A Discontinuous Galerkin Fast Spectral Method for the Full Boltzmann Equation with General Collision Kernels

Transcription

1 A Discontinuous Gaerkin Fast Spectra Method for the Fu Botzmann Equation with Genera Coision Kernes Shashank Jaiswa a, Aina A. Aexeenko a, Jingwei Hu b, Purdue University, West Lafayette, IN 4797, USA a Schoo of Aeronautics and Astronautics b Department of Mathematics Abstract The Botzmann equation, an integro-differentia equation for the moecuar distribution function in the physica and veocity phase space, governs the fuid fow behavior at a wide range of physica conditions, incuding compressibe, turbuent, as we as fows invoving further physics such as non-equiibrium interna energy exchange and chemica reactions. Despite its wide appicabiity, deterministic soution of the Botzmann equation presents a huge computationa chaenge, and often the coision operator is simpified for practica reasons. In this work, we introduce a highy accurate deterministic method for the fu Botzmann equation which coupes the Runge-Kutta discontinuous Gaerkin (RKDG) discretization in time and physica space (Su et a., Comp. Fuids, 19 pp , 15) and the recenty deveoped fast Fourier spectra method in veocity space (Gamba et a., SIAM J. Sci. Comput., 39 pp. B658 B674, 17). The novety of this approach encompasses three aspects: first, the fast spectra method for the coision operator appies to genera coision kernes with itte or no practica imitations, and in order to adapt to the spatia discretization, we propose here a singuar-vaue-decomposition based agorithm to further reduce the cost in evauating the coision term; second, the DG formuation empoyed has arbitrary order of accuracy at eement-eve, and has shown to be more efficient than the finite voume method; thirdy, the eement-oca compact nature of DG as we as our coision agorithm is amenabe to effective paraeization on massivey parae architectures. The sover has been verified against anaytica Bobyev-Krook-Wu soution. Further, the standard benchmark test cases of rarefied Fourier heat transfer at Knudsen numbers Kn =.4745 and 1.58, Couette fow at Kn =.5 and 1., osciatory Couette fow at Kn = 1., and thermay driven cavity fow at Kn = 1. and.5 have been studied and their resuts are compared against direct simuation Monte Caro (DSMC) soutions with equivaent moecuar coision modes. Keywords: rarefied gas dynamics, the fu Botzmann equation, deterministic sover, discontinuous Gaerkin method, fast Fourier spectra method. 1. Introduction In micro/rarefied gas fows, the gas moecue wa-surface interactions ead to the formation of Knudsen ayer (KL): a oca thermodynamicay non-equiibrium region extending O(λ) from the surface, where λ is the gas mean free path (MFP) [3]. The Knudsen number (Kn) is defined as λ/h, where H is the characteristic ength of the system. The cassica constitutive reations of the Navier-Stokes-Fourier equations fai to predict noninear behavior in the KL and deviations are significant in the sip (1 3 < Kn < 1 1 ) and transition fow regimes (1 1 < Kn < 1) [3 5]. The Botzmann equation, an integrodifferentia equation for the moecuar distribution function in the physica and veocity phase space, governs the fuid fow behavior for a wide range of Knudsen numbers and physica conditions, incuding compressibe, turbuent, as we as fows Corresponding author. Emai addresses: jaiswa@purdue.edu (Shashank Jaiswa), aexeenk@purdue.edu (Aina A. Aexeenko), jingweihu@purdue.edu (Jingwei Hu) invoving further physics such as non-equiibrium interna energy exchange and chemica reactions. Accurate physica modes and efficient numerica methods are required for soving the Botzmann equation so as to predict the non-equiibrium phenomenon encountered in such rarefied fows. The approaches for numerica soution of the Botzmann equation date back to as eary as 194s [6] using, for exampe, the now widey used direct simuation Monte Caro (DSMC) method [7, 8]. The DSMC method, based on the kinetic theory of diute gases, modes the binary interactions between partices stochasticay. However, it is this stochastic nature of the method that introduces high statistica noise in ow-speed fows, and imposes strict constraints on ce-size and time-step. Moreover, the forma accuracy of partice time-stepping is inear. The stiffness properties of the Botzmann equation further aggravates the time-step constraints. To overcome these imitations, improved partice-based approaches have been proposed [9], incuding hybrid continuum/partice sovers [1, 11], variance reduction methods [1], and simpified Bernoui trias [13]. Preprint submitted to Journa of - March 8, 18

2 It is to be noted that the assertion that DSMC soves the actua fu Botzmann equation is not stricty vaid. Indeed, the DSMC method can be derived rigorousy as the Monte Caro soution of the N-partice master kinetic equation [14]. Wagner et a. [15] estabished convergence proof for Bird s DSMC method for the Botzmann equation in the imit of infinite number of partices, N. Moreover, the proof has inherent assumptions on the boundedness of the coision operator which is ceary highighted in Wagner s work (see section 5 in [15]). The deterministic soutions based on discretization of governing differentia equations on representative grids is centra to computationa fuid dynamics (CFD). However, the mutidimensiona nature of the Botzmann coision integra becomes a botteneck resuting in excessive use of time and computing resources. To bypass this issue, simpified Botzmann equation variants such as inearized-botzmann (LB) [16], Bhatnagar-Gross-Krook (BGK) [17], and eipsoida Bhatnagar-Gross- Krook (ES-BGK) [18] are used. These simpified modes perform better at ow Knudsen number fows in sip and eary transition regimes. Yet they often fai to capture the physics at high Knudsen numbers as we as for diffusion dominated fows at even ow Knudsen numbers (see [19, ]). This forces us to switch back to DSMC. Over the past decades, the deterministic methods that sove the fu Botzmann equation have undergone considerabe deveopment. Without being exhaustive, we refer to [1, ] for a comprehensive review. In this work, we empoy the recenty deveoped fast Fourier spectra method [] to sove the Botzmann coision operator. Compared with other deterministic methods such as the discrete veocity modes (DVM), the Fourier spectra method can provide significanty more accurate resuts with ess numerica compexity; compared with DSMC, it produces smooth, noise-free soutions and can simuate ow-speed fows such as those encountered often in micro-systems. On the other hand, the Fourier spectra method is sti computationay demanding, as it requires O(N 6 ) memory to store precomputed weights and has O(N 6 ) numerica compexity, where N is the number of discretization points in each veocity dimension [3, 4]. The main contribution in [] is a ow-rank strategy to acceerate the direct Fourier spectra method so that it requires ony O(MN 4 ) memory to store precomputed weights (no precomputation is needed in certain cases) and has O(MN 4 og N) compexity, where M is the number of discretization points on the sphere and M N. Furthermore, the fast method appies to arbitrary coision kernes and can be easiy extended to genera coision modes incuding the muti-species and ineastic Botzmann operators. We mention that there is another ine of research that deveops the fast Fourier spectra method based on Careman representation of the coision operator [5]. The compexity of the method is O(MN 3 og N). However, its appicabiity is imited to hard sphere moecues. The method has been extended to genera coision kernes in [6], but it reies on an ad hoc modification of the kerne and parameter fitting. A of the former approaches have reied on ow-order (up to second-order) finite voume (FV) or finite-difference (FD) methods for spatia discretization of the Botzmann equation. In this work, we empoy the discontinuous Gaerkin (DG) method for spatia discretization, a cass of high order method widey used for time dependent muti-dimensiona hyperboic equations [7 31]. Compared to high-order FV/FD methods, DG provides easy formuation on arbitrary meshes, high-order fux reconstruction, straightforward impementation of boundary conditions, high-order accuracy, as we as strong inear scaing on parae processors due to the compactness of the scheme [31]. DG has been empoyed for soving the BGK and ES-BGK equations for D/1D [3], and D [1, 33] fow probems. To the best of our knowedge, DG discretization in physica space hasn t been appied for soving the fu Botzmann equation ti date. To summarize, we present a 1D/D-3V fu Botzmann equation sover by couping the Runge-Kutta discontinuous Gaerkin (RKDG) discretization in time and physica space [1] and the fast Fourier spectra method in the veocity space []. The method is high-order in both physica space and time, and spectray accurate in veocity space. There are stricty no ad-hoc adjustments/assumptions or parametric fitting invoved in our present formuation for soving the coision operator. The RKDG formuation on unstructured grids for the fu Botzmann is found out for the first time. Moreover, our singuar vaue decomposition (SVD) variant of the agorithm for evauating weak form of the coision term is nove and unique to DG formuation. In the section that foows, we give a brief overview of the Botzmann equation and the coision kerne invoved. Section 3 presents a brief overview of the DG method in genera, and describes weak DG formuation of the Botzmann equation, incuding the direct, and SVD variant of the proposed agorithm for evauating the coision term. Numerica experiments and resuts are discussed in section 4. Concuding remarks are presented in Section 5. A brief description of the fast Fourier spectra method is provided in the Appendix.. The Botzmann equation The Botzmann equation for a singe-species, mono-atomic gas without externa forces can be written as (cf. [34]) f t + c x f = Q( f, f ), t, x Ω x, c R 3, (1) where f = f (t, x, c) is the one-partice distribution function of time t, position x, and partice veocity c. f dx dc gives the number of partices to be found in an infinitesima voume dx dc centered at the point (x, c) of the phase space. Q( f, f ) is the coision operator describing the binary coisions among partices, which acts ony in the veocity space: Q( f, f )(c) = B(c c, σ)[ f (c ) f (c ) f (c) f (c )] dσ dc, R 3 S () where (c, c ) and (c, c ) denote the pre and post coision veocity pairs, which are reated through momentum and energy conservation as c = c + c + c c σ, c = c + c c c σ, (3)

3 with the vector σ varying over the unit sphere S. The quantity B ( ) is the coision kerne depending ony on c c and the scattering ange χ (ange between c c and c c ), and can be expressed as B(c c, σ) = B( c c, cos χ), cos χ = σ (c c ). (4) c c Given the interaction potentia between partices, the specific form of B can be determined using the cassica scattering theory (cf. [35]): B( c c, cos χ) = c c Σ( c c, χ), (5) where Σ is the differentia cross-section given by Σ( c c, χ) = b b sin χ χ, (6) with b being the impact parameter. With a few exceptions (e.g. hard sphere moecues), the expicit form of Σ can be hard to obtain since b is reated to χ impicity. To avoid this compexity, phenomenoogica coision kernes are often used in practice with the aim to reproduce the correct transport coefficients. Koura et a. [36] introduced a scattering mode so caed as Variabe Soft Sphere (VSS) by assuming an expicit cosine dependence between scattering ange and impact parameter, defined as: χ = cos 1 {(b/d) 1/α }, (7) where α is the scattering parameter, and d is the diameter borrowed from Bird s [5] Variabe Hard Sphere (VHS) mode, which is defined as d = d ref [( 4RTref c c ) ω.5 ] 1/ 1. (8) Γ(.5 ω) Here R = k B /m is the gas constant (k B is the Botzmann s constant and m is the singe partice mass), Γ denotes the usua Gamma function, d ref, T ref, and ω are, respectivey, the reference diameter, the reference temperature, and the viscosity index. The diameter d and exponent α are determined so that the transport (viscosity and diffusion) coefficients of VSS are consistent with experimenta data [37, 38]. Substituting the eqns. (6)-(8) into (5), we obtain the genera form of B as B = b ω, α c c (1 ω) (1 + cos χ) α 1, (9) where b ω, α is a constant given by b ω, α = d ref 4 (4RT ref) ω.5 1 Γ(.5 ω) α. (1) α 1 In particuar, the VHS kerne is obtained when ω [.5, 1] and α = 1 (ω = α = 1 corresponds to the Maxwe moecues, and ω =.5, α = 1 to the hard spheres); and the VSS kerne is obtained when ω [.5, 1] and α (1, ]. Note that the approach described above for coision kerne is generay appicabe for arbitrary interaction modes. The corresponding scattering cross-section and ange vaues shoud be used in eqns. (6) and (7) with tabuated poynomia approximations avaiabe, for exampe, for Lennard-Jones interactions in [39] Non-dimensionaization To reduce the parameters, it is convenient to non-dimensionaize a variabes and functions. We first choose the characteristic ength H, the characteristic temperature T, and the characteristic number density n. Then define the characteristic veocity u = RT and the characteristic time t = H /u. Now we rescae t, x, c, and f as foows t = t t, x = x H, c = c u, f = and the coision kerne B as B = then the equation (1) becomes B 1 ω πd ref (4RT ref) ω.5 u (1 ω) f t f, (11) n /u 3, (1) + c x f = 1 Kn Q( f, f ), (13) with the coision operator Q( f, f )( c) = B( c c, cos χ)[ f ( c ) f ( c ) R 3 S f ( c) f ( c )] dσ d c, where B( c c, cos χ) = (14) α ω+α Γ(.5 ω)π c c (1 ω) (1+cos χ) α 1, and the Knudsen number Kn is given by Kn = (15) 1 π n d ref (T ref/t ) ω.5 H, (16) which is the ratio between the MFP and the characteristic ength (consistent to eqn. (4.65) in [5, 8]). Henceforth, we wi aways refer to the non-dimensionaized Botzmann equation (13)-(16) in our presentation, and wi be dropped for simpicity. It is worth emphasizing that athough the coision kerne (9) is adopted in this work for easy comparison with DSMC soutions, the fast spectra method for the coision operator appies to any kernes of the form (4), see Appendix. 3. Discontinuous Gaerkin formuation 3.1. Brief overview The Runge Kutta discontinuous Gaerkin (RKDG) method [7 31] is a cass of finite eement methods couping RK discretization in time and DG discretization in space which provides high-order numericay accurate soutions to governing partia differentia equations. Higher order accuracy is desirabe for simuating fows with strong gradients, dropet coisions as in muti-phase fows, combustion-modeing, reactors,

4 and micro-mechanica systems. RKDG can recover fow properties at the domain boundaries with the same high-order accuracy as in the interior of the domain. In the Botzmann equation simuations, the computationa domain consists of physica and veocity domains. We propose to use the RKDG method in time and physica space and the Fourier spectra method in the veocity space. Hence the veocity space is partitioned using the Cartesian type grid point (with reasons to be expained in section 3.3), and the physica space is spit up into a set of ine segments (in 1D), trianges/quadriateras (in D), and tetrahedras/prisms/hexahedras (in 3D) for instance. In particuar for D grids of quadriatera ces, each ce in the physica space has four faces. The ce connectivity is such that a ce face is either interna and intersects two ces ony, or comprises part of an externa boundary and beongs to singe ce ony. In such a grid system, the DG method is deveoped to sove the Botzmann equation at each veocity grid point c j. Within a given spatia eement i, the distribution function f is approximated as a inear combination of orthogona basis functions φ i (x) as f j i = F i, j φ i (x), (17) =1 where K is the number of unknowns in the eement aso known as oca degree of freedom. The task is to determine the coefficients F i, j of the expansion for a eements. Therefore, the compexity of the probem is proportiona to the number of veocity nodes, the number of spatia eements, the order of basis functions, and the number of time integration steps. Due to the muti-dimensionaity of the probem, and the typica size of phase space considered in the current work (order of miions), parae computation is highy desirabe. In finite eement setting, the information is exchanged between two-adjacent eements using the shared nodes between them. The DG method, in contrast to the cassica finite eement method that reies on goba stiffness matrices, dupicates the vaues that are shared between the eements. To determine a unique vaue at the shared nodes, DG introduces monotone interface fux (as in finite voume method). It is this fux that aows eement-to-eement decouping, maintain unique-soution vaue at shared nodes, and aows for expicit time stepping (See [31]). It is this eement-to-eement decouping and eement oca-nature of the DG method that makes it amenabe to strong scaing on parae processors, and therefore our choice of spatia discretization scheme. 3.. Discretization in the physica space Assume that the Botzmann equation is posed in the domain Ω x with boundary Ω x in the physica space. We decompose Ω x into I variabe sized eements : Ω x I. (18) i=1 4 In each eement, we approximate the function f (t, x, c) by a poynomia of order N p : x : f i (t, x, c) = =1 F i (t, c) φi (x), (19) where φ i (x) is the basis function supported in Di x, K is the tota number of terms in the oca expansion, and F i (t, c) is the eementa degree of freedom. In genera K depends on eementashape. In 1D, K = N p + 1. In D, K = (N p + 1) for quadriatera eements, and K = (N p + 1)(N p + )/ for trianguar eements. In 3D, K = (N p + 1) 3 for hexahedra eements, and K = (N p + 1)(N p + )(N p + 3)/6 for tetrahedra eements. We now present a genera 3D spatia weak DG formuation for the Botzmann equation. Reduction to the D case can be achieved by choosing a two-dimensiona basis, and ignoring the z-axis dependence. Simiary for the 1D case. Time and veocity space are eft as continuous at the moment. We first form the residua by substituting the expansion (19) into the equation (13): R i = =1 φ i t F i + =1 F i c xφ i 1 Kn 1 =1 =1 Q ( F i 1, F i ) φ i 1 φ i, () where we used the quadratic property of the coision operator. We now require that the residua is orthogona to a test functions. In the Gaerkin formuation, the test function is the same as the basis function, thus R i φ i m dx =, 1 m K, (1) in each eement. Substituting () into (1) and appying the divergence theorem, we obtain ( =1 = + 1 Kn φ i m φ i m φ i dx ) t F i 1 =1 =1 ( F ˆn i) dx =1 F i φ i x (c φ i m) dx ( ) Q(F i 1, F i ) φ i m φ i D i 1 φ i dx, x () where ˆn i is the oca outward pointing norma and F denotes the numerica fux. Specificay, the surface integra in the above equation is defined as foows ( φ i m F ˆn i) ( ) dx = F e ˆn i e dx, (3) e φ i m e with ˆn i e and F e being the outward norma and numerica fux aong the face e. In our impementation, we choose the upwind fux: F c f i (t, x e = e, int( ), c), c ˆn i e c f i (t, x e, ext( ), c), c ˆn i (4) e <

5 where int and ext denote interior and exterior of the face e respectivey. Note that the second term in equation () can be expanded as φ i x (c φ i m) dx = c 1 + c 3 φ i φ i φ i m x dx + c φ i φ i m D i y dx x φ i m z dx, (5) where c 1, c, c 3 are the three components of c. Finay, et us define the mass matrix M m, stiffness matrices S x m, Sy m,sz m, and the tensor H m 1 as M m = φ i m(x) φ i (x) dx, (6) S x m = S y m = S z m = H m 1 = Then the equation () can be recast as M m t F i c 1 =1 =1 = e φ i m e φ i (x) x φi m(x) dx, (7) φ i (x) y φi m(x) dx, (8) φ i (x) z φi m(x) dx, (9) φ i m(x) φ i 1 (x) φ i (x) dx. (3) S x m F i c ( F e ˆn i e) dx + 1 Kn =1 1, =1 S y m F i c 3 =1 S z m F i H m 1 Q ( F i 1, F i ) (c), (31) for 1 m K. Equation (31) is the DG system we are going to sove in each eement of the physica space Discretization in the veocity space To further discretize the system (31) in the veocity space, we empoy a finite difference (or discrete veocity) discretization. Each veocity component c i (i {1,, 3}) is discretized uniformy with N points in the interva [ L, L]. The grid points are chosen as L + ( j 1/) c, with j = 1,..., N and c = L/N (the choice of L is given beow). For brevity we wi use c j to denote the 3D veocity grid point. The reason of using the uniform veocity grid is because our fast agorithm for the coision operator is based on Fourier transform, which is naturay done on a uniform mesh (see Appendix for detais). Simpy speaking, it takes the function vaues at the grid points as input, does the cacuation (incuding forward and backward FFTs) in a back box sover, and outputs the vaues of the coision operator at the same grid points. Inside the sover, it assumes the distribution function 5 has a compact support, and chooses a reativey arge computationa domain encosing this support, then periodicay extends the function to the whoe space R 3. As such, the method can achieve spectra accuracy (subject to domain truncation error which is usuay very sma); furthermore, the simpe mid-point rue woud aso aow one to construct the moments with spectra accuracy. To determine the domain size L, we first choose the maximum temperature T max and veocity U max specified at a boundaries, normaize them by non-dimensiona temperature and mostprobabe speed respectivey, and estimate σ such that the interva [c min, c max ] defined as c max, c min = U max ± σ T max, (3) can produce the correct vaues of T max and U max (i.e., it is arge enough to encose the Gaussian characterized by T max and U max ). Finay, L is chosen as L =.5 max( c max, c min ), (33) which is a reativey safe choice to avoid aiasing effect ([3]). In genera, the parameter σ ranges between 3 to 6. With the above setup, we just need to sove the system (31) at each veocity grid c j and in each spatia eement. The macroscopic quantities: density, buk veocity, temperature, stress tensor, and heat fux vector in the spatia eement can be recovered using numerica integration (mid-point rue) of the distribution function over the entire veocity grid: n i (t, x) = f i (t, x, c j ) c, j U i (t, x) = 1 n i f i (t, x, c j )c j c, j T i (t, x) = 3n i f i (t, x, c j ) c j U i c, j P i (t, x) = f i (t, x, c j )(c j U i ) (c j U i ) c, j q i (t, x) = f i (t, x, c j )(c j U i ) c j U i c, j (34) where c = c 3. Note that n i (t, x), U i (t, x), T i (t, x), P i (t, x), q i (t, x) are the poynomias defined in each eement since f i (t, x, c j ) are poynomias Evauation of the coision term We are now eft with the issue of evauating the coision term in (31): H m 1 Q ( ) F i 1, F i (c). (35) 1, =1 The method we use was proposed in []. Given a function f at N 3 veocity grid, it produces Q( f, f ) at the same grid with O(MN 4 og N) compexity, where M is the number of discretization points on the sphere and M N. In the Appendix, we give a brief description of this method for evauating the operator of the form Q( f, g). Compared to the origina method in

6 [], we improve the accuracy and efficiency by using a different quadrature on the haf sphere. Equipped with the fast coision sover, the tota computationa cost of evauating (35) woud be O(K MN 4 og N) for a given m, where the extra factor is due to the outer quadratic summation. For (reativey) high order poynomia approximations, K can be arge. To further reduce the cost, here we propose a simpe approach based on singuar vaue decomposition (SVD). For each fixed m, we precompute the SVD of the matrix (H m 1 ) K K as R H m 1 = U m 1,r Vm r,, (36) r=1 where R is the rank of the matrix (R K). Substituting it into (35) yieds R 1, =1 r=1 with U m 1,r Vm r, Q(F i 1, F i )(c) = f i r := 1 =1 R r=1 U m 1,r F i 1, g i r := Q ( f i r, g i r) (c), =1 V m r, F i. (37) Note that the functions f i r and g i r can be computed in a different oop. Therefore, the tota cost of evauating (35) is reduced to max{o(rmn 4 og N), O(RKN 3 )} Discretization in time Once the spatia and veocity discretization is done, the time discretization can be performed by simpy appying an expicit Runge-Kutta method to the system (31). Here we adopt the widey used strong-stabiity-preserving (SSP) RK schemes [4]. For notationa simpicity, we rewrite the system (31) as t F i = L (F i ), (38) and use F i to denote the soution vector with components Fm, i 1 m K. Then the SSP-RK scheme of nd order is given by v (1) = F i + tl (F i ), F i,new = 1 F i + 1 v(1) + 1 tl (v(1) ); and the SSP-RK scheme of 3rd order is given by v (1) = F i + tl (F i ), v () = 3 4 F i v(1) tl (v(1) ), F i,new = 1 3 F i + 3 v() + 3 tl (v() ) Initia and boundary conditions The initia vaue of the distribution function is set to Maxweian at given initia macroscopic conditions n (x), T (x), and U (x): f (x, c) = n (πt ) exp [ (c U ) ]. (39) 3/ T 6 For the test cases considered in the current work, the fuy diffusive Maxwe boundary condition is assumed at the wa [41]. Consider a wa moving with veocity U w (t, x), and is at temperature T w (t, x), the infow boundary condition at x Ω x with the oca outward pointing norma ˆn is given by with f (t, x, c) = n w f w, (c U w ) ˆn <, (4) f w (t, x, c) = exp [ (c U w) T w ], (41) and n w is determined from conservation of mass as (c U n w = (c U w ) ˆn w) ˆn f dc (c Uw (c U ) ˆn< w) ˆn f w dc. (4) Detais about other boundary conditions can be found in [1, 5, 34]. 4. Numerica experiments and resuts In this section, we evauate the accuracy of the proposed discontinuous Gaerkin fast spectra method, which we sha denote by the acronym DGFS in the foowing. Standard benchmark cases of Bobyev-Krook-Wu (BKW) soution [4, 43] for Maxwe moecues, panar Fourier heat transfer at Kn =.4745, 1.58, Couette fow at Kn =.5, 1., osciatory Couette fow at Kn = 1., and thermay driven cavity fow at Kn = 1.,.5 have been considered in the present work. The resuts are compared with those obtained from the DSMC method [5] with equivaent moecuar coision modes, and anaytica soutions wherever appicabe Sover configurations SPARTA [19] has been empoyed for carrying out DSMC verifications in the present work. It impements the DSMC method as proposed by Bird [5]. The sover takes into account the rotationa kinetic energy associated with the moecuar rotationa veocity and its moment of inertia, in addition to the transationa kinetic energy. The sover has been benchmarked [19] and widey used for studying hypersonic, subsonic and therma [44 47] gas fow probems. In this work, ce size ess than λ/3 has been ensured in a the test cases. A minimum of 3 DSMC simuator partices per ce are used in conjunction with the no-time coision (NTC) agorithm. Each steady-state simuation has been averaged for a minimum 1, steps so as to minimize the statistica noise. Argon gas with mass m = kg, reference viscosity of N/m s at reference temperature T ref of 73K is seected. The moecuar diameters are seected so as to maintain the reference viscosity: d ref = 4.59Å, ω = 1. for the Maxwe coision mode, and d re f = 4.11Å, ω =.81 for the VHS coision mode. These vaues are consistent for both DSMC and DGFS in a test cases uness otherwise expicity stated.

7 4.. Hardware configurations Seria impementations are run on Inte E5-68 Xeon(TM) Processor v.8 GHz (Conte custer at Purdue). The operating system used is 64-bit RHEL 6.7. Parae impementations which incude MPI versions of DGFS and SPARTA sovers are run on the same machine with each core having the same processor configurations. Different versions of the sovers have been written in C++ and are compied using OpenMPI mpic , g with OpenMP-4. support, and third eve optimization fags. A the simuations are done with doube precision foating point vaues D case: BKW soution For constant coision kerne B = 1/(4π) and Knudsen number Kn = 1, an exact soution to the spatiay homogeneous Botzmann equation can be constructed (see [4, 43]). The nondimensionaized equation (13) simpifies to f t and the anaytica soution is given by f (t, c) = = Q( f, f ), (43) ) ( 1 exp ( c 5K(t) 3 (πk(t)) 3/ K(t) K(t) + 1 K(t) ) c, K (t) (44) where K(t) = 1 exp( t/6). The initia time t must be greater than 6 n(.5) for f to be positive. An arbitrary time of t = 5.5 has been picked in the present work Time evoution of the distribution function Figure 1 iustrates the time evoution of the distribution function siced aong the veocity domain centerine, i.e., f (c 1, N/, N/), where N denotes the number of points in each dimension of the veocity mesh. The smooth anaytica soution is obtained by discretizing the veocity space with N = 56 points. The numerica soution is evauated by discretizing the veocity space with N = 16, 3 and 64 respectivey. M = 6 spherica design quadrature points are used on the haf sphere in a cases. It is observed that: a) with increase in N, the numerica soution moves coser to the smooth anaytica soution at different time instants; b) as time goes by, the distribution function tends toward the Maxweian Time evoution of the entropy The H-theorem states that the entropy is aways decreasing (the physica entropy is increasing), which can be expressed mathematicay as t f n f dc = R 3 Q( f, f ) n f dc, R 3 (45) where f n f dc is the so-caed H-function or entropy. The R entropy can 3 be a powerfu quantity for verification of numerica soutions in rarefied fows. Figure iustrates the time evoution of the entropy for numerica and anaytica soutions. In particuar, for N = 16 veocity grid, athough the reative error 7 f (c1, c = c, c3 = c) N=16 N=3 N=64 Exact M= c 1 t=5.6 t=6.5 t=7. t=8.5 Figure 1: Comparison of BKW anaytica and numerica soutions over time. The anaytica soution is evauated by discretizing the veocity space with N = 56 points. The numerica soution is evauated by discretizing the veocity space with N = 16, 3 and 64 as indicated in the pot. M = 6 is used on the haf sphere in a cases. The cassica fourth order Runge-Kutta method with t =.1 has been empoyed for time integration. is order of 1 3 as shown in Figure 3, the entropy in this case is significanty ower than anaytica entropy, and aso vioates the second aw of thermodynamics due to insufficient veocity mesh resoution. The resuts are improved if we use arger veocity grids N = 3 and 64, and are indeed cose within the error imit of the anaytica soution. These soutions foow the second aw of thermodynamics. We beieve that this comparison of anaytica/numerica entropy is fairy significant for estabishing the importance of the fast spectra method. H-function Time Exact N=16, M=6 N=3, M=6 N=64, M=6 Figure : Time evoution of BKW anaytica and numerica entropy. The anaytica soution is evauated by discretizing the veocity space with N = 56 points. The numerica soution is evauated by discretizing the veocity space with N = 16, 3 and 64 as indicated in the pot. M = 6 is used on the haf sphere in a cases Normaized error Figure 3 iustrates the time evoution of normaized error in L norm between the numerica and anaytica soutions with

8 ogarithmic y-axis. Note that both numerica and anaytica soutions are evauated at N = 16, 3 and 64 in contrast to the previous cases where the anaytica soution was evauated at N = 56 points. An exceent agreement between anaytica and numerica soutions is ceary evident from the Figure 3. It is aso observed that the differences between M = 6 and M = 16 soutions are sma (quantitativey on order of 1 4 ). The sight increase of the error in the cases of N = 3 and 64 is due to the aiasing effect of the spectra method as discussed in [3]. fnumerica fexact L / f exact L 1e- 1e-3 1e-4 1e-5 1e-6 1e Time N=16, M=6 N=16, M=16 N=3, M=6 N=3, M=16 N=64, M=6 N=64, M=16 Figure 3: Comparison of BKW anaytica and numerica soutions over time with ogarithmic y-axis. Symbos have usua meanings as defined previousy D case: Fourier heat transfer y x T = 63K T r = 83K Figure 4: Numerica setup for 1D Fourier/Couette fow case. Distance between the was is fixed as H. Note that the ces are finer in the near-wa region. For non-constant coision kernes, anaytica soutions for the Botzmann equation do not exist. Therefore, we compare our resuts with DSMC which soves the Botzmann equation stochasticay. In the current test case, the coordinates are chosen such that the was are parae to the y direction and x is the direction perpendicuar to the pates. The schematic as we as boundary conditions appied at boundary patches have been shown in Figure 4. The two parae pates, at rest, have been set H distance apart. The reference (T ref ), eft-wa (T ), and right-wa (T r ) temperatures are 73K, 63K, and 83K, respectivey. The simuation is carried out at two different Knudsen numbers namey Kn =.4745 and Kn = 1.58 by varying the density whie keeping the H fixed. Argon with Maxwe coision mode is taken as the working gas (see [48] for additiona DSMC conditions) Vaidation: SVD v.s. direct agorithm Figure 5 iustrates the normaized temperature profie aong the domain ength obtained using the SVD and direct variants of the coision agorithm. It is observed that the corresponding two curves are inextricabe which verifies that both agorithms evauate the same Botzmann operator. T (K) SVD Direct Figure 5: Variation of temperature aong the domain ength, obtained using SVD and direct agorithm variants of DGFS at Kn = 1.58 using Maxwe coision mode for Argon moecues. The physica space consists of ces and poynomia order of 1, and veocity space consists of 3 points Temperature at different Knudsen numbers Figure 6 iustrate the temperature profie aong the domain ength for different Knudsen numbers obtained using the SVD variant of the agorithm. The resuts are compared against the DSMC data [19], where our DGFS impementation captures the noninear [49] nature of temperature profies in the near wa region, i.e., the Knudsen ayer. T (K) DSMC (Gais et a. ) DGFS (s=, k=1, v=) Kn= Figure 6: Variation of temperature aong the domain ength for Kn =.4745 and 1.58 using Maxwe coision mode for Argon moecues obtained with DSMC and DGFS. The physica space consists of ces and poynomia order of 1, and veocity space consists of 3 points. 8

9 4.5. 1D case: steady Couette fow We now consider the effect of veocity gradient on the soution. The geometry remains the same as in previous case. The eft and right parae was move with a veocity of U = (, ±5, ) m/s, and the reference (T ref ), eft-wa (T ), and rightwa (T r ) temperatures are set to a constant vaue of 73K. The simuation is carried out at two different Knudsen numbers namey Kn =.5 and Kn = 1. by varying the density whie keeping the H fixed. Argon with VHS coision mode is taken as the working gas (see [5] for additiona DSMC conditions). Figure 7 iustrates the veocity aong the domain ength. The deterministic soution is in exceent agreement with the DSMC soution [5], and again our mode captures the noninearity in the near-wa region. N avg. It is observed that keeping N avg fixed, with decrease in PPC, the sampe size decreases and consequenty the statistica noise increases as iustrated in Figures 8, 9 and 1, 11. Keeping PPC fixed, with increase in N avg, the sampe size increases and consequenty the statistica noise decreases, but the simuation ags behind in time as a resut of high N avg. These observations are depicted in Figures 1 and 11. Through Figures 9, 1, 11, we want to emphasize the smooth time accurate resuts obtained from DGFS, and the we-known stochastic nature of DSMC soutions. In present case, we used as arge as 1 partices per ce for obtaining time accurate resuts. In arge scae simuations, 1 partices per ce might not be feasibe computationay, and hence the resuts from DSMC woud aways be inaccurate in those cases. 3 DSMC (Gu et a. 9) DGFS (s=, k=, v=) 4 3 t=.t t=.4t t=.6t t=.8t t=1.t 1 1 v (m/s) v (m/s) T = sec Figure 7: Variation of veocity aong the domain ength for Kn =.5 and 1. obtained with DSMC and DGFS using VHS coision mode for Argon moecues. The physica space consists of ces and poynomia order of, and veocity space consists of 3 points. Figure 8: Time evoution of veocity aong the domain ength for osciatory Couette fow at Kn = 1. using VHS coision mode for Argon moecues. We use 5 ces, 1 PPC, and 1 N avg. Symbos and ines denote DSMC and DGFS resuts respectivey. For DGFS, the physica space consists of ces and poynomia order of, and veocity space consists of 4 3 points D unsteady case: osciatory Couette fow In present case, to estabish the time accuracy of discretization, we consider the effect of time varying veocity gradient on soution. The geometry and fow parameters remains the same as in previous case, except that the eft wa is at rest, and the right wa moves with a veocity of U = (, 5, ) sin(ζt) m/s, where ζ = π/5e s 1. The simuation is carried out at Kn = 1.. Specificay for DSMC simuations, the domain is discretized into 5 ces with 1 partices per ce (PPC) and the resuts are averaged for every 1 (N avg ) time steps. Figure 8 depicts the time evoution of veocity aong the domain for both DSMC and DGFS resuts. Since the present case is unsteady, high statistica noise is observed in case of DSMC resuts. In contrast, DGFS produces a sufficienty smooth soution. Nevertheess, both the resuts are in fair agreement with each other. Further, we observe high amount of sip ( %) at the eft wa since the fow is in transition regime. An accurate unsteady DSMC resut is inherenty tricky. We carried out set of simuations by varying PPC, ce-count, and v (m/s) t=.t t=.4t t=.6t t=.8t t=1.t -3 T = sec Figure 9: Time evoution of veocity aong the domain ength for osciatory Couette fow at Kn = 1. using VHS coision mode for Argon moecues. We use 5 ces, 1 PPC, and 1 N avg. Symbos and ines denote DSMC and DGFS resuts respectivey. For DGFS, the physica space consists of ces and poynomia order of, and veocity space consists of 4 3 points. 9

10 4 3 t=.t t=.4t t=.6t t=.8t t=1.t 4 3 t=.t t=.4t t=.6t t=.8t t=1.t 1 1 v (m/s) -1 v (m/s) T = sec Figure 1: Time evoution of veocity aong the domain ength for osciatory Couette fow at Kn = 1. using VHS coision mode for Argon moecues. We use 5 ces, 1 PPC, and 1 N avg. Symbos and ines denote DSMC and DGFS resuts respectivey. For DGFS, the physica space consists of ces and poynomia order of, and veocity space consists of 4 3 points. -3 T = sec Figure 11: Time evoution of veocity aong the domain ength for osciatory Couette fow at Kn = 1. using VHS coision mode for Argon moecues. We use 5 ces, 1 PPC, and 1 N avg. Symbos and ines denote DSMC and DGFS resuts respectivey. For DGFS, the physica space consists of ces and poynomia order of, and veocity space consists of 4 3 points D case: thermay driven cavity fow In present case, we consider the effect of fow induced due to therma gradients. We consider a square box of ength L = meters. The eft and right was are fixed at T c = 63K. At the top and bottom was, we introduce a ineary increasing temperature (from T c to T h = 83K) in eft haf of domain, and a ineary decreasing temperature (from T h to T c ) in the right haf. The schematic of the domain have been iustrated in Figure Boundary condition At the top and bottom was, given T c, T h, and the position vector of end-points r c and r h : 1. DGFS: Using the Lagrangian poynomia, we interpoate the temperature vaues at the known set of DG soution/quadrature points on the surface. Once the temperature T i is known at a given quadrature point, we then define a Maxweian wa distribution around T i for that particuar soution point.. DSMC: Given a partice on boundary with some position vector r i, we interpoate the temperature ineary using three-dimensiona equation of ine. And, then we emit the partice with the Maxweian defined around T i (interpoated temperature for partice with position vector r i ) Temperature Figure 13 iustrates the side-by-side comparison of iso-contours of temperature in the domain interior. Ignoring the statistica noise, we observe a fair agreement between DSMC and DGFS resuts. 5. Concusions We have presented a deterministic numerica method for the fu Botzmann equation. The method combines the discontinuous Gaerkin discretization in the physica space and the fast Fourier spectra method in the veocity space to yied highy accurate numerica soutions. The DG-type formuation empoyed in the present work has advantage of having arbitrary order accuracy at the eement-eve, and its eement-oca compact nature (and that of our coision agorithm) enabes effective paraeization on massivey parae architectures. Our fast spectra method for evauating the Botzmann coision operator does not rey on any ad hoc adjustment or parameter fitting of the coision kerne in contrast to the previousy proposed methods in iterature. Further, we have proposed a nove SVD based coision agorithm, through which we can decrease the computationa cost of simuation by an order of number of points in the seected quadrature. To verify the proposed DGFS method, we carried out rarefied gas fow simuations for spatiay homogeneous, Fourier, Couette, osciatory Couette, and thermay driven cavity fows at different Knudsen numbers. Each of these cases have been run with different coision kerne to highight the genera nature of our coision agorithm. We concude that the resuts obtained with our deterministic sover and DSMC are inextricabe ignoring the statistica noise and the errors therein. The deterministic soution of the Botzmann equation by the DGFS method, in particuar, is suitabe for studying ow-speed and unsteady fows. Appendix In this appendix, we give a brief description of the fast Fourier spectra method proposed in []. Our impementation here differs from [] in mainy two aspects: 1) the symmetrized 1

11 T f T c y T f T h x T c Temperature profie : T f Figure 1: Numerica setup for thermay driven cavity fow case. The representative ineary-graded mesh is shown with dotted ines. version of the coision kerne is used which aows the integration to be performed on haf sphere rather than whoe sphere; ) a different spherica quadrature is adopted which shows better numerica performance. First of a, from (14) and (3), it is easy to see that one can repace the coision kerne by its symmetrized version: B sym ( c c, cos χ) = B( c c, cos χ) + B( c c, cos χ). (46) Second, from the discussion in Section 3.3.1, a we need is to evauate an operator of the form Q( f, g)(c) = R 3 B sym ( c c, cos χ)[ f (c )g(c ) S f (c)g(c )] dσ dc. (47) The main steps of the Fourier spectra approximation of (47) can be summarized as foows: Change the variabe c to u = c c : Q( f, g)(c) = B sym ( u, σ û)[ f (c )g(c ) R 3 S f (c)g(c u)] dσ du, where û is the unit vector aong u, and (48) c = c u + u σ, c = c u u σ. (49) Determine the computationa domain D L = [ L, L] 3 as described in Section 3.3, and periodicay extend f, g to R 3. Truncate the integra in u to a ba B R with R = 4 3+ L (criterion based on [3]). 11 Figure 13: Contour of temperature in the square-box for Kn =.5 and Kn = 1. obtained with DSMC (eft) and DGFS (right) using VHS coision mode for Argon moecues. For DGFS, the physica space consists of 56 ces and poynomia order of 3, and veocity space consists of 4 3 points. Approximate f, g by truncated Fourier series f N (c) = N/ 1 k= N/ ˆ f k e i π L k c, g N (c) = N/ 1 k= N/ Note here k is a three-dimensiona index. ĝ k e i π L k c. (5) Substitute f N, g N into (48), and perform the standard Gaerkin projection ˆQ k : = 1 Q( f N, g N )(c)e i π (L) 3 L k c dc D L B R = N/ 1,m= N/ +m=k [G(, m) G(m, m)] ˆ f ĝ m, (51) where k = N/,..., N/ 1, and the kerne mode G is given by G(, m) = B sym ( u, σ û) e i π +m L u+i π m L u σ dσ du. S (5) It is cear that the direct evauation of ˆQ k (for a k) woud require O(N 6 ) compexity. But if we can find a ow-rank, separated expansion of G(, m) as G(, m) R α r ( + m) β r () γ r (m), (53) r=1

12 then the gain term (positive part) of ˆQ k can be rearranged as R ˆQ + k = α r (k) r=1 N/ 1, m= N/ +m=k ( βr () ˆ f ) (γr (m)ĝ m ), (54) which is a convoution of two functions β r () fˆ and γ r (m)ĝ m, hence can be computed via FFT in O(RN 3 og N) operations. Note that the oss term (negative part) of ˆQ k is readiy a convoution and can be computed via FFT in O(N 3 og N) operations. In order to find the approximation in (53), we simpify (5) as (using the symmetry of the kerne) R ( π G(, m) = F( + m, ρ, σ) cos S + L ρ m ) σ dσ dρ, (55) where S + denotes the haf sphere, and ( π F( + m, ρ, σ) := ρ S B sym (ρ, σ û) cos + L ρ + m ) û dû. (56) Now using the fact that cos(α β) = cos α cos β + sin α sin β, if we approximate the integra in (55) by a quadrature, we obtain [ ( π G(, m) w ρ w σ F( + m, ρ, σ) cos L ρ ) ( π ) σ cos L ρm σ ρ,σ ( π + sin L ρ σ ) ( π )] sin L ρm σ, (57) which is exacty in the desired form (53). In the impementation, we use the Gauss-Legendre quadrature for ρ. As the integrand osciates on the scae of O(N) in the radia direction, the tota number of quadrature points needed shoud be O(N). For the integration on the haf sphere, we choose to use the spherica design [51], which is the near optima quadrature on the sphere [5]. Other quadratures are possibe, for exampe, the Lebedev quadrature as used in []. Through numerica tests, we found that the spherica design usuay yieds better resuts than Lebedev, probaby due to the fact that the quadrature points are more uniformy distributed. Therefore, assuming M quadrature points are used on the haf sphere (in practice M N ), the fina computationa cost of evauating ˆQ k is reduced from O(N 6 ) to O(MN 4 og N). Acknowedgments J. Hu s research was supported by NSF grant DMS-165 and NSF CAREER grant DMS Support from DMS : RNMS KI-Net is aso gratefuy acknowedged. References [1] W. Su, A. A. Aexeenko, G. 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