Cubature rules for weakly and fully compressible off-lattice Boltzmann methods
Dominik Wilde, Andreas Krämer, Mario Bedrunka, Dirk Reith, Holger Foysi
CCubature rules for weakly and fully compressible o ff -lattice Boltzmann methods Dominik Wilde a,b, ∗ , Andreas Kr¨amer c , Mario Bedrunka b,a , Dirk Reith b,d , Holger Foysi a a Chair of Fluid Mechanics, University of Siegen, Paul-Bonatz-Straße 9-11, 57076 Siegen-Weidenau, Germany b Institute of Technology, Resource and Energy-e ffi cient Engineering (TREE),Bonn-Rhein-Sieg University of Applied Sciences, Grantham-Allee 20, 53757 Sankt Augustin, Germany c Department of Mathematics and Computer Science, Freie Universit¨at Berlin, Arnimallee 6, 14195 Berlin, Germany d Fraunhofer Institute for Algorithms and Scientific Computing (SCAI), Schloss Birlinghoven, 53754 Sankt Augustin, Germany
Abstract O ff -lattice Boltzmann methods increase the flexibility and applicability of lattice Boltzmann methods by decoupling the discretiza-tions of time, space, and particle velocities. However, the velocity sets that are mostly used in o ff -lattice Boltzmann simulationswere originally tailored to on-lattice Boltzmann methods. In this contribution, we show how the accuracy and e ffi ciency of weaklyand fully compressible semi-Lagrangian o ff -lattice Boltzmann simulations is increased by velocity sets derived from cubature rules,i.e. multivariate quadratures, which have not been produced by the Gauß-product rule. In particular, simulations of 2D shock-vortexinteractions indicate that the cubature-derived degree-nine D2Q19 velocity set is capable to replace the Gauß-product rule-derivedD2Q25. Likewise, the degree-five velocity sets D3Q13 and D3Q21, as well as a degree-seven D3V27 velocity set were successfullytested for 3D Taylor-Green vortex flows to challenge and surpass the quality of the customary D3Q27 velocity set. In compress-ible 3D Taylor-Green vortex flows with Mach numbers Ma = { .
5; 1 .
0; 1 .
5; 2 . } on-lattice simulations with velocity sets D3Q103and D3V107 showed only limited stability, while the o ff -lattice degree-nine D3Q45 velocity set accurately reproduced the kineticenergy provided by literature. Keywords:
Lattice Boltzmann method, Cubature, Semi-Lagrangian, Gauss-Hermite quadrature, Compressible
1. Introduction
The lattice Boltzmann method (LBM) [1, 2, 3, 4, 5] is ane ffi cient approach for the numerical simulation of fluids. Com-pared to other methods discretizing the Boltzmann equation,such as discrete-velocity models [6, 7] or (unified) gas-kineticschemes [8, 9], the LBM exhibits three key properties. First,the distribution functions are integrated along characteristicsin time, leading to the well-known 0.5-shift in the relaxationtime [10]. Second, the equilibrium distribution function is ex-pressed as Hermite series, mostly of degree two, but higher forcompressible [11] and thermal cases [12]. Third, the velocityspace is integrated via quadrature rules, in particular by Gauß-Hermite quadratures [12, 13]. By the latter, the unbounded ve-locity space of the Boltzmann equation is expressed by only alimited number of weighted particle velocities, tied together asa velocity set. The type of numerical integration with the un-derlying weight function exp (cid:16) −| x | (cid:17) does not only appear in theLBM, but for instance also in Kalman filters [14]. Therefore,the literature on numerical integration is a good starting pointfor the search of well-suited velocity sets [15, 16, 17]. Morespecifically, the literature specifies two main approaches to de-rive the respective rules: Gauß-product rules and non-productrules [15]. The literature denotes multivariate non-product rulesalso as cubature rules [16, 17]. ∗ Corresponding author: [email protected]
The simpler and widely-used approach of the above-mentioned ones is the Gauß-product rule, which is calculated bythe outer product of a 1D quadrature. For example, the D1Q3velocity set is essentially the 1D Gauß-Hermite quadrature con-structed by the roots of the third order Hermite polynomial [18].By applying the Gauß-product rule, the D1Q3 is turned into aD2Q9 in two dimensions, and by applying the same rule oncemore the D3Q27 velocity set is derived [18]. Since the num-ber of particle velocities exceeds the number of the encodedphysical moments, pruning the D3Q27 velocity set leads to theD3Q15 and D3Q19 sets with di ff erent high-order truncation er-rors [19, 20, 21]. A modification of the regular D3Q27 is the re-cently introduced crystallographic LBM [22], which expressesthe particle distributions in a body centered cubic arrangementof grid points. All above mentioned velocity sets fit into a reg-ular grid, which is another key asset of the well-establishedon-lattice Boltzmann method, with the vector of the particlevelocities ending on one of the neighbouring grid points. Forweakly compressible flows, this collection of velocity sets incombination with a second-order polynomial expansion of theMaxwell-Boltzmann equilibrium is widely used, although notperfect in terms of Galilean invariance, due to well-known er-rors in the stress tensor [23]. However, when a higher degreeof precision for the numerical integration is needed [24, 25], inparticular for compressible [26] or dilute flows [27], the result-ing velocity sets become unfeasible, for two reasons. Firstly,Gauß-product rules su ff er from the ”curse of dimensionality”, Preprint submitted to Journal of Computational Science January 12, 2021 a r X i v : . [ phy s i c s . c o m p - ph ] J a n .e. the number of abscissae rapidly increases especially forhigh spatial dimensions with high degree of precision. Sec-ondly, when derived from Hermite polynomials the degree-nineone-dimensional velocity set D1Q5 holds abscissae, which donot fit on a regular grid, so that an even larger velocity set D1Q7with equidistant abscissae must be used to obtain a su ffi cientlyhigh integration order [28]. These multivariate lattices withequidistant nodes are constructed by solving the orthogonalityrelations [13, 24, 29].Contrary to Gauß-product rules, the abscissae of cubaturerules are not derived by quadratures of lower dimension. Theyare rather found individually for a certain pair of degree ofprecision and dimension. Due to this freedom of velocity dis-cretization, o ff -lattice Boltzmann methods are required to applythem in the LBM. There is only a very limited number of publi-cations, which display simulations using o ff -lattice velocity sets[30, 31, 32].The present manuscript therefore derives velocity sets fromcubature rules and explores them using the semi-Lagrangianlattice Boltzmann method (SLLBM, [33]), with applicationsto both weakly and fully compressible flows. In contrastto Eulerian time integration schemes like finite di ff erence[34, 35], finite volume [36, 37], discontinuous Galerkin LBMschemes[38, 39], the SLLBM inherits the Lagrangian time in-tegration along characteristics from the LBM and recovers theo ff -lattice distribution function values by interpolation. Recentworks [40, 41, 42, 43, 44, 45, 46] provided evidence that theSLLBM is a promising o ff -lattice Boltzmann method for thesimulation of both weakly and compressible flows. However,these works used either on-lattice velocity sets, e.g. D2Q9, orD3Q27, or velocity sets that had been derived by the Gauß-product rule, e.g. D2Q25 from 1D Gauß-Hermite quadrature.This gap is closed by the present article. This work shows thatcubature-based velocity sets significantly enhance o ff -latticeBoltzmann simulations in terms of both e ffi ciency and accu-racy.The remainder of this manuscript is structured as follows.Quadrature and cubature in the LBM are briefly recapitulatedin section 2 with a list of all investigated velocity sets in this ar-ticle. Section 3 details the semi-Lagrangian lattice Boltzmannmodel for both weakly and fully compressible flows. The re-sults section 4 studies three test cases: a compressible two-dimensional shock-vortex interaction and the three-dimensionalTaylor-Green vortex both in the weakly and in the fully com-pressible regime, each of them requiring di ff erent equilibria andvelocity sets. Sections 5 and 6 provide discussion and conclu-sion.
2. Cubature in lattice Boltzmann methods
Following [17], we consider the approximation of an integralof function F I ( F ) = (cid:90) Ω ω ( x ) F ( x ) d x , (1)with Ω ⊂ R D , weight function ω ( x ) ≥ D ≥ C ( F ) = Q − (cid:88) i = w i F ( x i ) , (2)with number of abscissae Q and discrete weights w i .The degree of a quadrature is defined as the largest integer d that yields I ( F ) = C ( F ) for all monomials D − (cid:89) i = x j i i with D − (cid:88) i = j i ≤ d . (3)of degree ≤ d .These cubature formulas are applied to the distribution func-tion of the force-free BGK-Boltzmann equation ∂ f ∂ t + ξ · ∇ f = − λ (cid:0) f − f eq (cid:1) , (4)with (unshifted) relaxation time λ , particle distribution func-tion f , equilibrium distribution function f eq , and particle veloc-ities ξ . To that end, the Hermite moments a ( n ) are gained in theform [12, 13] a ( n ) = (cid:90) R D f H ( n ) d ξ , (5)where H ( n ) is a nth-order Hermite polynomial. The distributionfunction f is expressed as a finite Hermite series f ≈ f N = ω ( ξ ) N (cid:88) n = n ! a ( n ) H ( n ) , (6)with truncation order N [13] with the result that moments oforder M are exactly represented, if M ≤ N . The velocity spaceis discretized by replacing the integral of Eq. (5) by a weightedquadrature leading to a ( n ) = (cid:90) R D ω ( ξ ) f ω ( ξ ) H ( n ) d ξ (7) = Q − (cid:88) i = w i f ( ξ ) H ( n ) ( ξ i ) ω ( ξ i ) = Q − (cid:88) i = f i H ( n ) i , (8)with the replacements f i = w i f ( ξ ) /ω ( ξ i ) and H ( n ) ( ξ i ) = H ( n ) i .From [47] it is known that moments of order M can be ex-actly determined by quadrature or cubature rules of degree d ≥ N + M . In particular, for weakly compressible flows the or-der of the moments is usually limited to N =
2, although N = N =
4. Byovercoming the restriction that abscissae ξ i need to match aregular grid, cubature rules become applicable, provided theyapproximate the weight function ω ( ξ ) = π ) D / e −| ξ | / . (9)In the literature, cubature rules are usually listed for the weightfunction ˜ ω ( x ) = exp (cid:16) −| x | (cid:17) . To obtain a velocity set to approx-imate the moments in Eq. (7), the cubature’s abscissae have to2 − − − y Figure 1: Shape of the D2Q19 velocity set. be scaled by √
2. The lattice speed of sound c s of these velocitysets is—unless otherwise specified—unity c s =
1, which alsoimplies the reference temperature T = • First, the newly introduced degree-nine D2Q19 in com-parison to the D2Q25 derived from Gauß-product rule forfully two-dimensional compressible flows. The shape ofD2Q19 is shown in Fig. 1 • Second, in three dimensions velocity sets for both weaklyand fully compressible flows were chosen. The D3Q13based on a icosahedron was already used in a weaklycompressible finite di ff erence LBM with a triangular mesh[32], whereas the D3Q21 derived from a dodecahedron hasnot been examined so far. Both platonic solids, shown inFig. 2, possess a high geometric isotropy, i.e. the flow in-formation encoded into the distribution function values istransported by regularly spaced abscissae ending on thesurface of a sphere. The weights and abscissae of theD3Q13 and D3Q21 velocity sets are listed in Tables 2 and3, respectively. Both velocity sets were compared to thecustomary D3Q27 on-lattice velocity set and in additionto a degree-seven velocity set, which we call D3V27 [15].This velocity set is not a ff ected by the cubic error in thestress tensor that troubles degree-five velocity sets. Al-though in the same spirit, it is not identical to the velocityset presented by Yudistiawan et al. [31]. • Last, for three-dimensional compressible flows, a recentlyintroduced D3Q45 velocity set was applied [48], based ona cubature rule by Konyaev [49, 50]. This o ff -lattice veloc-ity set was compared in an o ff -lattice Boltzmann simula-tion to the state of the art on-lattice counterparts D3Q103and D3V107 applied to compressible on-lattice Boltzmannsimulations.Table 1 sums up the velocity sets, which were used for thesimulations in this article. The additional notation E QD , d followsthe literature on multivariate quadratures integrating the weightfunction ˜ ω ( x ) = exp (cid:16) −| x | (cid:17) , listing dimension D , degree d and a) b) Figure 2: Shape of icosahedron a) and dodecahedron b), which are the basesfor the velocity sets D3Q13 and D3Q21 used in this work.Table 1: Quadrature and cubature rules utilized in the present article. Notation E QD , n according to Stroud [15] with number of abscissae Q , dimension D , anddegree of precision d . Name E QD , d Remarks SourcesD2Q19 E , Cubature rule by Haege-mans and Piessens [52],Appendix CD2Q25 E , Gauß-product rule from1D degree-nine Gauß-Hermite quadrature [28]D3Q13 E , Derived by the verticesof an icosahedron [15, 32],Table 2D3Q21 E , Derived by the verticesof a dodecahedron [15],Table 3D3Q27 E , Doubly applied Gauß-product rule from 1DGauß-Hermite quadra-ture [53]D3V27 E , Cubature rule by Stroudand Secrest. [15]D3Q45 E , Cubature rule byKonyaev. [49, 50],Appendix ED3Q103 E , On-lattice velocity set [25]D3V107 E , On-lattice velocity set [30]number of abscissae Q . To identify the suitability of the intro-duced velocity sets beforehand, the symmetry conditions in Ap-pendix B were successfully tested [51]. For example, a degree-five velocity set only reproduces the symmetry condition up tofifth order, whereas degree-nine velocity sets will also correctlyreproduce the symmetry conditions up to ninth order.
3. Model description
The lattice Boltzmann equation reads f i ( x + ξ i δ t , t + δ t ) = f i ( x , t ) − τ (cid:16) f i ( x , t ) − f eq i ( x , t ) (cid:17) (10)with relaxation time τ = ν/ ( c s δ t ) + . ν , and time step size δ t .This work applies the discrete equilibrium distribution func-tion based on an expansion in terms of Hermite polynomials3 able 2: Abscissae ξ i and weights w i for the D3Q13 velocity set, based on anicosahedron [15, 32, 4] with r = (5 + √ / s = (5 − √ /
2. The latticespeed of sound is c s = i w i ξ i / , , , . . . , /
20 (0 , ± r , ± s )5 , . . . , /
20 ( ± s , , ± r )9 , . . . ,
12 1 /
20 ( ± r , ± s , Table 3: Abscissae ξ i and weights w i for the D3Q21 velocity set, based ona dodecahedron [15] with φ = (1 + √ /
2. The lattice speed of sound c s is c s = √ / i w i ξ i / , , , . . . , /
100 ( ± , ± , ± , . . . ,
12 3 /
100 (0 , ± φ, ± /φ )13 , . . . ,
16 3 /
100 ( ± /φ, , ± φ )17 , . . . ,
20 3 /
100 ( ± φ, ± /φ, f eq , Ni ( x , t ) = w i N (cid:88) n = n ! a ( n ) eq ( x , t ) : H ( n ) i , (11)with ”:” denoting full contraction, w i the discrete weights and N the expansion order. Both the moments a ( n ) eq and the Hermitetensors H ( n ) i are listed in the Appendix. In the weakly com-pressible case the local temperature θ = T / T is set to θ = ρ = Q − (cid:88) i = f i (12) ρ u = Q − (cid:88) i = ξ i f i . (13)Compared to the standard LBM, the SLLBM replaces the node-to-node streaming step by a cell-wise interpolation procedureusing interpolation polynomials ψ of interpolation order p toobtain the departure points, whose locations are dictated by therespective reversed particle velocities f i ( x − δ t ξ i ) , i.e. for all M points x in each cell Ξ : f i ( x , t ) = M (cid:88) j = ˆ f i Ξ j ( t ) ψ Ξ j ( x ) , (14)with ˆ f i Ξ j denoting the distribution function value at the sup-port points in each cell.Gauß-Lobatto-Chebyshev points were used for the distribu-tion of support points in the cells [44, 33]. For the compressible test cases in Sections 4.1 and 4.3 weemployed the recently introduced compressible SLLBM [44]with the following extensions. A second distribution function g is introduced to enable the calculation of flows with vari-able heat capacity ratio γ [54], following the same collide-and-stream algorithm as applied to the distribution function f . g i ( x + ξ i δ t , t + δ t ) = g i ( x , t ) − τ (cid:16) g i ( x , t ) − g eq i ( x , t ) (cid:17) (15)The equilibrium of g is determined by the relation g eq i = θ (2 C v − D ) f eq i , (16)with heat capacity at constant volume C v . To complement Eqs.(12) and (13), the local temperature θ is determined by2 ρ C v θ = Q − (cid:88) i = (cid:32) | ζ i | f i T + g i (cid:33) , (17)with peculiar particle velocity ζ i = ξ i − u .The local speed of sound c ∗ s consequently depends on the localrelative temperature θ and on the heat capacity ratio γ , via c ∗ s = c s (cid:112) θγ. (18)The relaxation time τ in the compressible case is dependent onthe local pressure P , i.e. τ = µ/ ( c s δ t P ) + . µ . In addition, a quasi-equilibrium approach was usedto vary the Prandtl number Pr. More details about the com-pressible SLLBM solver are listed in [44]. The NATriuM solverwas used for the o ff -lattice Boltzman simulations [45], which isbased on the finite element library deal.ii [55]. The on-latticeBoltzmann simulations were performed using the lettuce soft-ware [56, 57], being based on the GPU-accelerated machinelearning toolkit PyTorch [58].
4. Results
The present work considers three test cases to evaluate theintroduced velocity sets: the two-dimensional shock-vortex in-teraction, comparing the degree-nine velocity sets D2Q19 andD2Q25. This flow challenges the solver by acoustic emis-sions by the vortex downstream the shock. Next, the weaklycompressible three-dimensional Taylor-Green vortex [59] wassimulated by various 3D velocity sets, e.g. D3Q13, D3Q21,D3V27 and D3Q27 to explore the capability of the velocity setsto deal with underresolved flows. Finally, simulations of fullycompressible 3D Taylor-Green vortex flows were explored withMach numbers 0.5, 1.0, 1.5, and 2.0. This test case exhibitsshocklets and viscous e ff ects and features a standardized ini-tialization. The on-lattice simulations were performed using theD3Q103 and D3V107 velocity sets, while the o ff -lattice simu-lations were run by the D3Q45 velocity set. The compressible two-dimensional shock-vortex interactionwas tested as a first test case to compare the two velocity sets42Q19 and D2Q25. This benchmark was intensively stud-ied by Inoue and Hattori [60], with the following setup. Tworegions with Ma a = . b determined by Rankine-Hugoniot conditions are divided by a steady shock. TheReynolds number is defined as Re = c ∗ s , ∞ R /ν ∞ with subscript ∞ denoting the inflow conditions. The flow field of the vortexis given by u θ ( r ) = (cid:112) γ T Ma v r exp((1 − r ) / , (19)where Ma v denotes the Mach number of the vortex. The initialpressure and density field are P ( r ) = γ (cid:32) − γ −
12 Ma v exp(1 − r ) (cid:33) γ/ ( γ − (20)and ρ ( r ) = (cid:32) − γ −
12 Ma v exp(1 − r ) (cid:33) / ( γ − . (21)The resolution was 256 ×
256 cells with polynomial order p = × R × R . Initially the centre of the vortex was located at x v = x shock =
30. Likewise as in [44]we used a stretched grid to spatially resolve the shock region.The time step size was δ t = .
002 with characteristic time t (cid:48) = R / c ∗ s , ∞ . The Prandtl number was Pr = .
75 in all cases. Fig. 3depicts the density contours at t = v = .
5, Re =
400 and provides evidence that there is no significant di ff erencebetween the simulations run by the D2Q19 and D2Q25 velocitysets.The evaluation of the sound pressure at Ma v = .
25 andRe =
800 confirms this observation, being shown in Fig. 4. Thesound pressure ∆ P = ( P − P B ) / P B , with subscript B denotingthe pressure downstream, was measured at time t = ◦ inclined radius with respectto the x-axis. As expected, the sound pressure of both SLLBMsimulations coincided well with the reference solution by Inoueand Hattori [60]. These findings approve the cubature-derivedD2Q19 as a potent alternative for compressible o ff -lattice sim-ulations to the D2Q25 being based on the Gauß-product ruleused in previous works [41, 44]. The three-dimensional Taylor Green vortex served as a testcase to apply the degree-five velocity sets D3Q13, D3Q21, anddegree-seven D3V27 in comparison to the customary degree-five D3Q27 using an isothermal configuration θ = N = N =
3. The Mach number was Ma = .
1. Thefollowing initial conditions were used u ( x , = sin( x )cos( y )cos( z ) − cos( x )sin( y )cos( z )0 , (22) Figure 3: Density contours of the shock-vortex interaction with Ma v = . Re =
400 in the range ρ ∈ [0 . , .
55] in 119 steps with velocity setsD2Q19 (top) and D2Q25 (below). No significant di ff erence is visible despitethe D2Q19’s 24 percent reduction in computational cost. P ( x , t = =
116 (cos(2 x ) + cos(2 y ))cos(2 z + , (23)The kinetic energy of the flow is defined as k = π ) π (cid:90) (cid:90) (cid:90) | u | dxdydz (24)and the enstrophy as a measure of the dissipation with respectto the resolved scales [61] E = π ) π (cid:90) (cid:90) (cid:90) ( ∇ × u ) dxdydz . (25)First, a well-resolved flow was tested by setting the Reynoldsnumber to Re = cells with order of finite element p =
4, resulting in 128 gridpoints. The velocity sets were scaled in such a way that thetime step size was δ t = . dk / dt and the scaled enstrophy ν E of the fourfocused velocity sets in comparison to the reference by Brachet5 2 4 6 8 10 12 r/R − . − . − . − . . . . ∆ P Inoue and Hattori, 1999SLLBM D2Q19SLLBM D2Q25
Figure 4: Comparison of D2Q19 and D2Q25 in terms of the sound pressure ∆ P = ( P − P B ) / P B of the shock-vortex interaction with Re = , Ma v = . t (cid:48) = (cid:48) measured from the center of the vortex along the radius with135 ◦ with respect to the x-axis. The SLLBM match the reference by Inoue andHattori [60] well. et al. [59]. It is shown that all velocity sets match the refer-ence solution for both dissipation and scaled enstrophy. Thus incase of well-resolved simulations, even the lean D3Q13 veloc-ity set commended itself as a good choice for three-dimensionalflow simulations. In case of the underresolved simulations withReynolds number Re = δ t = . ff erent though, displayed in Fig. 6showing di ff erent evolutions of dissipation and scaled enstro-phy for each of the velocity sets. Fig. 7 compares the scaledenstrophies revealing that the enstrophy of the common D3Q27most drastically underestimated the enstrophy in this config-uration. The degree-seven D3V27 as well as the degree-fiveD3Q21 both more accurately predict the scaled enstrophy. As a final test case the compressible 3D Taylor-Green vortexwas simulated. Recently, this test case has been by extensivelyinvestigated by Peng and Yang [62]. The Reynolds number was Re = γ = .
4, and the Prandtlnumber was Pr = .
75. The initial velocity field correspondsto Eq. (22), whereas the pressure is obtained by P = ρ T with T =
1, while the density is given by ρ ( x , t = = . + C
16 (cos(2 x ) + cos(2 y ))cos(2 z + . (26)This relation of temperature and density corresponds to the con-stant temperature initial condition (CTIC) detailed in Peng andYang [62]. The nominator C can be either set to C = Ma orto C =
1; this work made use of the latter case. Table 4 liststhe time step sizes of the respective simulations with the veloc-ity sets D3Q103, D3V107, and D3Q45. The time step sizes ofthe on-lattice velocity sets were dictated by the configuration, 0 . . . . . . . . . . D i ss i p a ti on a) D3Q13 Dissipation – dk / dt Scaled Enstrophy ν E Reference0 . . . . . . . . . . D i ss i p a ti on b) D3Q210 . . . . . . . . . . D i ss i p a ti on c) D3V270 . . . . . . . . . . D i ss i p a ti on d) D3Q27 Figure 5: Dissipation and scaled enstrophy over time from simulations of thethree-dimensional Taylor-Green vortex at Reynolds number Re =
400 usingthe velocity sets D3Q13 a), D3Q21 b), D3V27 c), and D3Q27 d). For thiswell-resolved simulation at N =
128 both the dissipation − dk / dt and the scaledenstrophy ν E matched the reference solution well for all velocity sets consid-ered. . . . . . . . . D i ss i p a ti on a) D3Q13– dk / dt ν E Reference0 . . . . . . . . D i ss i p a ti on b) D3Q210 . . . . . . . . D i ss i p a ti on c) D3V270 . . . . . . . . D i ss i p a ti on d) D3Q27 Figure 6: Dissipation − kE / dt and scaled enstrophy over time from simulationsof the three-dimensional Taylor-Green vortex at Reynolds number Re = N =
128 reveals di ff erences for the dissipationand scaled enstrophy for the respective velocity sets. . . . . . S ca l e d E n s t r ophy ν E D3Q13D3Q21D3V27D3Q27Ref.
Figure 7: Comparison of the scaled enstrophies ν E shown in Fig. 6.Table 4: Time step sizes of the compressible Taylor-Green vortex simulations.Unstable simulations marked by asterisks. Mach number D3Q103 D3V107 D3Q45Ma = . = . = . = . obtained by the velocity sets D3Q103, D3V107and by the SLLBM D3Q45 simulation. It displays that the ref-erence by Peng and Yang was well captured for Mach numbersMa = . = . = . = .
5. Discussion
The discretization of the velocity space by quadrature is akey asset of the lattice Boltzmann method. However, the cou-pling of the velocity space discretization with the spatial andtemporal discretization in on-lattice Boltzmann methods is ob-structive to find su ffi ciently small velocity sets especially forhigh quadrature order. Even a recent systematic study as done7 k i n a) D3Q103on-latticeRef.Ma=0.5Ma=1.0Ma=1.5Ma=2.0 E k i n b) D3V107on-latticeRef.Ma=0.5Ma=1.0Ma=1.5Ma=2.00 2 4 6 8 10t0 . . . . . . . . E k i n c) D3Q45SLLBMRef.Ma=0.5Ma=1.0Ma=1.5Ma=2.0 Figure 8: On-lattice simulations utilizing the D3Q103 a) and the D3V107 incomparison to the SLLBM D3Q45 simulation c). All velocity sets accuratelyreproduced the kinetic energies for the medium Mach numbers Ma = . = .
0, while the on-lattice simulations failed for the higher Mach numbersMa = . = .
0. By contrast, the flexible time step size enabledsuccessful SLLBM simulations even at the highest Mach numbers. by Spiller and D¨unweg did not yield smaller velocity sets thanthe already known D3Q103 [64], whose number of abscissaetherefore appears to be the lower limit at this quadrature de-gree with equidistant nodes. The enormous size of these high-degree sets prevented their application in actual simulations, atleast up to the present study. The simulations of the fully com-pressible Taylor-Green vortex in the present work showed on-lattice velocity sets are generally capable for compressible sim-ulations, but they lack stability at Mach numbers beyond unity.The biggest issue, from our point of view, is the fixed time stepsize of most compressible on-lattice Boltzmann methods. Anexception to this are hybrid lattice Boltzmann methods, whichsolve the energy equation by finite volume or finite di ff erencemethods [65, 66]. These methods are capable to adjust the timestep size by changing the reference temperature, but they su ff erfrom restrictions in the stability regions [67]. Despite the suc-cesses of recent works [26, 42, 68, 69, 70], the applicability ofcompressible on-lattice Boltzmann methods remains vague. Al-though shifted stencils have proven to extend the Mach numberrange of on-lattice Boltzmann methods [71, 72] in certain sit-uations, the Taylor-Green vortex at high Mach numbers wouldstill not be stable with static reference frames. Solely dynamicshifts of the reference frame as presented by Coreixas and Latt[73] might be an alternative for on-lattice Boltzmann methods,although they cause additional computational costs and requirefurther investigation.Compared to on-lattice Boltzmann methods, the discretiza-tion of the velocity space is a largely unexplored field ofresearch in o ff -lattice Boltzmann methods. We were able,however, to identify a large potential in equipping o ff -latticeBoltzmann methods by velocity sets with significantly di ff erentshapes. Instead of deriving stencils case-by-case, the researchon multivariate quadrature rules yielded suitable cubature rulesas templates for velocity sets. Consequently, this work’s mainpurpose was to explore these velocity sets by o ff -lattice Boltz-mann simulations.The results in Section 4 clearly indicate that the spatial free-dom obtained in the collocation of abscissae, as done in cu-bature rules, reduces the number of discrete velocities signif-icantly. Simultaneously, the computational costs halved, nar-rowing the gap between on-lattice and o ff -lattice Boltzmannmethods in e ffi ciency. For example, the D2Q19 degree-ninevelocity set is approximately half the size of its on-lattice coun-terpart D2V37 with equal quadrature order [24]. The di ff erencebetween the D3Q45 and the D3Q103 velocity set is even larger,but the size of the D3Q45 is also superior compared to a re-cently utilized degree-nine D3Q77 o ff -lattice velocity set forcompressible finite volume LBMs [74]. The D3Q45 simula-tions of the Taylor-Green vortex proved to be stable and accu-rate even for high Mach number flows. This can be attributedto the temporal discretization of o ff -lattice Boltzmann methods,which is independent of the discretization of space and velocityspace.Di ff erences were also identified in terms of accuracy. TheD3Q13 and D3Q21 based on icodahedra and dodecahedra,shown in Fig. 2, proved to be both e ffi cient and accurate o ff -lattice velocity sets for weakly compressible flows. Both ve-8ocity sets showed a better enstrophy agreement with the ref-erence solution in underresolved flow simulations. The reasonfor this are the fewer advection equations that have to be solvedconnected with fewer interpolation steps, leading to less artifi-cial di ff usion. Moreover, due to their derivation from platonicsolids, these velocity sets exhibit a high geometric isotropy,which is favourable for the simulation of turbulent flows. Inaddition, a degree-seven velocity set D3V27 with the aim toeliminate the cubic errors in the model proved to be more ac-curate than the on-lattice D3Q27, but coming at a comparableprice.To sum up, o ff -lattice Boltzmann methods relax the LBM interms of the velocity space discretization. The development ofsparse numerical cubatures is still ongoing. The present worklays the foundations to directly utilize new cubature rules inCFD simulations with o ff -lattice Boltzmann methods.
6. Conclusion
This paper studied the utilization of cubature rules in o ff -lattice Boltzmann methods for weakly and fully compressibleflows. The deduced o ff -lattice velocity sets presented in thisarticle reduce the number of discrete velocities and increase theaccuracy of the simulation. In addition, fully compressible o ff -lattice simulations with degree-nine velocity sets feature betterstability compared to the corresponding on-lattice counterparts.Taken together, cubature rule-derived o ff -lattice velocity setsare a viable alternative to the customary on-lattice Boltzmannvelocity sets and should have priority in o ff -lattice Boltzmannsimulations. Acknowledgements
We gratefully acknowledge support for D.W. by German Re-search Foundation (DFG) project FO 674 / Appendix A. Hermite polynomials and moments
The scaled Hermite polynomials with ˆ ξ i = ξ i / c s up to fourthorder read H i (0) = H i α (1) = ˆ ξ i α c s H i αβ (2) = ˆ ξ i α ˆ ξ i β − δ αβ c s H i αβγ (3) = ˆ ξ i α ˆ ξ i β ˆ ξ i γ − ( ˆ ξ i α δ βγ + ˆ ξ i β δ αγ + ˆ ξ i γ δ αβ ) c s H i αβγδ (4) = ˆ ξ i α ˆ ξ i β ˆ ξ i γ ˆ ξ i δ − T i + ( δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ) c s , with T i = ˆ ξ i α ˆ ξ i β δ γδ + ˆ ξ i α ˆ ξ i γ δ βδ + ˆ ξ i α ˆ ξ i δ δ βγ + ˆ ξ i β ˆ ξ i γ δ αδ + ˆ ξ i β ˆ ξ i δ δ αγ + ˆ ξ i γ ˆ ξ i δ δ αβ . The moments of the Boltzmann equation up to fourth orderare a (0) eq = ρ a (1) α, eq = ρ u α a (2) αβ, eq = ρ ( u α u β + T ( θ − δ αβ ) a (3) αβγ, eq = ρ (cid:104) u α u β u γ + T ( θ − δ αβ u γ + δ αγ u β + δ βγ u α ) (cid:105) a (4) αβγδ, eq = ρ [ u α u β u γ u δ + T ( θ − δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ) T ( θ − + δ αβ u γ u δ + δ αγ u β u δ + δ αδ u β u γ + δ βγ u α u δ + δ βδ u α u γ + δ γδ u α u β )] . Appendix B. Symmetry conditions
The symmetry conditions from zeroth to ninth order are (cid:88) i w i = (cid:88) i w i ξ i α = (cid:88) i w i ξ i α ξ i β = c s δ αβ (cid:88) i w i ξ i α ξ i β ξ i γ = (cid:88) i w i ξ i α ξ i β ξ i γ ξ i δ = c s ∆ αβγδ (cid:88) i w i ξ i α ξ i β ξ i γ ξ i δ ξ i (cid:15) = (cid:88) i w i ξ i α ξ i β ξ i γ ξ i δ ξ i (cid:15) ξ i ζ = c s ∆ αβγδ(cid:15)ζ (cid:88) i w i ξ i α ξ i β ξ i γ ξ i δ ξ i (cid:15) ξ i ζ ξ i η = (cid:88) i w i ξ i α ξ i β ξ i γ ξ i δ ξ i (cid:15) ξ i ζ ξ i η ξ i θ = c s ∆ αβγδ(cid:15)ζηθ (cid:88) i w i ξ i α ξ i β ξ i γ ξ i δ ξ i (cid:15) ξ i ζ ξ i η ξ i θ ξ i ι = , with ∆ αβγδ = δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ , as well as ∆ αβγδ(cid:15)ζ = δ αβ ∆ γδ(cid:15)ζ + δ αγ ∆ βδ(cid:15)ζ + δ αδ ∆ βγ(cid:15)ζ + δ α(cid:15) ∆ βγδζ + δ αζ ∆ βγδ(cid:15) , ∆ αβγδ(cid:15)ζηθ = δ αβ ∆ γδ(cid:15)ζηθ + δ αγ ∆ βδ(cid:15)ζηθ + δ αδ ∆ βγ(cid:15)ζηθ + δ α(cid:15) ∆ βγδζηθ + δ αζ ∆ βγδ(cid:15)ηθ + δ αη ∆ βγδ(cid:15)ζθ + δ αθ ∆ βγδ(cid:15)ζη . References [1] Guy R. McNamara and Gianluigi Zanetti. Use of the boltzmann equationto simulate lattice-gas automata.
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