PHYSICS OF FLUIDS VOLUME 1, UMBER 1 OCTOBER Analysis of disretization in the diret simulation Monte Carlo iolas G. Hadjionstantinou a) Department of Mehanial Engineering, Massahusetts Institute of Tehnology, Cambridge, Massahusetts 139 Reeived 15 Otober 1999; aepted 7 June We propose a ontinuous-time formulation of the diret simulation Monte Carlo that allows the evaluation of the transport oeffiient dependene on the time step through the use of the Green Kubo theory. Our results indiate that the error exhibits quadrati dependene on the time step, and that for time steps of the order of one mean free time the error is of the order of 5%. Our preditions for the transport oeffiients are in good agreement with numerial experiments. The alulation of the ell size dependene, first obtained by Alexander et al. Phys. Fluids 1, 154 1998, is reviewed and a orretion is pointed out. Amerian Institute of Physis. S17-66311-5 I. ITRODUCTIO The diret simulation Monte Carlo DSMC has been a very useful tool for the simulation of equilibrium and nonequilibrium gas flows. 1, It is very attrative beause it suessfully oarse grains the moleular desription to the hydrodynami regime, thus offering substantial omputational effiieny advantages over brute fore moleular dynamis simulations. DSMC simulations have been shown 3 to onverge to Boltzmann equation solutions in the limit of infinite number of partiles, and vanishing ell size and time step. However, like most omputational methods, in the limit of finite disretization, numerial error ontaminates the solution. In a reent paper, Ohwada 4 has shown that the DSMC proedure viewed as a splitting method applied to the time integration of the Boltzmann equation results in a first-order aurate desription of the distribution funtion. Ohwada finds that due to error aumulation, the distribution funtion diverges from the orret time-evolving distribution funtion proportionally to the time step t in transient problems. This is in ontrast to the results of Bogomolov 5 that had previously shown a seond-order behavior. In this paper we fous on a different lass of problems. We onsider problems that are steady with respet to the moleular time sale, and thus the distribution funtion has relaxed to a steady state. We quantify the deviation of this state from the orret nonequilibrium state by obtaining measures of the deviation of the transport oeffiients from their theoretial values. We will refer to the deviation of the transport oeffiients from the exat Enskog values for dilute gases whih DSMC reprodues in the limit of an infinite number of partiles and vanishing time step and ell size as the trunation error. As we now disuss, DSMC introdues a trunation error due to disretization both in spae and time. Trunation error from disretization in spae results a Eletroni mail: ngh@mit.edu from the seletion of ollision partners from ells of finite size (L). In a previous paper 6 it was shown that the resulting error is proportional to the square of the ell size, with the onstant of proportionality suh that ell sizes of the order of one mean free path result in errors of the order of 1%. In this paper we show that DSMC ommits a disretization rime by performing an instantaneous ollision of the moleules that would have normally ollided within a time step t. This shift in the ollision times is shown to have two ontributions to the transport oeffiients by omparing DSMC with a model where ollisions take plae in a ontinuous fashion. First, the moleules that are lose enough to be onsidered for ollision during the ollide part of the algorithm in the same ell would have been at a finite distane if they ollided at the proper time; this is a soure of error similar to the finite ell size effet. Seond, extra fluxes are generated due to the altered trajetories of the moleules that, as a result of the disretization, ollide at times other than their proper ollision time. Alexander et al. 6 have shown that the effets of disretization in spae an be alulated by appliation of the Green Kubo formulation 7 when the ollision proess is assumed to be ontinuous (t ). Unfortunately, DSMC does not lend itself naturally to the appliation of this formulation for the alulation of the trunation error due to a finite time step beause DSMC is a disrete-time model. In order to apply the Green Kubo proedure we will first formulate a ontinuous-time model that is dynamially equivalent to DSMC. In Se. II we give an overview of the dynamial steps of the DSMC algorithm and review the alulation for the effet of finite ell size. In Se. III we quantify the two soures of error arising from a finite time step and show how the error for transport oeffiients an be alulated as a funtion of the time step. In Se. IV we ompare our preditions with simulation results, and finish with some onluding remarks in Se. V. 17-6631//1(1)/634/5/$17. 634 Amerian Institute of Physis Downloaded 6 ov 7 to 18.8..8. Redistribution subjet to AIP liense or opyright, see http://pof.aip.org/pof/opyright.jsp
Phys. Fluids, Vol. 1, o. 1, Otober Analysis of disretization in DSMC 635 FIG. 1. Shemati of a representative ollision between partiles in DSMC for ell size L. The partiles ollide at t/. II. DSMC ALGORITHM AD CELL SIZE DEPEDECE OF TRASPORT COEFFICIETS In what follows we will limit our illustration to two dimensions; generalization to three dimensions follows diretly. Consider two partiles, i and j, that ollide at time step n Fig. 1; let their initial loations at the start of the time step be (x i,y i ) and (x j,y j ). Let be the time variable over whih DSMC oarse grains, that is mod(t,t), where tnt is time. The above partiles travel with veloities v i and v j, and at t/ when ollisions take plae will be in the same ell so that they are hosen as ollision partners. After olliding, the partiles aquire new veloities v i and v j and travel ballistially for the remainder of the time step t/ to their final positions (x f i,y f i ) and (x f j,y f j ). ote that in our notation a time step starts 1/t before the instantaneous ollision proess and ends 1/t after the instantaneous ollision proess. This being purely a matter of onvention has no effet on the DSMC dynamis: partiles travel ballistially between ollisions for a time t we exlude the presene of external fields that may introdue partile aelerations and then instantaneous ollisions aounting for the whole time interval t take plae. Our onvention merely introdues a shift in the global time ounter with ollisions taking plae at t t/, 3t/, 5t/,..., with ballisti motion in between, instead of t, t, t,..., with ballisti motion in between. In the limit that t we an neglet the error from the disrete nature of this algorithm and use the Green Kubo theory 7 to evaluate the effet of ell size L on the transport oeffiients. We review here the alulation of the visosity oeffiient as a funtion of the ell size in order to point out a orretion. Following Alexander et al. 6 we define the stress tensor J xy (t) for a hard sphere system, J xy tm u i v i i1 u i y i y j tt, 1 where is the number of moleules, m is the mass of the moleules, u i, v i are the x and y omponents of the veloity of moleule i, u i is the hange in the veloity of partile i in the x diretion during ollision, and denotes the sum over all ollisions, eah ollision ourring at t and involving moleules i and j. The distane (y i y j ) between the two moleules at ollision introdues a potential ontribution that is absent in the Enskog theory of dilute gases. The visosity is alulated using the Green Kubo formulation as applied to hard spheres by Wainright, 7 VkT 1 ds 1 t t s sdt Jxy tj xy ts, where V is the volume, and t s is a suffiiently long smoothing time several time steps. Substitution of Eq. 1 into Eq. yields 6 K C P, 3 where the supersripts K, C, P denote kineti, ross, and potential terms. The kineti term is the Chapman Enskog visosity of a dilute gas hard spheres of diameter ) 6 K 5 16 m, 4 where n kt/m is the ollision rate per unit volume, and 1/(n ) is the mean free path. The ross term is zero beause partile veloities are unorrelated with their positions. 6 The potential term is given by 6 P m kt y iy j u i m kt y iy j u i, 5 where denotes average over ollisions. For a gas of uniform density, (y i y j ) L /6. We would like to point out that (u) 3(kT/m) 4 instead of the original value of 8 9(kT/m) given in Alexander et al., 6 and thus the visosity as a funtion of ell size in DSMC is given by 5 mkt 16 1 16 L. 45 The expression for the thermal ondutivity given in the paper by Alexander et al. is not affeted by this orretion. III. TIME-STEP DEPEDECE OF TRASPORT COEFFICIETS A. Equivalent ontinuous-time model We now proeed to examine the ase where t is finite and thus the DSMC algorithm beomes disrete in time. For the above formalism to be applied we propose a ontinuoustime model that is dynamially equivalent to DSMC. Consider a model where the partiles that aording to DSMC would have ollided at t/ that is, at t/ were in the same ell instead ollide eah olliding pair independently at t that lies uniformly between and t. The distribution funtion for the partile loation relative to the DSMC ollision point at ollision (t ) de- 6 Downloaded 6 ov 7 to 18.8..8. Redistribution subjet to AIP liense or opyright, see http://pof.aip.org/pof/opyright.jsp
636 Phys. Fluids, Vol. 1, o. 1, Otober iolas G. Hadjionstantinou pends on the differene between the ollision time t and the DSMC ollision time t/: We an write the single partile distribution funtion in one dimension in the ase that the ollision ell is entered on x as L/ f x;t px qxx ;t dx, 7 L/ where p(x ) is the probability uniform in this approximation for a partile to be in the ell at x, and q(xx ;t ) is the probability for a partile that is at x at t/ to be found at x at t. Due to the Maxwellian veloity distribution q(xx ;t ) has a partiularly simple form that leads to m L/ 1 f x;t ktt t/ L/ L exp mxx dx, 8 ktt t/ where k is Boltzmann s onstant, and T is the temperature. The above expression redues to a Maxwellian distribution entered at x) if the ell size L. We an see that a finite ell size modifies the partile distribution funtion and, thus, stritly ouples to the finite time-step trunation error. In the interest of simpliity the effet of finite ell size an be fully quantified if the algebrai omplexity is undertaken and beause the error due to a finite ell size in the limit t is known, 6 we will take L. In this limit the relative distane in the y diretion between a ollision pair at ollision an be written as y i y j t t g y, 9 where gäv i v j is the preollision relative veloity of the partiles. Figure indiates that for this model to be dynamially idential to the DSMC implementation, olliding partiles need to be shifted after their ollisions by an amount that orrets for their ballisti motion with postollision veloities instead of the preollision veloities for the appropriate amount of time (t t/). The shift ensures that the moleules have final positions (x f i,y f i ) and (x f j,y f j ) that are the same as for DSMC Fig. 1. The amount eah moleule needs to be shifted is given by r i t t v i v i. 1 FIG.. Shemati of a representative ollision between partiles in the ontinuous-time model. The disontinuous jump in the trajetories represents the shift ourring along with the ollision. This shift introdues mass, momentum, and energy fluxes that ontribute to enhaned transport oeffiients as shown below. B. Calulation of transport oeffiients We now present the alulation of the visosity of the DSMC-equivalent ontinuous-time model as a funtion of the time step. Appliation of the Green Kubo theory implies that steady problems are onsidered, at least at the autoorrelation deay time sale. This is not a very restritive assumption sine typial hydrodynami evolution takes plae at time sales muh longer than the latter time sale. The momentum flux resulting from the partile shift an be written as J s xy m m u i y i u j y j tt t t v i g x tt, 11 where gv i v j is the postollision relative veloity between partile i and partile j. Shifting the partiles before or after ollision results in the same total potential ontribution to the stress tensor: J xy tm i1 u i v i u i y i y j u i y i uy j j tt 1 m i1 u i v i u i y i y i y j y j u i y i u j y j tt 13 m i1 u i v i t t u i g y v i g x tt. 14 Downloaded 6 ov 7 to 18.8..8. Redistribution subjet to AIP liense or opyright, see http://pof.aip.org/pof/opyright.jsp
Phys. Fluids, Vol. 1, o. 1, Otober Analysis of disretization in DSMC 637 where kt/m is the most probable speed. The alulation of other transport oeffiients follows along the same lines. Potential ontributions are proportional to (t t/) (t) /1 and ross ontributions vanish beause DSMC is entered in time. The pressure is unaffeted by the time step as we would normally expet: The virial u i (y i y j )u i y i uy j j is proportional to (t t/). A similar alulation for the thermal ondutivity yields 75k kt 1 64 m 64 t 675. The diffusion oeffiient an be alulated using the Einstein formula D 1 t 1 i x i tx i, 1 FIG. 3. Error in oeffiient of visosity as a funtion of the saled by / ) time step from Ref. 8. Cirles denote the normalized error in momentum flux (E v in the simulations of Garia and Wagner Ref. 8, and the solid line is the predition of 19. Here v i is the hange in veloity of partile i in the y diretion due to the ollision. Appliation of the Green Kubo formula again results in kineti, ross, and potential terms. The kineti term is again the dilute gas term Eq. 4. The ross term is zero beause DSMC is entered in time: The ontribution is proportional to (t (t/) and hene zero beause t t t dt. 15 The potential ontribution an be written as P m kt t t u i g y v i g x. 16 This expression assumes that there is no orrelation beyond the first ollision between moleules. The uniform ollision in time gives t t t 1, 17 whereas ug y vg x 16 5 kt m. 18 The expression for the visosity inluding the time-step ontribution is thus, 19 5 mkt 16 1 3 t 15 where a long time limit is assumed. Applying this formula to the ontinuous-time model reveals that the shift Eq. 1 for both olliding partiles required for every ollision, leads to a potential-like term r i (tt )(t t/)v i (t t ) that yields an error proportional to (t). The resulting expression for the diffusion oeffiient D is D 3 kt 1 8n m 4 t 7, where n is the number density. IV. COMPARISO WITH UMERICAL SIMULATIOS Garia and Wagner 8 performed steady state and transient numerial simulations in a variety of onfigurations to measure the trunation error as a funtion of the time step. In those simulations the ell size was taken to be L/5 so that the ell size ontribution is negligible, and the time step was varied from t/( )tot16/. The error is defined as the normalized deviation in the flux orresponding to the transport oeffiient with respet to the exat result. The exat result is taken to be a very aurate simulation with t/(8 ). Figure 3 shows the omparison between the numerial results of Garia and Wagner for the visosity oeffiient and the theoretial predition Eq. 19 for steady Couette flow. The agreement is very good for t. For t/ the error deviates from the quadrati time step dependene. Garia and Wagner 8 point out that this is due to the upper bound set on the transport oeffiients by the ollisionless limit that is indiated on the same graph. The preditions given here for the thermal ondutivity Eq. and diffusion oeffiient Eq. are also in very good agreement with the numerial results of Garia and Wagner 8 in steady state. The transient alulations of Garia and Wagner also exhibit a quadrati error in the time step; however, no omparison with our results was presented. V. COCLUDIG REMARKS We have presented a formulation that allows the alulation of the transport oeffiient dependene on the time step. The alulations predit that the error in the transport oeffiients is of order t, and that for t/ the trun- Downloaded 6 ov 7 to 18.8..8. Redistribution subjet to AIP liense or opyright, see http://pof.aip.org/pof/opyright.jsp
638 Phys. Fluids, Vol. 1, o. 1, Otober iolas G. Hadjionstantinou ation error is approximately 5%, whih onfirms the empirial observations that aurate solutions require a time step that is a small fration of the mean free time. The theory presented relies on the appliation of the Green Kubo theory and is thus valid for problems that appear steady at the autoorrelation deay time sale. The results of this paper have been verified by extensive numerial simulations 8 of steady state onfigurations; the agreement between simulations and theory is very good for t. For t/ the error approahes the value set by the ollisionless limit. ACKOWLEDGMETS The author would like to thank Alej Garia and Frank Alexander for helpful disussions. 1 G. A. Bird, Moleular Gas Dynamis and the Diret Simulation of Gas Flows Clarendon, Oxford, 1994. F. J. Alexander and A. L. Garia, The diret simulation Monte Carlo method, Comput. Phys. 11, 5881997. 3 W. Wagner, A onvergene proof for Bird s diret simulation Monte Carlo method for the Boltzmann equation, J. Stat. Phys. 66, 111199. 4 T. Ohwada, Higher order approximation methods for the Boltzmann equation, J. Comput. Phys. 139, 11998. 5 S. V. Bogomolov, Convergene of the method of summary approximation for the Boltzmann equation, USSR Comput. Math. Math. Phys. 8, 79 1988. 6 F. J. Alexander, A. L. Garia, and B. J. Alder, Cell size dependene of transport oeffiients in stohasti partile algorithms, Phys. Fluids 1, 154 1998. 7 T. Wainright, Calulation of hard-sphere visosity by means of orrelation funtions, J. Chem. Phys. 4, 93 1964. 8 A. L. Garia and W. Wagner, Time step trunation error in diret simulation Monte Carlo, Phys. Fluids 1, 61. Downloaded 6 ov 7 to 18.8..8. Redistribution subjet to AIP liense or opyright, see http://pof.aip.org/pof/opyright.jsp