Draft version December 19, 2014Preprint typeset using LATEX style emulateapj v. 8/13/10

NUMERICAL CONVERGENCE IN SMOOTHED PARTICLE HYDRODYNAMICS

Qirong Zhu1,2, Lars Hernquist3, Yuexing Li1,2,1Department of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA

2Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA and2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA

Draft version December 19, 2014

ABSTRACT

We study the convergence properties of smoothed particle hydrodynamics (SPH) using nu-merical tests and simple analytic considerations. Our analysis shows that formal numericalconvergence is possible in SPH only in the joint limit N →∞, h→ 0, and Nnb →∞, where Nis the total number of particles, h is the smoothing length, and Nnb is the number of neighborparticles within the smoothing volume used to compute smoothed estimates. Previous workhas generally assumed that the conditions N →∞ and h→ 0 are sufficient to achieve conver-gence, while holding Nnb fixed. We demonstrate that if Nnb is held fixed as the resolution isincreased, there will be a residual source of error that does not vanish as N → ∞ and h → 0.Formal numerical convergence in SPH is possible only if Nnb is increased systematically as theresolution is improved. Using analytic arguments, we derive an optimal compromise scaling forNnb by requiring that this source of error balance that present in the smoothing procedure. Fortypical choices of the smoothing kernel, we find Nnb ∝ N0.5. This means that if SPH is to beused as a numerically convergent method, the required computational cost does not scale withparticle number as O(N), but rather as O(N1+δ), where δ ≈ 0.5, with a weak dependence onthe form of the smoothing kernel.

1. INTRODUCTION

Smoothed particle hydrodynamics (SPH) has be-come a popular tool for studying astrophysical flowssince its introduction in the mid-1970s by Lucy (1977)and Gingold & Monaghan (1977). Instead of solvingthe equations of hydrodynamics on a mesh, SPH oper-ates by associating fluid elements with particles thatcharacterize the flow. Local properties are estimatedby “smoothing” the attributes of particles (neighbors)near a given point. Typically, the particles carry afixed mass (unless converted to another form, suchas collisionless stars), and the smoothing is done in aspherically symmetric manner using an interpolationfunction known as the smoothing kernel. Because itis gridless, SPH is naturally spatially adaptive if thescale of the smoothing, set by the smoothing lengthh, varies as the particles move with the flow. If in-dividual particle timesteps are employed, the schemewill also be temporally adaptive.

SPH has been used with success especially in inves-tigations of the formation and evolution of galaxiesand large scale structure, where spatial and temporaladaptivity are essential because of the large range inscales present. Along with the successes has come anunderstanding of various limitations with SPH. Ex-amples include: an artificial clumping instability forparticular choices of the smoothing kernel (Schuessler& Schmitt 1981); a lack of energy or entropy conser-vation for various forms of the equations of motionwhen the smoothing lengths vary spatially (Hern-quist 1993); and a poor handling of instabilities atshearing interfaces owing to artificial surface tension

effects (Agertz et al. 2007). Many of the problemshave already been overcome through a variety of im-provements to the original formulation of SPH, suchas the introduction of generalized smoothing kernels(Dehnen & Aly 2012), fully conservative equationsof motion based on variational principles (Springel &Hernquist 2002), and various methods to eliminate ar-tificial surface tension effects (Price 2008; Read et al.2010; Hopkins 2013; Saitoh & Makino 2013).

However, some issues remain unresolved, which hasled to numerous misconceptions within the commu-nity of users of SPH. For example, because densi-ties in SPH are calculated using smoothed estimates,rather than by solving the mass continuity equation,the flow of the fluid on small scales is not describedcorrectly, as emphasized by Vogelsberger et al. (2012).One claimed advantage of SPH over grid based meth-ods is that the previous history of fluid elements canbe determined straightforwardly because individualparticles retain their identity over time. With meshcodes, this can be accomplished only by introducingtracer particles into the flow so that the evolution ofthe density field can be reconstructed (Genel et al.2013). In fact, this “advantage” of SPH owes to thefact that this method does not solve the equations ofmotion correctly on the smoothing scale.

Here, we focus on another issue with SPH thathas received little attention in the literature, butfor which confusion abounds: the numerical conver-gence of this technique. A commonly used approachfor making SPH spatially adaptive is to allow thesmoothing length of each particle to decrease or in-crease depending on the local density of particles.

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One possible choice is to scale the smoothing lengthsin proportion to the mean interparticle separation ac-cording to h ∝ N−1/3 (Hernquist & Katz 1989). Abenefit to this scaling is that the number of particlesused to compute smoothed estimates (particles neigh-boring a given point), Nnb, is roughly constant, opti-mizing the efficiency of the method from one region toanother. But, this choice is somewhat arbitrary andleads to difficulties that are not widely recognized.

In SPH, local quantities are estimated by smooth-ing continuous fluid variables with an interpolationkernel. The resulting convolutions are evaluated nu-merically by approximating the integrals with discretesums. These sums are calculated over Nnb fluid ele-ments (particles) near a given point. For the scalingnoted above Nnb ≈ constant, and is typically limitedto relatively small values Nnb ≈ 30 − 100 to maxi-mize spatial resolution. However, the approximationof the convolution integrals by discrete sums entailsan error that depends directly on Nnb and not N or h.Therefore, as the resolution is improved by increasingN and reducing h, numerical convergence is not pos-sible because the error in the discrete estimates doesnot vanish if Nnb is held constant.

Systematic studies of the convergence properties ofSPH are lacking, but evidence that this source of errorexists is present in the literature. For example, in hisreview of SPH, Springel (2010b) performed a series oftests to empirically determine the convergence rate ofSPH. The behavior we suggest can be seen clearly inthe Gresho vortex problem, as in Fig. 6 of Springel(2010b). Here, various runs were performed at im-proving resolution by increasing the total number ofparticles, N , and reducing the smoothing lengths, h,but holding the number of neighbors, Nnb fixed. Thesolid curves in this figure compare the error in eachcase to a known analytic solution. For small particlenumbers, increasing N decreases the error, but even-tually the error saturates and plateaus at value thatdepends in detail on the choice of artificial viscosity.However, if the runs with varying N are compared tothe highest resolution versions (dashed curves), theerror shows a monotonically decreasing trend with N ,for all choices of the artificial viscosity. This behaviorcan be explained only by the presence of a residualsource of error that does not depend directly on Nand h. In what follows, we show that this error arisesin the discrete sums used to approximate the localsmoothing convolutions and that it depends directlyonly on the number of neighbors in these sums, Nnb.

For a random distribution of particles, this resid-ual error essentially arises from shot noise in the dis-crete estimates and is expected to scale as ∝ N−0.5

nb .However, as emphasized by Monaghan (1992), SPHparticles are usually not randomly distributed but areinstead arranged in a “quasi-regular” pattern becauseof local forces between them. An analysis of the er-ror in this situation suggests that the noise shouldscale with Nnb as ∼ (d − 1)N−1

nb logNnb, where d isthe number of dimensions. (Note that to this orderof accuracy, the error estimate vanishes for d = 1,

meaning that 1-dimensional tests cannot be used togauge the numerical convergence of SPH for d > 1,contrary to common belief.)

Using simple numerical tests, we verify that theexpected ∼ N−1

nb logNnb behavior is a reasonablecharacterization of the discreteness error for multi-dimensional simulations with SPH. Therefore, thediscreteness error, errord, can be bounded by the es-timates N−1

nb < errord < N−0.5nb . Using this scaling,

we suggest an optimal compromise between accuracyand efficiency by requiring that this error decline atthe same rate as the error associated with the smooth-ing procedure itself, which directly involves only thesmoothing lengths, h. In detail, the outcome dependson the form of the smoothing kernel, but for typi-cal choices we find that the number of neighbors inthe smoothed estimates should increase with the to-tal particle number as Nnb ∝ N0.5. In this case, thesmoothing lengths would scale as h ∝ N−1/6, ratherthan with the average separation between particles ash ∝ N−1/3, in which case h represents the geomet-ric mean of the size of the system and the averageinterparticle separation.

Our empirical tests verify that numerical conver-gence can indeed be obtained with SPH, if the neigh-bor number grows systematically with improved reso-lution. Of course, this comes with the consequence ofincreased computational expense relative to holdingNnb fixed, and implies an unfavorable cpu scaling ofO(N3/2) compared to grid codes which are typicallyO(N). This should not be surprising, since SPH isa Monte Carlo-like algorithm and it is well-known inother contexts that, while being flexible and general,Monte Carlo methods typically have unfavorable con-vergence properties relative to other more specializedtechniques. Ultimately, applications with SPH musttherefore choose between accuracy and computationalefficiency. What is clear, however, is that if SPH isused with constant Nnb, numerical convergence is notpossible.

We emphasize that this issue with the convergenceof SPH is well-known in some disciplines (e.g. Robin-son & Monaghan 2012). However, the specific prob-lem we analyze here has been virtually ignored in dis-cussions of the use of SPH in applications to cosmol-ogy and galaxy formation.

2. CONSISTENCY AND SELF-CONSISTENCY

In SPH, densities are estimated from the particlesby smoothing. This involves convolving a continu-ous field quantity, A(r), with a smoothing functionW (r, h) through

As(r) =

∫A(r)W (r− r′, h)dr′, (1)

where A(r) and As(r) denote the true field and itssmoothed version. The smoothing length h repre-sents a characteristic width of the kernel over whichthe desired quantity is spread. The smoothing kernelis normalized to unity and asymptotically approaches

2

a Dirac δ-function when h→ 0. This can be achievedwith a sufficiently large number of particles to de-scribe the continuous system as N →∞.

Assuming that the volume within the smooth-ing kernel is sufficiently sampled with these discretepoints, we can further approximate the above inte-gral by discrete summations. The “volume” element∆r′ is estimated from mb/ρb where mb and ρb are themass and density of particle b and we have

Ad(r) =∑b

Abmb

ρbW (r− rb, h). (2)

With this approximation, we immediately obtainthe density estimate in its discrete form ρd(r) as usedin SPH according to:

ρd(r) =∑b

mbW (r− rb, h), (3)

where the sum is over all the particles within the vol-ume element centered at r. A smoothing functionwith compact support is usually used to limit the“volume” to finite extent in order to minimize com-putation time. Equation (3) now operates on a finitenumber Nnb of particles near a given point. In or-der to satisfy the consistency of this step, which is toapproach the continuous limit with such finite sum-mations, the condition Nnb →∞ is required.

Moreover, in applications with finite N and Nnb,there is a lack of self-consistency based on the aboveapproach. If we would represent a scalar field A(r)with a constant value, say 1, the following condition

1 =∑b

mb

ρbW (r− rb, h), (4)

is generally not satisfied with the density field esti-mated from (3).

This lack of self-consistency is most extreme wherelarge density gradients are present, e.g. between twophases of flow (Read et al. 2010) or close to a bound-ary (Liu & Liu 2006). In other words, the volume es-timate in SPH from equation (3) with a finite Nnb isnot an accurate partitioning of space. In order to sat-isfy the condition of consistency and self-consistency,the following conditions

N →∞, h→ 0, Nnb →∞ (5)

indeed should be met if no other fixes such as nor-malization are applied.

Next we discuss the consistency of the Euler equa-tions that SPH actually solves. The original Eulerequations in Lagrangian form describe the evolutionof density, momentum, and internal energy accordingto:

dρ

dt= −ρ∇ · v (6)

dv

dt= −∇P

ρ(7)

du

dt= −P

ρ∇ · v (8)

where ρ, v and u represent the density, velocity, andinternal energy per unit mass. The equation of state

P = u(γ − 1)ρ (9)

is a closure equation for the above system, where γ isthe adiabatic index of the gas. Following the deriva-tion in Springel & Hernquist (2002), the equations ofmotion can be discretized as

dρadt

= fa∑b

mb(va − vb) · ∇aW (ha) (10)

dvadt

= −∑b

mb(faPaρ2a

∇aWab(ha)+fbPbρ2b

∇aWab(hb))

(11)

duadt

= faPbρ2b

∑b

ma(va − vb) · ∇aW (ha), (12)

where the factors fa and fb depend on derivatives ofthe density with respect to the smoothing lengths. Inpractice instead of u, we can also use a variable A(s)from P = A(s)ργ to solve the energy equation as inSpringel & Hernquist (2002). These two approachesare equivalent in principle.

Read et al. (2010) have calculated the errors in thecontinuity and momentum equations with the aboveSPH formulation assuming a finite Nnb (ignoring thefactors fa, which spoils the fully conservative natureof the equations) as

dρadt≈ −ρa(Ra∇a) · va +O(h) (13)

dvadt≈ − Pa

hρaE0,a −

(Va∇a)Paρa

+O(h) (14)

where V, R are matrices close to the identity matrix Iand E0 is a non-vanishing error vector. V, R and E0

are all determined by the particle distribution withinsmoothing length h (see Read et al. (2010) for detaileddefinitions). As a result, these errors will remain atthe same level while we increase N if Nnb is fixed.However, a consistent scheme would require all theerror terms in the above two equations go down ash→ 0. Similar conclusions can also be drawn for theenergy equation.

Based on the above discussions of the density esti-mate and the discretized Euler equations in SPH, thecondition Nnb → ∞, N → ∞ and h → 0 has to bemet in order to have a consistent and self-consistentscheme. Moreover, reducing h will lead to finer timesteps according to the CFL condition. If the magni-tude of the error from a fixed Nnb remains constant,the overall error will build up more rapidly in a highresolution run than a lower resolution one.

3. ERROR ESTIMATE OF DENSITY ESTIMATE

3

The error in SPH from the smoothing procedure(independent of particle distribution) can be approx-imated to lowest order by:

errors ∝ hα. (15)

For the usual B-splines (Monaghan 2005; Price 2012b)and the Wendland functions (Dehnen & Aly 2012),this is a second order term where α = 2. Higherorder precision can be achieved by constructing dif-ferent classes of smoothing kernels. However, not allare appropriate since smoothed estimates of positivedefinite quantities can take on negative values.

The error from discretizing the integral convolu-tions in the smoothing procedure depends on whetheror not particles are ordered in space. Optimistically, ifthe particles are quasi-ordered and not randomly dis-tributed, it has been conjectured that the discretiza-tion error can be approximated by (Monaghan 1992)

errord ∝logNnbNnb

. (16)

This estimate is based on the study of the complex-ity associated with the low discrepancy sequence innumerical integration by Wozniakowski (1991). Thisindicates that the actual number of particles withinthe volume of each SPH particle should match theexpected number of particles. Clearly, this is highlydependent on the randomness in the actual distribu-tion of SPH particles.

The pressure force between SPH particles alwaystends to regulate the distribution of the SPH neigh-bors into an isotropic distribution. However, this“restoring” pressure force does not push the particlesinto an exact desired distribution since the number ofneighbors is finite. Hence, the velocity field exhibitsnoise on small scales. Such velocity noise will furtherinduce fluctuations in the density field. In realisticapplications, especially with highly turbulent flows,it is questionable whether the pressure forces can ef-fectively regulate SPH particles into a quasi-orderedconfiguration.

On the other hand, if the distribution of particles istruly random, the convergence of SPH should follow(slow) Monte-Carlo behavior:

errord ∝1√Nnb

. (17)

We further parameterize the dependence of dis-cretization error on the number of neighbors fromthese two extreme situations as

errord ∝ N−γnb , (18)

where 0.5 < γ < 1.From the perspective of convergence, we require

that as the total number of SPH particles is madelarger, we will be able to simulate finer scale struc-ture. The smoothing error is already consistently re-duced by making h smaller. At the same time, wemust also use more neighbors in the discrete sums tocombat the discretization error, which will eventually

dominate if no action is taken. We can write theseconditions as

N →∞, h→ 0, Nnb →∞. (19)

If we parameterize the dependence of smoothinglength h on N written by

h ∝ N−1/β , (20)

then the relation between Nnb and N is just

Nnb ∝h3

V/N∝ N1−3/β . (21)

The requirements of convergence impose the followconditions on the dependence of β on N :

β > 0, so h→ 0 as N →∞, (22)

andβ > 3, so Nnb →∞ as N →∞. (23)

For any case β > 3, there is a power-law dependenceof Nnb on N . In order to balance the smoothing anddiscretization errors, i.e. errord ∼ errors, we haveα ∼ γ(β − 3).

The dependence of of h and Nnb can be expressedas,

β ∼ α

γ+ 3, h ∝ N−γ/(3γ+α), Nnb ∝ Nα/(3γ+α). (24)

For commonly used smoothing kernel forms, wehave α = 2. Recall for random distributions we haveγ = 0.5 and for quasi-ordered distributions γ = 1, soβ for these two extremes is β = 7 and β = 5. The de-pendence of Nnb on N is thus between [N0.4, N0.57].This suggests a simple intermediate choice is givenby:

β ∼ 6, h ∝ N−1/6, Nnb ∝ N0.5. (25)

In detail, other choices would follow if higher-ordersmoothing kernels are used, which would require aneven stronger scaling of Nnb with N . This can bereadily seen from the expression of Nnb in equation24, where the power-law index α/(3γ+α) approaches1 for sufficiently large α if γ fixed. For example wehave α = 4 for the smoothing kernel functions con-structed by Monaghan (1985), and then Nnb shouldvary between [N0.57, N0.73]. This indicates that theregularity in the particle distribution, which poses astrong limit for numerical interpolation, has to be im-proved to be much better than a quasi-ordered con-figuration if higher order smoothing kernels are em-ployed.

We note that an earlier suggestion of increasing Nnbto reduce the error was proposed by Quinlan et al.(2006) from 1-D error analysis of SPH. On the sur-face, their analysis on the arbitrary spaced particles issimilar to our result of quasi-order distribution. Thedifference here is that the discretization error in SPHin multi-dimensions is much worse than the trunca-tion error inferred from the second Euler-MacLaurinformula in 1-D given by Quinlan et al. (2006).

4

4. STATE OF THE ART SPH: WENDLAND FUNCTIONWITHOUT PARING INSTABILITY AND INVISCID

SPH

A direct application with the scaling in equation 24will not work with classical SPH codes. If more than∼ 60 neighbors are used in the discrete estimates,the SPH particles will quickly form close pairs, asdemonstrated in Springel (2011) and Price (2012b).Such a configuration actually makes the discretizationerrors worse, which is opposite to our purpose.

For the kernel functions widely used in SPH codes,the function ∇W tends to flatten when two parti-cles are close together. Consequently, the net repul-sive force also diminishes between close pairs. Whena large number of neighbors Nnb is used, particleswill be less sensitive to small perturbations withinthe smoothing kernel. In reality, the density estimateerror fluctuates around the true value as a functionof Nnb as shown in Dehnen & Aly (2012). The wholesystem thus favors a distribution of particles withclose pairs which minimizes the total energy. Hencethe entire system would not achieve the resolutionone aims for. This clumping instability is the pri-mary reason that the convergence study with variableNnb is not possible in SPH simulations (Price 2012b;Bauer & Springel 2012; Vogelsberger et al. 2012; Hay-ward et al. 2014) with conventional smoothing ker-nels. With the cubic spline, sometimes additional re-pulsive forces are added in the inner smoothing lengthin the equations of motion to fight against the pair-ing instability, as in Kawata et al. (2013). Read et al.(2010) instead proposed a centrally peaked kernel tomitigate against clumping.

Dehnen & Aly (2012) have shown that Wendlandfunctions are free from the clumping instability forlarge Nnb when used to perform smoothing in SPH.The desirable property of the Wendland functions isthat they are smooth and have non-negative Fouriertransforms in a region with compact support. Sim-ilarly, Garcıa-Senz et al. (2014) has shown that thesinc family of kernels is able to avoid particle clump-ing. In what follows, we adopt the Wendland C4 func-tion, defined on [0, 1] as

W (q;h) =495

32πh3(1− q)6(1 + 6q +

35

3q2), (26)

in the P-Gadget-3 code (an updated version ofGadget-2 (Springel 2005)) to study the convergencerate of SPH.

We also update the viscosity switch in P-Gadget-3using the method by Cullen & Dehnen (2010) to ef-fectively reduce the artificial viscosity in shear flowswhile maintaining a good shock capturing capability.This effectively reduces the viscosity in the flows pro-ducing results closer to the inviscid case. This switchalso greatly enhances the accuracy of SPH in the sub-sonic regime especially where large shear is present.The time integration accuracy is also improved fol-lowing the suggestions by Saitoh & Makino (2009)and Durier & Dalla Vecchia (2012).

To verify the absence of the clumping instabilitywith our new code, we plot the distribution of dis-tances to the closest neighbor particle from a realisticsituation. In Figure 1, we compare the performanceof the Wendland C4 with the original cubic spline forthe Gresho vortex test (Gresho & Chan 1990). Thedetails of the set-up can be found in the following sec-tions. In the upper panel of Figure 1, we show thedistribution of the distance to the nearest neighborfor each SPH particle in a subdomain of the simula-tion at t = 1 with the cubic spline. This test involvesstrong shearing motions from a constant rotating ve-locity field. At t = 1, the vortex has completed almosta full rotation. The distance to the nearest neighboris further normalized to the mean interparticle sepa-ration in the initial conditions.

With Nnb = 50, the distribution of distances showsthat most of the particles are still well-separated fromone other. This situation quickly worsens as soon aswe use Nnb = 58, which is the critical number for thecubic spline reported by Dehnen & Aly (2012). A sig-nificant portion of the particles now have their closestneighbor within a factor of 0.2 of the initial separa-tion. For an even higher Nnb = 400, most of theparticles have formed close pairs as indicated by thepeak near 0. As a comparison, the lower panel of Fig-ure 1 shows that close particle pairs are not presentwith the Wendland C4 as we vary Nnb from 100 to200 (equivalent to ∼55 with cubic spline; see Dehnen& Aly 2012) to 500. This test confirms the good per-formance of SPH against the pairing instability withthe Wendland function and gives us confidence thatwe can use this code to perform further tests to studythe convergence rate with varying Nnb.

5. DENSITY ESTIMATE

5.1. A set of randomly distributed points

We use a randomly distributed particle set to testthe density estimate in SPH. 643 particles are dis-tributed randomly within a 3-D box with unit length.We use the density estimate routine in SPH with vary-ing Nnb from 40 to 3200 to derive the density field ac-cordingly. In Figure 2, we plot the histogram of thedensity estimated at the position of each particle forvarying Nnb. The variance within the density field islargest for small Nnb values, which are shown in thered colors. As the number of neighbors increases, thedistribution slowly approaches a Gaussian like distri-bution peaked at ρ = 1. This behavior shows thetrade-off between variance and bias in the SPH den-sity estimate. Larger Nnb hence larger h will increas-ingly smear out fluctuations on short scales. Howeverthis does not show the actual accuracy of the SPHdensity estimate.

In fact, the precise density field in this set-up isdifficult to give but the behavior of the density fieldcan be seen from a pure statistical argument. To bemore specific, the number of particles falling withina fixed volume associated with each particle follows aPoisson distribution. The expected number of suchoccurrences goes linearly with larger volume (larger

5

Fig. 1.— Upper: The distribution of distances to the nearestneighbor rij normalized to the initial particle spacing rij,0 in theGresho vortex test at t = 1, for the cubic spline in a portionof the simulation domain. Lower: The same histogram for asimulation with the Wendland C4 function. In the upper panel,once the number of neighbor exceeds a threshold, Nnb = 58for the cubic spline, close pairs of SPH particles form quickly.As a result, the volume associated with each SPH particle isnot well-sampled by their neighbors. This particle clumpingeffectively reduces the resolution of SPH. As a comparison, thisinstability is not observed for the simulation with the WendlandC4 function.

Nnb). As a result, the standard deviation of the den-sity distribution follows a N−0.5

nb trend. So should thetrend of the standard deviation of the density distri-bution given by SPH if the density estimate is accu-rate. However, this is not strictly true as we show inthe lower panel of Figure 2, where the black crossesare the measured standard deviation of the densitydistribution with varying Nnb. There is a significantspread that is much greater for small Nnb than theguiding line indicating N−0.5

nb . The deviation is al-ready significant for Nnb = 200, which is the recom-

mended number by Dehnen & Aly (2012) based ona shock tube test. In the upper panel of Figure 2,we can also see there is a long tail for small Nnb val-ues. This suggests there is an overestimate of densitywhere we suspect the self-contribution of each particleto the density estimates plays an exaggerated role.

In order to verity this, we carry out density esti-mates with self-contribution excluded for the sameconfiguration of particles. The standard deviation ofthe density distribution as a function of Nnb is shownin the red filled circles in the lower panel of Figure 2.This brings the relation much closer to the expectedN−0.5nb line where the deviation at small Nnb is greatly

reduced due to the subtraction of the self-contributionterm.

Excluding the self-contribution is not the usualpractice in modern SPH codes. Flebbe et al. (1994)actually advocated to use equation 3 with self-contribution excluded to give a better density calcula-tion based on a similar comparison in 2-D. Whitworthet al. (1995) however argues that the distribution ofSPH particles is essentially different from a randomdistribution. If particles are distributed randomly, in-dividual particles do not care about the locations ofother particles. In SPH, due to the repulsive pressureforce, each SPH particle is trying to establish a zonewhere other SPH particles cannot easily reside. How-ever, as we see in Figure 1, SPH forms close pairs sothat the argument by Whitworth et al. (1995) is nolonger valid. Note that it is not uncommon to see avalue above Nnb = 58 in the literature with the cu-bic spline. The SPH density estimate will then givea biased result for irregularly distributed particles.

This comparison actually favors the use of a largeNnb where the distribution is highly disordered. Al-though the above experiment is based on a situationwhere the particle distribution is truly random, therealistic situation is between such a truly random dis-tribution and a quasi-regular one. However, quasi-regular distributions are hard to achieve in multi-dimensional flows because of shear.

5.2. A set of points in a glass configuration

A random distribution of particles, as discussed inthe above section, is an extreme case for SPH sim-ulations. We relax these particles according to theprocedure described in White (1996) to allow themto evolve into a glass configuration. Given an r−2 re-pulsive force, the particles settle in a equilibrium dis-tribution which is quasi force-free and homogeneousin density. This mimics the other extreme case whereSPH particles are well regulated by pressure forces.Nevertheless, the outcome of this relaxation is notperfect where tiny fluctuations in the density field arestill present because of the discrete nature of the sys-tem.

We now measure the SPH density estimate on thisglass configuration for different Nnb and estimate thestandard deviation of the density for each Nnb. Asthe upper panel in Figure 3 shows, the density dis-tribution is much narrower than that for a random

6

Fig. 2.— Upper: The distribution of density field estimatesbased on an SPH smoothing routine with Wendland C4 func-tion on a random set of points with varying Nnb. The colorscheme represents different curves with increasing Nnb withthe color from red to purple. The standard deviation decreasesas we use a larger Nnb in the SPH density estimate. ForNnb > 1000, the result is close to a Gaussian peaked at ρ = 1.Lower: The standard deviation measured in the density dis-tribution as a function of varying Nnb. The expected trend ofσ ∝ N−0.5 is given by the dashed-dotted line. The SPH densityestimate with and without the self-contribution term is indi-cated by black crosses and red filled circles. Self-contributionproduces an overestimate of the density where particles arerandomly distributed.

set of points. For sufficiently high Nnb, the distri-bution approaches a Dirac-δ distribution. The con-vergence is also much faster than in Figure 2. Infact, we observe a N−1

nb trend for the glass configu-ration. For this set up, there is no need to excludethe self-contribution term, which is consistent withWhitworth et al. (1995).

This measured N−1nb rate is quite interesting in

itself. Monaghan (1992) conjectured that the dis-cretization error with SPH behaves as log(Nnb)/Nnb.

This means that the randomness in the distribution ofSPH particles is closer to a low discrepancy sequencerather than to a truly random one. Our experimentshows this N−1

nb (neglecting the log(Nnb)) rate is cer-tainly appropriate for a glass configuration, which iscarefully relaxed to an energy minimum state. How-ever, this rate may not hold for a general applications.

Thus, these two examples support the dependenceof the density estimate error on Nnb as in the previousanalysis. Namely, N−0.5

nb for truly random data points

and N−1nb for a distribution with good order. Realistic

applications will fall between these two extremes andcould be more biased towards the N−0.5

nb case.

5.3. Errors in the volume estimate

We have noted that a constant scalar field is not ex-actly represented by the SPH method. Based on ourdiscussion of the self-consistency of the SPH densityestimation, we can further measure the magnitude ofthis inconsistency and its dependence on the numberof neighbors Nnb. Define a parameter Q as

Qa =

Nnb∑j

mb

ρbWab(ha), (27)

which represents the SPH estimate of a constantscalar field A = 1. This parameter Q should follow apeaked distribution around 1 with some errors. Wecompute this parameter with a simple extra routineafter the density calculation in the code since this pa-rameter requires the SPH density field as input. InFigure 4, we plot the distributions of the calculatedvalues of Q for randomly distributed particles withseveral different Nnb values. The distributions of Qare indeed rather broad around 1. Similarly as the er-rors in the density estimate, the standard deviation ofthe distribution is consistently larger for smaller Nnbvalues. Some extreme values of Q with small Nnb areeven on the order of unity. The peaks of the distri-bution with small Nnb are actually slightly below 1,which indicates some bias towards a density overes-timate. This is consistent with the above discussionof the self-contribution to the density overestimate atsmall Nnb values.

For this random distribution of particles, increas-ing Nnb will lead to a more accurate volume estimateby reducing the inconsistency in the density estimate.For example, the spread of the distribution of Q forNnb = 800 is greatly reduced compared to Nnb = 48.As we repeat this experiment with even more neigh-bors, the distribution of Q approaches a Dirac-δ func-tion eventually with the error in the volume estimatereduced to zero. In this limit, the self-consistency ofSPH is finally restored. However, as in the lower panelof Figure 4, such an improvement depends slowly onthe number of number Nnb at a rate σ(Q) ∝ N−0.5

nb .We also consider the cubic spline in this test, which

is included in the lower panel of Figure 4. The de-pendence of Q on Nnb for a cubic spline follows thesame N−0.5

nb trend. The cubic spline shows a slightly

7

Fig. 3.— Upper: Similar to Figure 2, the distribution of den-sity field estimates based on an SPH smoothing routine withthe Wendland C4 function on a glass configuration data setfor varying Nnb. The distribution of the density estimates issignificantly narrower than the random point set in Figure 2.The color schemes represents different curves with increasingNnb for color going from red to purple. The standard deviationalso decreases as we use a larger Nnb in the SPH density esti-mate but at a much faster rate compared to Figure 2. Lower:The standard deviation measured in the density distributionsas a function of varying Nnb. A trend of σ ∝ N−1 rather thanσ ∝ N−0.5 is seen as indicated by the dashed-dotted line.

lower magnitude for the same Nnb compared with theWendland C4 function. This is because the Wend-land C4 is slightly more centrally peaked than thecubic spline. To have the same effective resolutionlength, a higher number of Nnb is required for Wend-land C4 (see the table of such scaling in Dehnen &Aly (2012)). The behavior of Q with respect of Nnbis consistent with the two smoothing functions.

We then repeat this calculation with the particlesin a glass configuration. The results of the WendlandC4 function and cubic splines are presented with the

solid symbols in Figure 4. As we see from the previoussection, the relaxed particle distribution reduces theerrors in the density estimate. The inconsistency inthe volume estimate, as measured in Q, declines as afunction of σ(Q) ∝ N−1

nb in this case. This is true forboth the Wendland C4 function and cubic spline.

In practice, when we use SPH, the true distributionis unknown. So the error in the density estimate isdifficult to quantify. We suggest a simple procedureto measure Q in simulations to give us an estimateof the overall quality of the density estimate. Basedon the comparison between the Q value for randomlydistributed particles and particles in a glass configu-ration, a typical number Nnb = 32 or 64 with cubicspline as often adopted in the literature leads to avolume estimate error between 1% and 10% from thetrue value. However, a glass configuration is a ratheridealized and we suspect that SPH particles will notmaintain such good order in most applications.

This error in the volume estimate, which is self-inconsistency in the SPH density estimate, also hasdynamical consequences as the volume estimate,

dr ≈ m

ρ, (28)

is used for the derivation of the equations of motion.In addition to the error terms derived by Read et al.(2010), an error on the order of a percent level is al-ready present from the density estimate step itself.The quality of the particle distribution is crucial tominimizing this error. However, once the particlesdeviate from a regular distribution, the errors in theequations of motion will pose another challenge forthe particles to return back to an ordered configura-tion. One way to impose a small error in the volumeestimate is to use a large Nnb. Increasing Nnb canregulate this error with a dependence between N−0.5

nb

and N−1nb .

6. DYNAMICAL TESTS

In this section, we perform dynamical tests withSPH for several problems to study the convergencerates in each situation. The evolution in multi-dimensional flows is more complex than in 1-D and, asnoted in §1, one-dimensional tests cannot be used tojudge convergence of SPH in multi-dimensions. Thetest problems are chosen so analytic behaviors areknown in advance (Springel (2010b); Read & Hay-field (2012); Owen (2014); Hu et al. (2014)). Also thetest problems consider both smooth flows and highlyturbulent flows. These test problems are purely hydroas no further compilations from other factors such asgravity are present. We also do not consider problemswith shocks for our tests as the presence of discontin-ues will reduce the code to being at best first orderaccurate.

6.1. Gresho Vortex problem

The first problem is the Gresho Vortex test (Gresho& Chan 1990). We quantify the L1 velocity error with

8

Fig. 4.— . Upper : The distribution of the volume esti-mate quality parameter Qa =

∑mb/ρbWab(ha) for the ran-

dom point set used in Figure 2 for differentNnb. This quantifiesthe deviation from an exact partition of unity. The distribu-tion of Q slowly approaches a narrower normal distributioncentered at Q = 1 with increasing Nnb. This indicates thatthe volume estimate error will eventually decrease to 0 as wehave Nnb → ∞ when the distribution of Q is essentially aDirac-δ function at 1. Lower : The relation between Q andNnb measured with Wendland C4 function and with a cubicspline for the random point set and for the glass configurationrespectively. For the glass configuration, the distribution of Qconverges to a Dirac-δ function at a rate of N−1 for both ofthe smoothing functions. However, the convergence rate goesas N−0.5 for both of the smoothing functions with the randompoint set.

SPH and its convergence rate as a function of the to-tal number of SPH particles. This test involves adifferentially rotating vortex with uniform density incentrifugal balance with pressure and azimuth veloc-

ity specified according to

P (r) =

P0 + 12.5r2 (0 ≤ r < 0.2)

P0 + 12.5r2 + 4

−20r + 4 ln(5r) (0.2 ≤ r < 0.4)

P0 + 2(2 ln 2− 1) (r ≥ 0.4)

(29)

and

vφ(r) =

5r (0 ≤ r < 0.2)

2− 5r (0.2 ≤ r < 0.4)

0 (r ≥ 0.4).

(30)

The value of P0 is taken to be P0 = 5 followingSpringel (2010b) and Read & Hayfield (2012), as inthe original paper of Gresho & Chan (1990). Dehnen& Aly (2012) and Hu et al. (2014) also investigate thebehavior by varying this constant background pres-sure P0. The velocity noise will decline for decreasedP0 such that the system is more stable. This testproblem is suitable for examining the noise in the ve-locity field since the density field is uniform. Thevortex should be time independent having the initialanalytical structure.

Springel (2010b) however found that this test posesa challenge to SPH since the vortex quickly breaksup with Gadget. Dehnen & Aly (2012), Read & Hay-field (2012), and Hu et al. (2014) have tested theirSPH formulation with the same problem and foundimprovements with a better parameterization of ar-tificial viscosity. The convergence rate reported byDehnen & Aly (2012) and Hu et al. (2014) is similarto the one given by Springel (2010b), N−0.7, whileRead & Hayfield (2012) give a rate of N−1.4. Thisdiscrepancy is due to a modified version of the equa-tions of motion by Read & Hayfield (2012). The set-up of this problem has three sharp transition pointswhich is problematic for SPH. This test is also verysensitive to the particle noise which will can induce afalse triggering of artificial viscosity.

We initialize the particles on an N×N×16 lattice.The number of particles vertically assures that wecan extend Nnb to several thousand for each parti-cle without overlap. The L1 velocity error as a func-tion of the number of effective 1-D resolution elementsN is shown in Figure 5. The black squares are thesimulations with Nnb = 120 and the blue filled cir-cles are the ones with Nnb = 200. For the red dia-monds, we run the simulation with increasing Nnb asNnb ∝ N3∗0.4 starting from Nnb = 120 at the low-est resolution N = 32. The convergence rates are asN−0.5, N−0.7 and N−1.2 respectively. The former twoare consistent with Springel (2010b), Dehnen & Aly(2012) and Hu et al. (2014). Also, a larger numberof neighbors gives a slightly faster convergence ratein this test (N−0.5 vs. N−0.7). For Nnb = 120 andNnb = 200, the L1 velocity error at the highest reso-lution at N = 500 is actually above the convergencerate. This is what is expected based on the argu-ment for consistency in the SPH method. A fixedNnb may work well for a range of different resolutions

9

until the resolution is ultimately so high that the er-rors from the density estimate, from the discretizedcontinuity equation and momentum equation, are notconsistently reduced according to the correspondingresolution length h.

The result with varying Nnb, on the other hand,shows a much faster convergence rate as N−1.2 withthe highest resolution at N = 320 with Nnb = 1900.This validates the arguments presented earlier andshows the result given by SPH does converge to theproper solution and the convergence rate is improvedin this test problem as we increase Nnb when h →0. Such a convergence rate is actually close to therate p = 1.4 measured with moving mesh code Arepo(Springel 2010a) for the same problem.

Fig. 5.— . Convergence rate of the velocity field in theGresho vortex test as a function of N1D with SPH. Bluefilled circles are the results with a fixed number of neighborsNnb = 200 while the black filled squares are the ones with asmaller number of neighbors Nnb = 120. The convergence rateshows a N−0.7 behavior for the former and N−0.5 for the lat-ter. The red squares are with increased Nnb as a function oftotal number of SPH particles N . The convergence rate forthis run scales almost as N−1.2. Such a convergence rate isclose to N−1.4 for the grid-based code in Springel (2010a).

We have also included two series of the Gresho vor-tex test with the cubic spline as in the green symbolsin Figure 5. The performance with the cubic splineis disappointing in this test. With Nnb = 48, the L1

error reaches to a minimum value at an intermediateresolution at N = 128. For Nnb = 96, where theclumping instability is reducing the effective resolu-tion, the minimum L1 error is achieved at N = 64.In terms of the absolute error, the highest resolutionactually fails to give the best results. This behavioris also observed in the Gresho vortex test and the KHinstability test in Springel (2010b).

6.2. Isentropic Vortex problem

This is another vortex problem described by Yeeet al. (2000) and employed by Calder et al. (2002)

and Springel (2011). It also has a time-invariant an-alytic solution. Unlike the previous test, the densityand velocity profiles are both smooth. This problemwill test the convergence rate of the solver for smoothflows. We generate the initial conditions in a box ofsize [−5, 5]2 with periodic boundary conditions. Sim-ilar to the previous test, several additional identicallayers are stacked along the z direction to get a thinslab configuration. The velocity field is set up as

vx(x, y) = −y β2π

exp(1− r2

2), (31)

vy(x, y) = xβ

2πexp(

1− r2

2). (32)

The distribution of the density field and the internalenergy per unit mass are calculated according to thefollowing temperature distribution

T (x, y) ≡ P/ρ = T∞ −(γ − 1)β2

8γπ2exp(1− r2), (33)

as ρ = T 1/(γ−1) and u = T/(γ−1). Consequently, theentropy P/ργ is a constant within the domain. As inSpringel (2011), we then measure the L2 error in thedensity field at t = 8.0 obtained with our code withrespect to the analytical density field for different res-olutions. The result if shown in Figure 6 where theblue filled circles are obtained with Nnb = 200 whilethe red diamonds are with varying Nnb. The dashedline is a N−2 trend with the dashed-dotted line takenfrom Springel (2011) measured with the Arepo code.

The overall magnitude of the error with SPH is alsoclose to the one for the Arepo simulation in Springel(2011). In fact, the result for Nnb = 200 with SPHshows a second order convergence for the first fourresolutions, but for the two highest resolution runsthe L2 density error starts to reach a plateau. Begin-ning with Nnb = 200 at N = 256, we re-simulate thetwo high resolution runs with Nnb ∝ N3∗0.5. As a re-sult, the overall error is reduced and is now consistentwith the expected rate of N−2.

The above result based on the isentropic vortexproblem shows that SPH can be a second-order accu-rate scheme as long as the other errors do not domi-nate over the “smoothing error” introduced from thedensity estimate. Again, the condition N →∞, h→0 and Nnb → ∞ needs to be satisfied. As for thefixed Nnb run, starting from N = 512 the other errorsrather than the “smoothing error” become dominantand start to degrade the performance of SPH.

In order to see the effects of the discretization er-ror, we have repeated this experiment with smallerNnb and also using the cubic spline. For the two se-ries of Nnb = 80 and Nnb = 48, as in green symbolsin Figure 6, the convergence rate systematically slowsdown from N−1.7 to N−1.3 from a second order con-vergence rate. The convergence rate with the cubicspine, where Nnb = 48, is the slowest one. Moreover,it shows signs of turning over at the highest resolutionwith the cubic spline.

10

Fig. 6.— . Convergence rate of the isentropic vortex test asa function of N1D with SPH code. The error is computed fromthe L2 norm of the density field at t = 8.0 when the vortexhas completed a full rotation. Blue filled circles are the resultfor a fixed number of neighbors Nnb = 200. The convergencerate exhibits a N−2 behavior except for the two high resolutionruns. The red filled diamonds are the two high resolution runsfor increasing Nnb and now the error is consistent with theexpected N−2 trend.

6.3. Subsonic turbulence

In this section, we move to a more complex prob-lem involving the chaotic feature of multi-dimensionalflows with SPH. In particular, we simulate subsonicturbulence in the gas (which is weakly compressible).Following Bauer & Springel (2012) and Price (2012a),we employ a periodic box with unit length filled withan isothermal gas γ = 1 with unit mean gas densityand unit sound speed. The initial positions of the par-ticles are set on a cubic lattice. The stirring force fieldis set up in Fourier space and we only inject energy be-tween k = 6.28 and k = 12.56. The amplitude of thesemodes follows a -5/3 law as: P (k) ∝ k−5/3, whilethe phases are calculated for an Ornstein−Uhlenbeckprocess

xt = fxt−∆t + σ√

(1− f2)Zn,

such that a smoothly varying driving field with fi-nite correlation time is obtained. The parameters inthe Ornstein−Uhlenbeck process are identical to theones adopted by Bauer & Springel (2012) and Price(2012a) as well as Hopkins (2013). A Helmholtz de-composition in k-space is applied before the Fouriertransformation to yield a pure solenoidal componentin the driving force field. At every time step, be-fore the hydro force calculation, this driving routineis called to compute the acceleration. The magni-tude of the driving force is adjusted such that ther.m.s. velocity is at ∼ 0.2cs once the system reachesa quasi-equilibrium state. Besides the routine to cal-culate the driving force, we have also included a func-tion to measure the velocity power spectrum on the

fly. The turbulent velocity power spectrum is calcu-lated whenever a new snapshot is written. We usethe “nearest neighbor sampling” method for our es-timate. Finally the 1D power spectrum is obtainedby averaging the velocity power with each componentin fixed |k| bins. We use the time-averaged velocitypower spectra between t = 10 and t = 20 for thefollowing comparison.

We emphasize that our SPH code is able to giveimproved performance in this test over the previousstudies of Bauer & Springel (2012) and Price (2012b).A snapshot of the velocity field is shown in Figure 7in a high resolution test with 2563 particles. OurSPH code is able to resolve finer structures comparedto the other two studies noted. The velocity fieldis qualitatively closer to the one simulated with gridbased codes. This is a result of the use of the artificialviscosity switch proposed by Cullen & Dehnen (2010)and the Wendland C4 smoothing kernel.

Four different resolutions, 643, 803, 963 and 1283,are used for a resolution study. We actually fix thesmoothing length h for all the simulations based onNnb = 200 for the 643 resolution run and subse-quently scale Nnb for the other three cases. Thisis done to directly test the effect of varying Nnb onthe hydrodynamics by controlling the same resolutionlength h so that the smoothing error is fixed. Theprojected velocity maps of these four resolutions areshown in Figure 8. The velocity fields are similar toone another between these four simulations.

In Figure 9, we compare the velocity power spec-trum for these four resolutions. We multiply thepower spectrum by k5/3 so that the inertial rangedescribed by P (k) ∝ k−5/3 shows up as a horizontalline. The power spectrum shows good agreement withthe k−5/3 law on large scales. The velocity power goesdown significantly below that once the wave numberk is above 40. However, there is significant powercontained on the smallest scales. The spatial extentwhere neighbor particles are searched corresponds tok ∼ 110, which is the range where the turn-over isobserved. This turn-over can be seen as the resultof dissipation of velocity noise on the resolution scale(Hopkins 2013).

Improvements with varying Nnb in this test prob-lems can be seen. The effect of varying Nnb on thevelocity power spectrum is visible on both large andsmall scales. With more neighbors, the velocity fieldon the kernel scale (smoothing length h) is better reg-ulated as the power is reduced by an order of magni-tude. This can be also seen in that the turn-over pointfor higher resolution increasingly shifts to a higher kvalue. On large scales where the modes are well re-solved, more neighbors also help to bring the powerspectrum into better agreement with the k−5/3 law.This can be verified by comparing the red and pur-ple lines for 1283 and 643 in Figure 9. The varianceon the power spectrum between k = 10 and k = 30is reduced for the latter case. The velocity powercontained within k = 30 and k ∼ 110 is also sig-nificantly higher in the 1283 run with Nnb = 1600

11

than 643 with Nnb = 200. The result is that thereis increasingly less power in the noisy motion on thekernel scale but more power in a coherent fashion.The conclusion is that Nnb does play an importantrole in this subsonic turbulence test as suggested byBauer & Springel (2012). This also agrees with theconsistency requirement for SPH in order to recoverthe continuous limit that the “discretization errors”also need to be reduced with h→ 0.

Figure 10 shows the value of Q, as defined above,in the turbulence tests at t = 15. We have also over-plotted the trend of Q from the static distribution ofrandom particles and the glass configuration in theprevious section. This allows us to use Q, which de-scribes the deviation from an exact partition of unity,to access the degree of disorder in realistic applica-tions of SPH. In fact, the distribution of Q for thesesix snapshots are between the two static cases. Thetrend of Q with respect to Nnb, however, is quite shal-low among these dynamical cases, indicating a N−0.5

nbtrend. SPH is not able to re-order the particles to adistribution with the degree of order characterized bythe glass configuration. The degree of disorder couldbe much higher in post-shock flows frequently encoun-tered in astrophysical simulations than what we havefound in these weakly compressible turbulence flowswith SPH since the density variation is much higherin the former.

Fig. 7.— . A high resolution subsonic turbulence test with2563 particles showing the magnitude of the velocity field att = 10.0 for a thin slice in the box. A cross comparison betweenthis figure and the ones in Bauer & Springel (2012) and Price(2012b) at the same resolution shows that our SPH code is ableto resolve the finest structures among all three. The velocityfield is qualitatively closer to the one simulated with a gridbased code. This is a result of the use of the artificial viscosityswitch proposed by Cullen & Dehnen (2010) and the WendlandC4 smoothing kernel. Nnb = 200 is used in this example.

7. CONCLUSIONS

Fig. 8.— Projected maps of velocity field for four subsonicturbulence tests. The total number of SPH particles is in-creased from 643 to 1283 while the number of number of neigh-bors Nnb is also increased accordingly in order to have the samesmoothing length h for all the simulations.

Fig. 9.— The velocity power spectrum in a driven subsonicturbulence test at different resolutions. The velocity power ismultiplied by k5/3, where k is the wave number, such thatthe inertial range will show up as a horizontal line. Differenttotal numbers of SPH particles are used while the number ofneighbors are set up such that the “smoothing length” h is thesame for all the simulations. Even with the same nominal “h-resolution”, the 1283 run shows an improvement over the 643

run, especially for the location of the point of turn-over fromthe minimum on small scales and the increased velocity poweron large scales.

The main findings of our study are as follows:

• We argue that the requirements of consistencyand self-consistency within the SPH frameworkrequire that N →∞, h→ 0 and Nnb →∞ need

12

Fig. 10.— The distribution ofQ =∑mb/ρbWij(ha) for sev-

eral snapshots at the same t in the subsonic turbulence tests inorder to access the degree of disorder in realistic SPH applica-tions. The trend of Q from the static random particles and theglass configuration are also over-plotted here. The distributionof Q for these six snapshots is between the two static cases.The trend of Q with respect to Nnb, however, is quite shallow

in these dynamical cases. These points roughly follow a N−0.5nb

trend. SPH is not able to reorder the particles to a distributionwith the degree of order characterized by a glass configuration.

to be satisfied to obtain the true continuum be-havior of a flow. We summarize the errors fromthe smoothing step and from the discretizationstep in the SPH density estimate and proposea power-law scaling between the desired Nnbgiven the total number of SPH particles N in or-der to have a consistent and convergent scheme.The dependence Nnb ∝ [N0.40, N0.57] is derivedbased on a balance between these two types oferrors for commonly used smoothing kernels.

• We verify the error dependence on Nnb for thediscretization error with a quasi-ordered glassconfiguration and a truly random one. Therange for the power-law index between Nnb andN is obtained based on these two extreme con-ditions. The discretization error in the densityestimate decreases roughly as log(Nnb)/Nnb forthe glass configuration and as N−0.5

nb for the ran-dom distribution.

• We propose a simple method to calculate a pa-rameter Q, which measures the deviation froman exact partition of unity in SPH. The error inthe volume estimate in SPH is further examinedfor a glass configuration and a random one. Inagreement with the behavior of the density es-timate error, this quantity also shows the samedependence on Nnb in these two situations. Inrealistic applications, the error introduced withthe density estimate step itself is on the order ofseveral percent for the Nnb value used in most

published work.

• We use the Wendland C4 function as a newsmoothing kernel to perform dynamical testswith a varying Nnb to avoid the clumping in-stability. We confirm that particle pairs do notform even with a large value of Nnb. The arti-ficial viscosity by Cullen & Dehnen (2010) alsogreatly improves the performance of our SPHcode. As a consequence, the results from ourstudy are not influenced by the numerical arti-facts from these two aspects as in some of theprevious studies.

• A fixed Nnb, which indicates an inconsistentscheme, gives a slow convergence rate as re-ported by previous studies in the two vortexproblems. As shown in the figures, the conver-gence rate levels off and eventually turns overfor sufficiently fine resolution. Also the errorobtained with the cubic spline is larger than forthe Wendland C4 function. The cubic splineactually exaggerates the discretization error asthe Wendland C4 function is much smoother.

• For smooth flows, as seen in the isentropic vor-tex problem, varying Nnb according to the pro-posed power-law dependence can give second-order convergence. The same improvement isseen in the Gresho vortex test, where severaldiscontinuities are included. The convergencerate p = 1.2 with varying Nnb is closer to therate p = 1.4 measured with the second-orderaccurate moving mesh Arepo code for the sameproblem.

• For highly turbulent flows, the velocity noiseposes another challenge in addition to the errorsin the density estimate for the SPH particles toretain an ordered configuration. Though ourcode is able to give a better result in the sub-sonic turbulence test compared with previousstudies, the velocity noise on the kernel scale isalways present. This velocity noise can be re-duced by an order of magnitude with a largerNnb, but the inertial range resolved with SPHcode only shows a slight improvement.

• We measure the randomness of SPH particles inthe above subsonic turbulence tests using thecalculation of Q. This further verifies our as-sumptions when deriving the dependence ofNnbon N . The distribution of SPH particles in thisexample is indeed between a truly random con-figuration and a quasi-ordered case. This ap-proach can be used in general to quantify therandomness in SPH in applications. Unfortu-nately, SPH is not able to rearrange the parti-cles into a highly ordered configuration charac-terized by a low discrepancy sequence.

• The usual practice of using a fixed Nnb indi-cates that the same level of noise will be present

13

even if higher resolution is used. This errorcan be stated as a “zeroth order” term whichis independent of resolution. Moreover, the er-rors introduced can be roughly approximated bya Gaussian distribution around the true valueprovided Nnb is large enough so that the self-contribution term is not significant even forhighly random distributions.

We emphasize that the inconsistency and the er-rors associated with particle disorder is not a spe-cific feature to a particular SPH code but is rather ageneric problem. In fact, the set of equations in Gad-get are fully conservative since they are derived fromthe Euler-Lagrange equations (see Springel & Hern-quist 2002, Springel 2010b and Price 2012b). To ourknowledge, this approach has not been fully adoptedby the community yet. The conservation of mass, lin-ear momentum, angular momentum and total energyhas been long recognized to be the unique advantageover other techniques. Indeed, its conservative na-ture, its robustness and its simple form have made ita popular tool for astrophysical modeling and in otherdisciplines. However, the errors associated with thefinite summation errord, usually termed as “noise”in the SPH literature, and its impact on the conver-gence of SPH has not been well understood so far. Aswe discussed earlier, such an error naturally and in-evitably emerges from the density estimate itself andalso from the momentum equation when evolving thesystem.

Obviously, one promising direction to improve theparticle method is to use a more accurate volume es-timate. It is possible to derive a numerical schemewhich can reproduce a polynomial up to the nth orderpolynomial exactly following the procedure by Liu &Liu (2006). In practice, it is sufficient to eliminate thezero-th order error by renormalization such that thecorrected scheme is second-order accurate by symme-try.

Recently, Hopkins (2014) proposed a new class ofgridless Lagrangian methods which use finite-elementGodunov schemes. These gridless methods dramati-

cally reduce the low-order errors and numerical vis-cosity in SPH method and significantly improve accu-racy and convergence. They show better performancein resolving fluid-mixing instabilities, shocks and sub-sonic turbulence, which is critical in a wide range ofphysical and astrophysical problems.

We caution against equating numerical precisionwith physical accuracy of a simulation because bothphysics and numerics play roles in the modeling of acomplex system. For example, we compared cosmo-logical hydrodynamical simulations of a Milky-Waylike galaxy using both the improved Gadget code(presented in this paper) and the gridless GIZMOcode by Hopkins (2014) with the same initial con-ditions, physics prescriptions and resolutions, andfound that both simulations produced similar galaxyproperties such as disk mass, size and kinematics,although GIZMO resolves the spiral structures bet-ter. These results suggest that the robustness ofastrophysical simulations depends not only on thenumerical algorithms, but also on the physical pro-cesses, as highlighted also in a number of recent pa-pers on galaxy simulations (e.g., Scannapieco et al.2012; Hopkins et al. 2014; Vogelsberger et al. 2013,2014a,b; Schaye et al. 2014).

ACKNOWLEDGEMENTS

We thank Volker Springel, Phil Hopkins, Mark Vo-gelsberger, Jim Stone, Andreas Bauer and DanielPrice for valuable discussions on this work. Wefurther thank Paul Torrey for his contributions tothe experimental setups in the early stage and sug-gestions throughout the project. LH acknowledgessupport from NASA grant NNX12AC67G and NSFgrant AST-1312095, YL acknowledges support fromNSF grants AST-0965694, AST-1009867 and AST-1412719. We acknowledge the Institute For Cyber-Science at the Pennsylvania State University for pro-viding computational resources and services that havecontributed to the research results reported in this pa-per. The Institute for Gravitation and the Cosmos issupported by the Eberly College of Science and theOffice of the Senior Vice President for Research at thePennsylvania State University.

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