values near $\alpha_{\rm SPH} = 0.1$ and $\beta_{\rm SPH} = 0.2$ are closer to optimal, and the best balance comes from linking $\beta_{\rm SPH}$ to a switched $\alpha_{\rm SPH}$, e.g. $\beta_{\rm SPH} = 2\alpha_{\rm SPH}$. If this is right, many SPH disc simulations have been dissipating more angular momentum and heat than intended, and the remedy is cheap: apply the existing viscosity switch to the quadratic term as well.

What carries the argument

The central measuring device is the steady-state accretion disc: mass is injected at a fixed rate and removed at inner and outer boundaries, so the surface density profile that forms is a direct readout of the total (physical plus numerical) viscosity acting in the simulation. The benchmark is the semi-analytic steady-state solution of the 1D diffusion equation (Eq. 4): numerical viscosity lowers the simulated surface density below this solution, and lower coefficients keep the disc closer to it. The key identity is the Meru–Bate ratio (Eq. 6), $\alpha_{\rm quad}/\alpha_{\rm lin} = (135/62\pi)(\beta_{\rm SPH}/\alpha_{\rm SPH})(\langle h\rangle/H)$, which the paper uses to show that when $\alpha_{\rm SPH}$ from a switch is $\approx 0.1$ and $\beta_{\rm SPH}$ stays at 2, the quadratic term dominates the linear one unless the disc is resolved with $\langle h\rangle/H \lesssim 0.1$ — a condition most simulations in the literature do not meet. This ratio is what motivates switching $\beta_{\rm SPH}$ as well.

What would settle it

Run the same steady-state disc setup at $\beta_{\rm SPH} = 0.2$ and $\beta_{\rm SPH} = 0.02$ while directly tracking particle interpenetration and measuring the shear stress: if lowering $\beta$ again further does not raise the surface density, or raises it only through particle-reordering dissipation, the claimed minimum at $\beta\approx 0.2$ is not the true minimum of numerical viscosity.

Watch

Extended reading notes

Core claim

The paper's central claim is that, for planar and circular discs, the default values of the numerical viscosity parameters in the phantom code can be too high, particularly the quadratic term $\beta_{\rm SPH} = 2$, and that the quadratic coefficient should be made time-dependent through a switch just as the linear coefficient already is. The evidence comes from steady-state disc simulations at three resolutions: when the switch is active, varying the minimum $\alpha_{\rm SPH}$ between 0 and 0.1 makes almost no difference because the shell-averaged $\alpha_{\rm SPH}$ sits near 0.1, whereas the choice of $\beta_{\rm SPH} = 2$ versus $\beta_{\rm SPH} = 0.2$ splits the results into two clear groups, with the high-$\beta$ runs falling well below the semi-analytic surface density. Runs with $\beta_{\rm SPH} = 2\alpha_{\rm SPH}$ reproduce the highest surface-density profiles of any configuration, showing that linking the quadratic term to the switched linear coefficient minimises numerical viscosity in smooth flows while preserving the ability to grow in shocks.

Load-bearing premise

The load-bearing premise is that the gap between the simulated steady-state surface density and the 1D semi-analytic solution is almost entirely due to numerical viscosity, with pressure gradients, boundary torques and finite-width mass injection contributing negligibly; the paper acknowledges these effects but does not quantify them.

Editorial extensions

If this is right

  • SPH disc runs that keep $\beta_{\rm SPH}=2$ constant (the phantom default with a switch on $\alpha$ only) dissipate more angular momentum and heat than necessary in smooth planar discs, pushing the steady-state surface density below the semi-analytic expectation.
  • Setting $\beta_{\rm SPH} = 2\alpha_{\rm SPH}$, with $\alpha_{\rm SPH}$ from a Cullen–Dehnen switch, reproduces the lowest numerical viscosity found in any configuration for smooth flows while keeping the viscosity available for strong shocks, and it removes one user-set parameter.
  • In well-resolved discs with switched $\alpha_{\rm SPH}\approx 0.1$ and constant $\beta_{\rm SPH}=2$, the quadratic term contributes more numerical viscosity than the linear term unless $\langle h\rangle/H \lesssim 0.1$, a condition few simulations satisfy, so the quadratic term should not be neglected when estimating numerical viscosity.
  • The 'artificial viscosity for a disc' method of Lodato & Price (2010) does not reproduce the steady-state surface density shape even at higher resolution; an explicit Navier-Stokes viscosity is the safer way to model a Shakura-Sunyaev $\alpha$.
  • As resolution increases, numerical viscosity falls below the physical viscosity ($\alpha_{\rm SS}=0.1$ in these runs), so at high resolution the choice of $\alpha_{\rm SPH}$ and $\beta_{\rm SPH}$ becomes secondary to resolving the disc scale height.

Reading between the lines

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

  • The same steady-state surface-density probe could be rerun for warped or eccentric discs, where the shear and compression structure differs; the planar circular optimum of $\alpha\approx 0.1$, $\beta\approx 0.2$ may not hold there.
  • A strong-shock test (Mach number $\gtrsim 10$) with $\beta_{\rm SPH} = 2\alpha_{\rm SPH}$ is the natural check of whether the switch raises the quadratic term quickly enough to prevent particle interpenetration; the authors note the conflict between Price & Federrath (2010) and Cullen & Dehnen (2010) on this point remains open.
  • The Meru–Bate ratio can be read as a resolution requirement: with constant $\beta_{\rm SPH}=2$ and switched $\alpha_{\rm SPH}\approx 0.1$, one needs $\langle h\rangle/H \lesssim 0.1$ for the linear term to dominate, which is much stricter than the usual $\langle h\rangle/H < 1$ guidance.
  • Measuring the residual numerical viscosity as a function of resolution at fixed $(\alpha_{\rm SPH}, \beta_{\rm SPH})$ and comparing it with the $\langle h\rangle/H$ scaling of the Meru–Bate ratio would give a direct calibration of how much switching $\beta$ actually saves.
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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 / 4 minor

Summary. This paper uses phantom SPH simulations of steady, planar, circular accretion discs with an explicit Navier-Stokes Shakura-Sunyaev viscosity (alpha_SS = 0.1) to study how the artificial-viscosity coefficients alpha_SPH and beta_SPH affect the effective numerical viscosity. By comparing the steady-state surface density profiles against a semi-analytic 1D solution at three resolutions (10^5, 10^6, 10^7 particles) and for a grid of constant coefficients and switched-alpha runs, the authors find that the default phantom value beta_SPH = 2 is unnecessarily dissipative in smooth discs, and that reducing beta (to ~0.2 or below) gives higher surface densities. They recommend applying a viscosity switch to the quadratic term by setting beta_SPH = 2 alpha_SPH with alpha_SPH from the Cullen-Dehnen switch, and they also recommend using an explicit Navier-Stokes viscosity rather than the 'artificial viscosity for a disc' method.

Significance. The paper addresses a practically important issue for the large community of phantom users: the default beta_SPH = 2 may dominate the numerical viscosity in well-resolved smooth discs. The study has strengths: it uses an external semi-analytic benchmark rather than fitting to simulation output, it covers three decades of particle number, and the qualitative ordering of models is consistent across resolutions. However, the main prescriptive result, beta_SPH = 2 alpha_SPH, is not new (it is already advocated in Morris & Monaghan 1997 and Cullen & Dehnen 2010), and the paper's smooth-disc experiments alone do not test the shock-capturing behaviour that motivates a time-dependent beta. The value of the paper is therefore primarily as a quantitative demonstration that the quadratic term is often the dominant numerical viscosity in disc simulations, rather than as a new numerical scheme.

major comments (3)
  1. [Section 5 (final paragraph), Section 6, footnote 9] The recommendation beta_SPH = 2 alpha_SPH is presented as a way to keep the quadratic viscosity low in smooth flows while allowing it to grow in strong shocks, but no shock test is provided. Footnote 9 explicitly concedes that Price & Federrath (2010) found beta = 2 insufficient with the Morris-Monaghan switch for strong shocks and that the question of whether the Cullen-Dehnen switch would behave differently is open. Because the smooth-disc results alone only show that constant low beta is preferable, the advantage of the proposed prescription over simply using beta = 0.02 rests on its shock behaviour, which is untested here. Please add a shock-tube or converging-flow test (e.g., the Price & Federrath 2010 setup) or clearly restrict the recommendation to smooth, planar discs.
  2. [Section 4.3, Eq. (4), Fig. 1] The inference that the surface-density deficit measures numerical viscosity assumes that other differences between the SPH simulations and the 1D semi-analytic model are subdominant. Pressure gradients, finite-width mass injection, and boundary torques are acknowledged in Section 4.1 but are not quantified, and no uncertainties are attached to the binned surface-density profiles. A control run with the physical viscosity disabled, or a residual plot normalised by the analytic profile at each resolution, would help separate the numerical-viscosity signal from these systematics; without it, the quantitative claim of having 'minimised' the numerical viscosity is not tightly constrained.
  3. [Table 1, Section 4.3, Section 6] The grid in beta_SPH (2, 0.2, 0.02) is coarse and no diagnostic of particle penetration or particle order is reported. The manuscript's abstract motivates an optimum by the competing effects of dissipation and particle disorder, but the simulations show a monotonic improvement down to the smallest tested beta (0.02), so the claimed optimum at alpha = 0.1, beta = 0.2 is not actually established by the data; it is consistent with earlier work but not demonstrated here. Please either extend the beta grid to lower values, add order/penetration diagnostics, or rephrase the conclusion as an upper bound rather than an optimum.
minor comments (4)
  1. [Footnote 11] The statement that the 'optimal' case alpha = 0.1, beta = 0.2 overlaps essentially exactly with A9 (alpha = 0.01, beta = 0.02) is confusing; if both give the same profile, the optimum is a plateau, and the ratio beta/alpha rather than the absolute values may be the controlling parameter. Please clarify.
  2. [Fig. 2, Footnote 10] The text says the beta = 2 alpha runs match the highest profiles, but footnote 10 reports that models B7 and B9 are slightly higher in R < 2; please adjust the wording to 'closely match' or quantify the difference.
  3. [Section 4.2 and Fig. 1] The grouping of models into subgroups is useful, but the plots omit several models (A5, A6, A8, B5, B7, etc.) that are discussed; consider including them in an appendix or providing a table of the disc masses for all models so the grouping can be verified.
  4. [Section 5, last paragraph] The phrase that both Morris & Monaghan (1997) and Cullen & Dehnen (2010) switches apply to both the linear and quadratic terms is correct for the original formulations, but phantom's default implementation only switches alpha; readers may benefit from an explicit note that the BB runs here required modifying the standard phantom setup.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: external steady-disk benchmark and prior, acknowledged switch prescription.

full rationale

Score 0: no significant circularity. The benchmark is an external semi-analytic steady-state disc solution (Nixon & Pringle 2021, Eq. 4), numerically integrated for the finite-width injection profile; the SPH runs are initialised from it and compared against it (cyan lines in Figs 1, 2 and A1), with no parameter of the benchmark fitted to the simulation output. The surface-density deficit is interpreted as numerical viscosity through the standard relation ν_total = ν + ν_num, which is an external inversion strategy rather than a definitional equivalence. The central recommendation, β_SPH = 2α_SPH with α_SPH from the Cullen-Dehnen switch, is explicitly imported from prior published switch formulations — the paper states it is 'as previously advocated in the literature' and made 'in agreement with Morris & Monaghan (1997) and Cullen & Dehnen (2010)' — and is tested in additional BB1/BB3 runs (Fig. 2), not fitted to the data that motivated it. Self-citations (Drewes & Nixon 2021 for the mass-injection method, Nixon & Pringle 2021 for the analytic solution, Lodato & Price 2010 for the Navier-Stokes viscosity scheme) supply tools and background rather than the paper's conclusion. Footnote 9 is a genuine limitation: 'It would therefore be worthwhile repeating the simulations of Price & Federrath (2010) with the Cullen-Dehnen switch to see if the requirement of a larger β_SPH value is necessary.' That concession concerns external validity in the shock regime, not circular derivation. The assumption in Sections 3 and 4.3 that the surface-density deficit is dominated by numerical viscosity is a modelling assumption, not a self-referential step, and it is accompanied by explicit discussion of resolution and boundary effects.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the validity of the 1D diffusion benchmark, the explicit Navier-Stokes implementation of the physical alpha = 0.1 viscosity, and the assertion that numerical viscosity dominates the discrepancy; no new physical entities are introduced.

free parameters (3)
  • beta_SPH / alpha_SPH ratio = 2
    The recommended ratio is set to 2 on the basis of prior literature (Monaghan 1997; Morris & Monaghan 1997; Price et al. 2018), not derived from the simulations presented here.
  • alpha_SPH maximum = 1
    The maximum switched alpha is chosen as 1 following phantom defaults; no calibration is performed for this value in the paper.
  • alpha_SPH minimum = 0
    The minimum switched alpha is chosen as 0 (models BB3, B3, B9) as suggested by Cullen & Dehnen; the paper shows results are insensitive to this choice in their test range.
assumptions (4)
  • domain assumption The 1D accretion disc diffusion equation with zero-torque boundary conditions accurately describes the axisymmetric evolution of the simulated 3D disc (Eq. 4, Nixon & Pringle 2021).
    Used to generate the semi-analytic surface density benchmark against which numerical viscosity is inferred; the paper acknowledges pressure-gradient and boundary-torque effects are neglected.
  • domain assumption The Shakura-Sunyaev alpha prescription with alpha_SS = 0.1, implemented as an explicit Navier-Stokes shear viscosity, is a correct model of the 'physical' viscosity in the simulations.
    The physical viscosity is set via the Lodato & Price method; the comparison assumes this contributes the known amount and any excess is numerical.
  • domain assumption The numerical viscosity terms are the dominant source of discrepancy between the SPH surface density and the semi-analytical solution.
    The paper argues higher resolution improves agreement, implying other differences are secondary, but this is not directly proven.
  • standard math Standard SPH and phantom code behavior (kernel, smoothing length, switches) is taken as given from Price et al. (2018).
    The paper relies on the phantom implementation details without re-deriving them.

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Pith. "Pith review of Minimising the numerical viscosity in Smoothed Particle Hydrodynamics simulations of discs." pith.science (2026). https://pith.science/paper/NXYKSM3K

@misc{pith2026250524343,
  author       = {Pith},
  title        = {Pith review of: Minimising the numerical viscosity in Smoothed Particle Hydrodynamics simulations of discs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXYKSM3K}},
  note         = {Machine review of arXiv:2505.24343}
}
abstract

Simulations using the Smoothed Particle Hydrodynamics (SPH) technique typically include numerical viscosity to model shocks and maintain particle order on the kernel scale. This numerical viscosity is composed of linear and quadratic terms, with coefficients $\alpha_{\rm SPH}$ and $\beta_{\rm SPH}$ respectively. Setting these coefficients too high results in excessive numerical dissipation, whereas setting them too low may lead to unwanted effects such as particle penetration, which also leads to excess dissipation. In this study, we simulate accretion discs using the SPH code {\sc phantom} to investigate the effective disc viscosity arising from numerical viscosity. We model steady-state coplanar and circular discs with different values of $\alpha_{\rm SPH}$ and $\beta_{\rm SPH}$, from which we determine the coefficients that lead to minimum levels of numerical viscosity by maximising the steady-state disc surface density for the same mass input rate. We find that, for planar and circular discs, the default values of the numerical viscosity parameters in the {\sc phantom} code can be too high particularly for the quadratic term. As higher values of the coefficients are required to adequately capture strong shocks in the flow, we suggest that the coefficient of the quadratic term should be time-dependent in a similar manner to the presently used ``switches'' on the linear term. This can be simply achieved by setting $\beta_{\rm SPH}$ to be a constant multiple of $\alpha_{\rm SPH}$ with $\alpha_{\rm SPH}$ determined by an appropriate switch, as previously advocated in the literature.

Figures

Figures reproduced from arXiv: 2505.24343 by the authors.

Figure 1
Figure 1. The surface density profiles of each model after they reach a steady state. The left panels contain models A1, A3, A4 and A9 and the right panels contain models B1, B3, B5 and B9. The top panels correspond to the lowest resolution simulations with Np, ini = 105 . The middle panels show the simulations with Np, ini = 106 and the lower panels are the simulations with Np, ini = 107 . The cyan line in each panel represe… view at source ↗
Figure 2
Figure 2. Dotted lines are surface density profiles of models B1(blue), B3(yellow), B7(green) and B9(red) with Np, ini = 106 . Blue squares and yellow circles are similar to models B1 and B3 except βSPH = 2 × αSPH. Dehnen (2010) and evidenced by [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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    " 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 7, 2026 · model on record in the stance chip above.