ON DSMC CALCULATIONS OF RAREFIED GAS FLOWS WITH

ON DSMC CALCULATIONS OF RAREFIED GAS FLOWS WITH SMALL NUMBER OF PARTICLES IN CELLS STEFAN K. STEFANOV

∗

Abstract. The direct simulation Monte Carlo (DSMC) analysis of two- and three-dimensional rarefied gas flows requires computational resources of very large proportions. One of the major causes for this is that, along with the multidimensional computational mesh, the standard DSMC approach also requires a large number of particles in each cell of the mesh in order to obtain sufficiently accurate results. This paper presents the development and validation of a modified simulation procedure which allows more accurate calculations with a smaller mean number of particles (hN i ∼ 1) in the grid cells. In the new algorithm, the standard DSMC collision scheme is replaced by a two-step collision procedure based on ”Bernoulli trials” scheme (or its simplified version proposed by the author), which is applied twice to the cells (or subcells) of a dual grid within a time step. The modified algorithm uses a symmetric Strang splitting scheme that improves the accuracy of the splitting scheme to O(τ 2 ) with respect to the time step τ making the modified DSMC method an effective numerical tool for both steady and unsteady gas flow calculations on fine multidimensional grids. The latter is particularly important for simulation of vortical and unstable rarefied gas flows. The modified simulation scheme might be useful also for DSMC calculations within the subcell areas of a multilevel computational grid. Key words. Direct Simulation Monte Carlo (DSMC) method, kinetic theory, rarefied gas flows, micro gas flows AMS subject classifications. 65C05, 76P05, 82C80

1. Introduction. For decades the Direct Simulation Monte Carlo (DSMC) technique [4, 5] has been regarded as a powerful numerical method for studying rarefied gas dynamics problems. The DSMC technique uses a finite set of model particles denoted by their positions and velocities {xi , ξi }, i = 1, . . . , N , that move and collide in a computational domain to perform a stochastic simulation of the real molecular gas dynamics. The basic concept of the method is built on a discretization in time and space of the real gas dynamics process and splitting the motion into two successive stages of free molecular motion and binary intermolecular collisions within the grid ∗ Institute

of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, Sofia

1113, Bulgaria ([email protected]). 1

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S. STEFANOV

cells each time step. A rigorous mathematical proof of the method’s convergence for large number of particles in cells to the Boltzmann equation was given by Wagner [27] in 1992. A recent discussion on the theoretical background of the method can be found in Sone’s monograph [21]. The second stage of modeling the binary collisions in cells is more complicated and over the years serious efforts have been made to improve the ”Time Counter” collision scheme originally proposed by Graham Bird [4]. The latter turned out to compute the binary collision frequency with a systematic error depending strongly on the inverse number of model particles occupying a grid cell each time step. Practically, it was established that one needs at least 20 - 30 or more particles per cell (depending on the simulated gas flow) in order to obtain reliable values for the macroscopic flow characteristics. Later, as a result of subsequent theoretical investigations, several collision schemes with better characteristics have been proposed:”Null-Collision” [14], ”Ballot-Box” [3, 28],”Modified-Nanbu” [16, 1], ”Majorant Collision Frequency” [13], and ”No Time Counter (NTC)”[5]. The most frequently used scheme has become the Bird’s NTC scheme and further in the text we will refer to it as the ”standard scheme”. All these approaches are based on the principle of estimating the maximum collision frequency in cells in order to define the number of particle pairs to be checked for collision. After that the real number of collisions in cells is specified by using the acceptance-rejection procedure. These schemes differ from one another mainly in the definition of the maximum collision frequency. In the cases of simulation of strong non-equilibrium flows such as shock waves etc., all of them use a large number of particles per cell (at least N = 10 − 20) in order to obtain reliable results. The same is true for the recently proposed Monte Carlo schemes for near-continuum [19, 17, 10, 8] and low speed [2, 12] gas flows which take into account the corresponding asymptotic properties of the Boltzmann equation. In particular, in papers [2, 12, 8] the authors suggest different interesting and promising approaches to achieve a variance reduction in the numerical solution by simulating only the deviation from the local equilibrium state. However, accurate simulation seems to need a large number of particles per cell to avoid the stochastic errors arising in the procedure of particle generation and cancelation that is an intrinsic part of the proposed algorithms. The source of these stochastic errors is closely related to

DSMC WITH SMALL NUMBER OF PARTICLES

3

the generation within each time step of random values from a small sample-size set of particles representing the local velocity distribution function within a cell. This leads to a systematic bias estimation of important parameters in the simulation. Similar considerations highlight the reasons for the limitations of the standard DSMC method when run with small numbers of model particles per cell. One could note here that having less particles per cell is not a very important consideration in DSMC which is well-known for requiring very little storage but being very CPU intensive and that the real improvement of the method lies in reducing the statistical fluctuations which are the primary source of computational cost. For one- or two-dimensional and simple three-dimensional flows, a particle simulation with large number of particles at relatively moderate Knudsen number is not a problem for present-day computers. For example, the memory requirements for simulation of a 3D problem at Knudsen number Kn = 0.03 − 0.05 on a fine enough mesh 100 × 100 × 100 cells with 50 particles per cell can be handled by a typical home PC. However, the simulation of the same flow at Knudsen Kn = 0.01 requires a mesh 300 × 300 × 300 and the memory requirements with 50 particles per cell cannot be handled by a single PC. Thus, the real problem arises when one wants to simulate a complicated multidimensional gas flow requiring a computational grid with a large number of cells. The vortical and unstable rarefied gas flows [23], taking place at small Knudsen number, are typical examples for such flows. In these cases, one unavoidably faces an insurmountable contradiction between the requirements of the method and computational efficiency and memory constraints. In practice, when the simulation is conducted with small number of particles per cell, a multigrid approach can be used appropriately in order to reduce the statistical fluctuations. Thus, the sampling is conducted on a coarser grid with larger cells containing a group of cells belonging to the basic finer grid. The multilevel grid scheme allows a correct calculation of the molecular process on a finer grid and at the same time provides a meaningful sample size of the macroscopic variables by averaging simultaneously over a group of cells. In case of a unsteady-state flow simulation the instantaneous fields are sampled by additional time averaging over an appropriate number of time steps τ (τ is less than the inverse value of mean collision frequency). Finally, the obtained instantaneous macroscopic fields are processed

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S. STEFANOV

by a simple average filter in order to remove the high-frequency statistical fluctuations from the simulation results. However, there is a reason according to which the macroscopic fluctuations should not be treated as undesirable noise produced by the simulation method. If one considers a gas flow in a very small domain, whose size is comparable to the molecular mean free path, the macroscopic fluctuations might be physically important and along with the basic macroscopic variables should also be investigated. This seems a versatile situation in many potential micro and nano gas flow considerations. As shown in section 2, the DSMC method is not only a computational tool for evaluation of the local macroscopic flow characteristics but also a method able to capture effects of local fluctuations and correlations of the macroscopic variables. Thus, one of the possible directions for the DSMC improvement is to find a way for reduction of the statistical fluctuations via sophisticated modifications of the algorithm taking into account some Boltzmann equation properties (see for example [12]). Another direction is to keep the features of the original method and improve its accuracy and efficiency by improving the stochastic properties of the simulation algorithm and splitting scheme. In the present paper we follow the second line of consideration. Being this in mind, the following question is natural: is it possible to modify the standard DSMC method so as to allow a particle simulation of rarefied gas flows with a smaller mean number of particles per cell? It should be noted that the question does not concern the total number of particles in the entire domain, which must be large enough. Before answering the question, let us analyze the possible sources of the stochastic errors. To this aim let us imagine that we have simulated a gas flow fulfilling all the standard DSMC method requirements for an accurate simulation. In other words, the simulation has used a fine enough computational grid, a small enough time step and a large enough number of particles per cell. Hence, as a result we have obtained a precise enough numerical solution of the considered problem. After that let us repeat the simulation refining the temporal and spatial resolution but keeping the total number of particles in the entire computational domain the same as in the first calculation. Surprisingly, we would fail to obtain a better numerical solution and instead could obtain a less accurate numerical solution. As shown later in the

DSMC WITH SMALL NUMBER OF PARTICLES

5

next section, a deeper analysis of the computational procedures shows that the less accurate result is a consequence of the violation of the requirement for a sufficient number of particles per cell. Insufficient number of particles per cell along with inappropriate temporal and spatial discretization are sources of stochastic errors, which violate the accuracy of the binary collision procedure. The stochastic errors result in an unacceptable increase in the number of repeated collisions in cells and unrealized potential collisions between particles at small separation distances situated in neighboring cells. Recently, Bird proposed a new version of the DSMC method with improved details of some of the procedures (see the materials of his short course presented at the DSMC conference in Santa Fe, 2007). In the improved collision NTC algorithm the problem with the insufficient number of unrealized potential collisions between particles situated in neighboring cells was partially solved within the subcell areas of the computational grid by using a sophisticated algorithm for finding partners for a collision among single particles belonging to different neighboring subcells. However, the realization of repeated collisions remained as a source of stochastic error. The convergence of the sophisticated Bird’s algorithm is studied by Gallis et. al. [11]. Their analysis showed that, in general, a smaller number of particles than 10 − 20 in cells worsens the accuracy of the sophisticated method. In the present paper, an alternative solution of the problem is proposed using a universal two-step collision procedure, which allows more accurate calculations with a smaller mean number of particles in the grid-cells. Considered from viewpoint of the general simulation algorithm the two-step collision procedure represents an intrinsic part of a symmetric Strang splitting scheme [25] that improves the accuracy of the splitting method to O(τ 2 ) with respect to the time step τ . The collision algorithm includes two new elements. First, the standard ”No Time Counter” collision procedure is replaced with the Bernoulli trials scheme, proposed by Yanitskiy [3, 28], or its simplified version, proposed by the author [24]. By using this procedure, we avoid the production of a part of the successively repeated collisions, which would otherwise occur if the ”No Time Counter” scheme was applied to cells with a small number of particles. Second, for each time step τ the latter procedure is applied twice in two successive half-time τ /2 steps on a dual grid: in the first step it is applied to all cells

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S. STEFANOV

of the standard grid; in the second step - to all cells of a grid shifted with respect to the initial grid in each coordinate direction by a distance approximately equal to a half-cell size h/2 (the distance is exactly h/2 for uniform orthogonal grids). By utilizing this second step, we make possible collisions of other pairs of particles, belonging to neighboring cells in the first half-time step and which are at a separation distance smaller than the cell size. In the shifted grid they become particles from one cell and, respectively, are checked for a possible collision. Thus, the new two-step collision procedure executed on this ”staggered” grid is able, compared to the standard ”No Time Counter” scheme, to avoid a part of the eventually repeated collisions and takes into account other collisions between particles from neighboring cells. The general algorithm becomes symmetric within a time step if the macroscopic variable sampling is settled between both half-time collision steps. In view of the general splitting algorithm we have obtained a Strang symmetric splitting scheme. According to Bobylev and Owhada [6] analysis of a generalized non-linear integro-differential operator equation the Strang splitting scheme improves the approximation accuracy of the method up to O(τ 2 ) with respect to the time step. The results from the calculations obtained by using the modified scheme with different time steps confirm their analysis (see section 5). Thus, the modified algorithm has two advantages over the standard one: first, the accuracy of the simulation is preserved when the mean number of particles in cells becomes very small - down to hN i ∼ 1 and might be less in many cases, and the second is the improved simulation accuracy with respect to time step. The disadvantage is that the computational efficiency of the Bernoulli trials collision scheme [28] used in the modified DSMC algorithm is inversely proportional to the number of particle pairs 2/Np = 2/N (N − 1) while the efficiency of the standard collision algorithm is proportional to 1/N . This disadvantage is partially avoided in the simplified version of the Bernoulli trials scheme, in which the number of pairs checked for collision is appropriately cut off so that the algorithm requires a number of operations proportional exactly to N − 1. Thus the latter scheme demonstrates a linear dependance on the number of particles likely to the standard algorithm. In the general case the simplified algorithm is more efficient than the original Bernoulli trials scheme but still slightly less efficient than the standard one in simulations using a larger number of

DSMC WITH SMALL NUMBER OF PARTICLES

7

particles per cell and keeping equal the other computational parameters. However, when the average number of particles per cell is small hN i ∼ 1 the modified DSMC method with implemented one of the modified collision algorithms demonstrates an efficiency approximately equal to that of the standard algorithm. Having in mind the latter notes, the modified algorithm should be preferred to the standard one in the following cases. First, when the DSMC simulation of a two- and three-dimensional rarefied gas flow requires a very fine uniform grid or a multilevel grid with subcell areas where the average number of particles per cell is unavoidably reduced to one-two particles per cell. The vortical and large gradient gas flows are typical flows of that kind whose accurate simulation requires a fine resolution mesh. Second, the improved accuracy with respect to time step makes the modified algorithm more suitable for simulation of unsteady-state multidimensional flows. In this paper, we present some validation results obtained from the simulation of the following problems: a one-dimensional shock wave formation in front of a moving supersonic speed piston [5] (see Fig. 1), a one-dimensional steady-state strong shock wave [5] and two-dimensional lid-driven cavity flow [15]. The new scheme has been applied successfully for the simulation of the three-dimensional Rayleigh-B´enard convection of a rarefied gas [22]. 2. The standard DSMC algorithm. Before proceeding to the description of the modified DSMC procedure we will review briefly the theoretical background of the DSMC method. In general, a detailed mathematical description of the motion of a rarefied gas system can be given by an evolutionary kinetic equation in the following non-closed form with respect to the velocity distribution function f (t, x, ξ): (2.1)

h i ∂ f (t, x, ξ) = −D[f (t, x, ξ)] + Q f (2) (t, x, ξ, x∗ , ξ∗ ) , ∂t

where f (t, x, ξ) = f (1) (t, x, ξ) and f (2) (t, x, ξ, x∗ , ξ∗ ) are one-particle and two-particle distribution functions of the particle velocities ξ and ξ∗ at time t and spacial coordinate x, D denotes a linear differential operator describing the free particle motion and Q is a non-linear integral operator describing the particle binary interactions. Equation (2.1) is nothing more than the last equation in the so-called BBGKY-hierarchy for an N-particle system with short-range intermolecular potential [7]. If σ denotes a collision

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S. STEFANOV

cross-section with finite diameter d of a cut-off intermolecular potential (for a hardsphere gas σ = πd2 , where d is the molecular diameter), in the Grad-Boltzmann limit (N → ∞, σ → 0, σN bounded) and under the assumption of molecular chaos, then (2.2)

f (2) (t, x, ξ, x∗ , ξ∗ ) = f (1) (t, x, ξ) f (1) (t, x∗ , ξ∗ ) ,

equation (2.1) is transformed into the well-known Boltzmann equation (2.3)

∂ f (t, x, ξ) = D[f (t, x, ξ)] + Q[f (t, x, ξ) , f (t, x, ξ))], ∂t

where operators D and Q take the form (2.4)

D = −ξ

∂ , ∂x

Z (f∗0 f 0 − f∗ f ) B (g, θ) dΩ(θ)dξ∗ .

Q= R3 ×S2

For simplicity, equation (2.3) is presented without an external force term. In Eq. (2.4), f∗ = f (t, x, ξ∗ ), f 0 = f (t, x, ξ 0 ) and f∗0 = f (t, x, ξ∗0 ). The post-collision particle velocities ξ 0 and ξ∗0 are expressed by the formulae: (2.5)

ξ 0 = ξ + θ[(ξ − ξ∗ ).θ] ξ∗0 = ξ∗ − θ[(ξ − ξ∗ ).θ].

The function B (g, θ) depends on the intermolecular potential, θ is a unit vector, expressing the variation of the particle velocity owing to a particle collision [21], g = |ξ − ξ∗ | is the relative molecular velocity, and dΩ is a solid angle element in the direction of θ. In general, function B is related to the collision cross-section σ in a straightforward way (see Refs. [7],[21]). A mathematical formulation of a rarefied gas problem based on Eq. (2.3) must be completed by corresponding initial and boundary conditions for the velocity distribution function defined respectively in the volume and at the boundaries of the computational domain D(x). Now, consider a computational domain D(x) divided into M cells by step h and let the time t be divided into small intervals by step τ (tk = kτ , k = 1, ..., K). Let the real gas in the domain be replaced by a finite set of N model particles presented by their positions and velocities that are changing over the simulated time period {X(tk ), Ξ(tk )} = {xi (tk ), ξi (tk )}, i = 1, . . . , N . The model particles are distributed M P into the grid cells so that N = N (l) (N (l) is the number of particles in cell (l)) l=1

DSMC WITH SMALL NUMBER OF PARTICLES (l)

9

(l)

and {xj (tk ), ξj (tk )}, j = 1, . . . , N (l) . The velocity distribution function f (t, x, ξ) is approximated in the following form (see Sone [21]) (l)

(2.6)

f (t, x, ξ) = ∆

M N P P

l=1 j=1

(l)

(l)

δ(x − xj (t))δ(ξ − ξj (t)),

(x ∈ D) ,

f (tk , X(tk ), Ξ(tk )) ≡ f (tk , x, ξ) where δ is the Dirac delta function, ∆ is a scaling factor. Calculating the mean value of f from (2.6) over the cell volume |dx(l) | = hα (α = 1, 2, or 3 is the volume dimension) we obtain a course-grained approximation of the velocity distribution function in cell (l) (2.7)

Z N (l) i ∆ ∆ X h (l) ¯ ¯ ¯ δ ξ − ξ (t) = f (t, x,ξ)dx, f (t, x(l) , ξ) = ¯¯ j ¯ ¯ (l) ¯ ¯dx(l) ¯ j=1 ¯dx ¯ (l) dx

where x(l) is a reference point in cell (l) (e.g. the geometric center of cell (l)). Formally, the standard DSMC method makes use of the following splitting scheme, applied to the discrete velocity distribution function known at time tk , for obtaining an approximation of the solution of Eq. (2.3) at tk+1 : (2.8)

τ,h f [tk + τ, X(tk ), Ξ(tk + τ )] = SQ {f [tk , X(tk ), Ξ(tk )]} ,

f (t0 , X(t0 ), Ξ(t0 )) ≡ f (0, x, ξ) (2.9)

τ f [tk + τ, X(tk + τ ), Ξ(tk + τ )] = SD {f [tk + τ, X(tk ), Ξ(tk + τ )]} .

For brevity hereafter, it is reasonable to assume that all mathematical conditions needed for a correct consideration are fulfilled. In equations (2.8) and (2.9) we denote τ,h τ and SD the numerical algorithms approximating the action of the by operators SQ τ,h collision and convective terms in Eq. (2.3), respectively. Operator SQ acts locally on τ the particle subset N (l) in each cell dx(l) , while operator SD acts on the total particle τ,h set within the entire computational domain D(x). A specific characteristic of SQ

is that the particle positions are neglected and a uniform random distribution of the particles in each cell is assumed. This assumption is an equivalent of the transformation to a course-grained distribution function given by Eq. (2.7). Actually, a rigorous consideration should take into account the fact that the DSMC collision algorithm τ,h SQ does not assume that the particle velocities are statistically independent i.e.

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S. STEFANOV

τ,h equality (2.2) must be fulfilled. Thus, operator SQ acts rather on a course-grained

two-particle velocity distribution function f (2) (t, x(l) , ξ, ξ∗ ) than on the one-particle τ distribution function given by Eq. (2.7). On other hand, the action of operator SD is

invariant with respect to both one- and two- particle distribution functions. Thus, the DSMC method is capable of estimating the evolution of a course-grained two-particle velocity distribution function, and consequently, of calculating corresponding local correlations and fluctuations of macroscopic variables ( see Ref. [23]). τ,h If we denote by SQ+D the operator evaluating the solution of (2.3) at tk + 1 from

the state at tk then the splitting method is expressed with the approximation τ,h τ τ,h SQ+D ≈ SD SQ .

(2.10)

Using the result obtained by Bobylev and Ohwada [6] and taking into account Eq. (2.7) one can show that the splitting method approximates the Boltzmann equation with accuracy O(τ + h). So far we was not concerned about the stochastic aspects of the DSMC method. To enlighten the possible stochastic errors and their sources some details of the stanτ,h τ and SD should be dard two-stage DSMC algorithm [5] presented with operators SQ

given. During a particle simulation the following two stages are performed over each time step (tk , tk+1 ), k = 1, ..., K : Stage 1. (Binary collision procedure) Three steps are included in the “No Time Counter” collision procedure performed in each cell l, l = 1, ..., M : • computing the number of particle pairs Nc to be checked for a collision from the formula ¯ ¯ D E ¯ (l) ¯ Nc = 1/2∆N (l) N (l) (σg)(l) τ / ¯dx ¯, max

(2.11) (l)

where (σg)max is a superior parameter, which is updated continuously during the simulation; • (acceptance-rejection method) each pair (i, j), 1 ≤ i < j ≤ N (l) , chosen at random from the particle subset N (l) , is checked for a collision with probability (2.12)

pij =

σij gij , (σg)max

DSMC WITH SMALL NUMBER OF PARTICLES

11

where σij is the effective collision cross-section of pair (i, j) and gij = |ξi − ξj |. • if the collision is accepted then the particle velocities are changed to their post-collision values (2.13)

ξ0i =

1 2

[(ξi + ξj ) + |ξi − ξj | ω]

ξ0j =

1 2

[(ξi + ξj ) − |ξi − ξj | ω] .

The system of equations (2.13) is an alternative of (2.5) where ω is a ¡ ¢ unity vector pointing out from new direction of relative velocity ξ 0 i − ξ 0 j . The end of vector ω is uniformly distributed on a sphere of unit radius. During stage 1, the particle positions are not changed. Stage 2. (Free particle motion) Each particle xi , ξi , i = 1, ..., N is moved over the time step τ to its new position: (2.14)

x0 i = xi + ξi τ.

The boundary conditions are also simulated within Stage 2. The macroscopic flow properties in each cell are sampled after a few time steps. This extended time interval is employed in order to avoid eventual correlations between successive samples. The stochastic nature of the procedure gives rise to stochastic errors that are in addition to the splitting scheme error. A deeper analysis of the stochastic errors requires rigorous mathematical treatment (see, for example, Dimov et. al. [9]) that might be a matter of supplementary studies beyond the scope of the present paper. Here the consideration is restricted to the examination of the stochastic errors caused by inappropriate spatial and temporal discretization and the appearance of areas of computational cells with insufficient number of model particles. The case of simulation with an insufficient total number of particles in the entire computational domain must also be considered. The above situations usually occur in large two- and threedimensional DSMC calculations or in simulations demanding grids with local subcell areas. Considering a grid with a small number of particles per cell, the conditions τ τ,h for correct simulation using the standard DSMC scheme SD SQ are violated in the

following two areas:

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S. STEFANOV

• the relative velocities between the small number of particle pairs in a cell forms a strongly degenerated probability distribution repeatedly used in the acceptance-rejection procedure (stage 1., Eq. (2.12)) within each time step ; • a significant portion of pairs of particles situated at a collision distance in different neighboring cells, that could contribute effectively to the local collision rate, is neglected. Such pairs are potential candidates for collision that cannot be realized by the standard DSMC scheme. It can be shown that the first situation leads to an unacceptable increase in the number of successively repeated collisions in cells while the second to an increase of the number of unrealized potential collisions between particles belonging to neighboring cells. The combined effect of both actions results in a local loss of potential collisions. This is evident for the pairs composed of particles belonging to neighboring cells. For example, some of the particles can exchange their positions during stage 2 without having been checked for collision. However, the assertion that the effect of successively repeated collisions is a reduction in the effective number of collisions must be clarified. Assertion can easily be proved for the elastic collision model with central symmetry of the interaction potential. Let us estimate the effect of n repeated collisions of the particle pair (i, j) belonging to a given cell. Let particles have initial velocities (ξi (0) , ξj (0) ). Applying equation (2.13) n times we obtain the post-collision velocities after n repeated collisions: ·µ ¶ ¯ ¸ ¯ (n−1) ¯ ¯ (n) ξ 0 i = 12 ξi (n−1) + ξj + ¯ξi (n−1) − ξj (n−1) ¯ ω (n) ¯ ·µ ¶ ¯ ¸ (2.15) ¯ (n−1) (n−1) ¯ (n) ¯ ω (n) . ξi (n−1) + ξj − ¯¯ξi (n−1) − ξj ξ0j = 21 ¯ (n−1)

(ξi (n−1) ¯+ ξj ) = (ξi (0) + ξj ¯Taking into account ¯ that ¯ ¯ (n−1) ¯ ¯ (n−1) ¯ ¯ξ i ¯ = ¯ξi (0) − ξj (0) ¯ we obtain − ξj ¯ ¯ ¯ ¯

(2.16)

ξ0j

(n)

(n)

) and

¸ ¯ ¯ ¯ ¯ + ¯ξi (0) − ξj (0) ¯ ω (n) ¯ ¶ ¯ ¸ ·µ ¯ (0) ¯ (0) = 21 − ¯¯ξi (0) − ξj ¯¯ ω (n) . ξi (0) + ξj ·µ

ξ0i

(0)

=

1 2

ξi (0) + ξj

(0)

¶

Thus the post-collision velocities obtained as a result of using n successively repeated elastic collisions are statistically distributed in the same way as if they are obtained by the realization of only one collision of the chosen pair. In the case of non-elastic

DSMC WITH SMALL NUMBER OF PARTICLES

13

repeated collisions, a reduction of the effect of collisions could also be made evident. As a consequence, the major effect on a standard DSMC simulation with small number of particles in cells is a reduction in the local collision frequency, which only converges to the Boltzmann collision frequency for large enough number of particles per cell. 3. Probability analysis of repeated collisions. Let us make an insight into the events of repeated collisions and discover the role of the acceptance-rejection procedure. The effect of the number of repeated collisions on the DSMC method accuracy has been studied numerically by Shevyrin et. al.[20]. Their analysis was based on the simulation of two classical one-dimensional rarefied gas flows: the plane Couette flow and the heat transfer between two parallel plates. The established correlation between the number of repeated collisions and the deviation of the computational results from the Boltzmann equation solution has demonstrated clearly a tangible effect of the repeated collisions on the simulation accuracy. In the present paper we go further analyzing the probability of repeated collisions of a certain pair (i, j) chosen from N ≥ 2 particles that occupy a cell within time step τ . Let within this time step m ≥ 2 be the maximum number of collisions, calculated from Eq. (2.11), that can be realized in the considered cell. The actual number of collisions is obtained by applying m times the acceptance-rejection procedure of the standard collision algorithm. The result of each iteration of the algorithm with respect to pair (i, j) can be described by the set of mutually exclusive and exhaustive events: A, the velocities of particles (i, j) are changed to their post-collision values (2.13) due to a collision between these particles; B, the velocity of particle i or particle j is changed due to a collision with other particle k 6= (i, j); C, the velocities of particles i and j are not changed (collisions with particles i and j are not occurred). Let PA , PB , PC be the corresponding probabilities for accurrence of one of A, B, or C events. The following equality is hold after each iteration of the collision algorithm: (3.1)

PA + PB + PC = 1.

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S. STEFANOV

For m iteration we obtain m

(3.2)

(PA + PB + PC )

The trinomial (3.2) expands as follows: X (3.3) (PA + PB + PC )m =

= 1.

µ

k1 +k2 +k3 =m

¶ m P k1 P k2 P k3 , k1 , k2 , k3 A B C

where each multinomial term of the expansion (3.3) µ ¶ m (3.4) Pm (k1 , k2 , k3 ) = P k1 P k2 P k3 , k1 , k2 , k3 A B C determines the probability of occurrence of a sequence of events with length m containing k1 events A, k2 events B and k3 events C. The multinomial coefficients µ ¶ m m (3.5) = k1 , k2 , k3 k1 !k2 !k3 ! count up the number of distinct ways to permute the sequence of m events, and k1 , k2 , k3 are the multiplicities of each of the distinct events. To compute the probability of occurring of one or more repeated collisions of a given pair (i, j) we must take into account all sequences containing at least two events A (k1 ≥ 2) and exclude from them the sequences in which every two events A are separated by at least one event B. Thus, sequences of type 0 AABC...0 , 0 ACCAB...0 , 0 BACAB...0 belong to the set of sequences containing repeated collisions while sequences of type 0 ABABABA0 t, 0

ACCBACBA0 must be excluded from this set. Our aim is to analyze short sequences

of events that corresponds to the calculations of collisions in cells with small number of particles by using the standard collision algorithm. Before calculating the probability of repeated collisions we must determine the probabilities PA , PB and PC . The acceptance-rejections procedure for calculations of a collision consists of two steps: first, choosing a pair (i, j) at random from Np = N (N − 1)/2 pairs with probability p1 = 1/Np = 2/N (N − 1) and second, colliding the pair (i, j) with probability pij (see Eq. (2.12)). Consequently, the probability PA for collision of the pair (i, j) is equal to PA = p1 pij = 2pij /N (N − 1). Similar reasonings are used for calculation of PB . Finally, we obtain

(3.6)

        

PA =

2pij N (N −1) ;

PB =

1 N (N −2) (

P k6=i,j

PC = 1 − PA + PB .

pik +

P k6=i,j

pjk );

DSMC WITH SMALL NUMBER OF PARTICLES

15

The consideration of event sequences with maximum number of collisions m = 2 and m = 3 is enough for a qualitative analysis of the probability of repeated collisions. For m = 2 equality (3.3) is reduced to (3.7)

(PA + PB + PC )2 = PA2 + PB2 + PC2 + 2(PA PB + PA PC + PB PC ).

Solely the sequence of events 0 AA0 leads to a repeated collision of pair (i, j) i.e. the probability PR that a collision is repeated is equal to PR = P2 (2, 0, 0) = PA2 .

(3.8) For m = 3 equality (3.3) reads

(3.9)

(PA + PB + PC )3 = (PA3 + 3PA2 PB + 3PA PB2 + PB3 + 3PA2 PC + 6PA PB PC + 3PB2 PC + 3PA PC2 + 3PB PC2 + PC3 ).

Now the probability that at least one repeated collision occurs is equal to   P 3 + 3P 2 PC = P 2 (1 + 2PC ), PB = 0, N = 2, A A A (3.10) PR =  P 3 + 2P 2 P + 3P 2 P = P 2 (1 + P + 2P ), N ≥ 3. B C A A B A C A The probability of repeated collisions is derived from the probabilistic description of multiple iterations of the acceptance-rejection procedure used in the standard collision algorithm. The analysis of equations (3.8) and (3.10) shows that the probability of repeated collisions is of order O(p2ij /Np2 ) and increases quickly when the number of particles in cell decreases. There is a quadratic dependence of PR on the relative velocity because pij is proportional to gij (see Eq. (2.12)). As a result a dominated part of repeated collisions is due to pairs with larger relative velocities. This leads to an extra distortion of the collision process in the range of high relative velocities. The effect is clearly seen in DSMC calculations of strong shock waves (for example see the computational results for shock wave formation given in section 5. In order to avoid these negative effects in the case of small number of particles in cells the acceptancerejection procedure should be replaced with other collision procedure, which does not produce repeated collisions within time step τ . It is worth noting that the attempt to solve the problem by doing the time step smaller does not improve significantly the collision process. A raise of repeated collisions still exists because many particle

16

S. STEFANOV

pairs remain in the same cells within two or more successive time steps and, consequently, might realize repeated collisions. In order to avoid repeated collisions in this case one should also refine the spatial resolution. In the next section we present a modified algorithm that meets the above requirements for accurate simulation with small number of particles in cells. 4. Two-step collision procedure for the DSMC method. In order to improve the stochastic properties of the binary collision process in cells or subcells of a local refined grid containing small number of particles per cell, two new elements are included in the modified collision algorithm (stage 1 of the general scheme). First, we have replaced the standard binary collision procedure proposed by Bird with the “Bernoulli trials” scheme proposed by Yanitskiy [28] or its simplified version [24] . By using one of these procedures, we avoid the production of at least part of the successively repeated collisions, which occur in the Bird’s “No Time Counter” scheme when it is applied to cells with a small number of particles. Second, the collision process is accomplished by successively sweeping the cells (or subcells) of a dual grid - each time within a half-time step τ /2. The dual grid is constructed as follows: in the first half-time step, following the standard approach, the grid of cells is mapped exactly onto the considered domain. In the second half-time step the grid configuration is defined by lines connecting geometric centers of cells used in the first half-time step. In the case of an orthogonal grid system, the second grid is obtained by translation of the first at distance equal to a half-cell size h/2 in each coordinate direction. This second step allow us to consider possible collisions of other pairs of particles, which are at a separation distance smaller than cell size h but have belonged to neighboring cells in the first half-time step. In the shifted grid they become particles from one cell and are checked for a possible collision. Thus, the new two-step collision procedure is able to avoid part of the eventual repeated collisions in Bird’s standard procedure and take into account other collisions, which cannot happen, when the standard scheme is used, because the particles belong to different grid-cells. For resolving the latter situation in the areas with subcells, Bird [5, 11] suggested a search procedure for collision partners within the neighboring subcells. This algorithm partially improves the collision process but it costs additional computational time. The new two-step colli-

DSMC WITH SMALL NUMBER OF PARTICLES

17

sion scheme on a dual grid also increases the computational time per time step τ but this disadvantage is compensated by the use of smaller average number of particles in each cell. Taking into account the remarks above, the modified two-step collision algorithm for a dual grid is realized as follows: The collision process (stage 1 of the general DSMC algorithm) within each basic time step τ is split into two successive steps - each with duration τ /2: • First step: For each cell l (l = 1, ..., M ) (with volume V (l) = |dx(l) | = hα and reference point x(l) ) of the grid mapped exactly on the considered domain the following steps are carried out: – all particles are given local indices with regard to their cell disposition; – in each cell the following Bernoulli trials algorithm [28] (see also [21]) (algorithm BT-A) is realized:

Algorithm BT-A (Bernoulli trials): – the combinations of all pairs (i, j) are chosen sequentially from N (l) particles in cell l as follows: - the first particle of each pair is the particle with index i = 1, . . . , (N (l) − 1) chosen in strict order from the particle list created for cell l; £ ¤ - the second particle is the particle with index j ∈ i + 1, N (l) chosen in strict order from k = (N (l) − i) particles taking place in the list after particle i. - particle pair (i, j) with velocities (ξi , ξj ), i < j = 1, . . . , N (l) is checked for collision with probability (4.1)

Wij = 4

σij gij τ /2 , V (l)

where Wij must satisfy the condition (4.2)

Prob{Wij ≥ 1} → 0.

- if the collision is accepted the velocities (ξi , ξj ) are changed to their

18

S. STEFANOV

post-collision velocities (ξ 0 i , ξ 0 j ) according to Eq. (2.13), otherwise the velocities are left unchanged. 2

The algorithm BT-A requires number of operations O(N (l) ) in cell l. • Second step: The grid is reconfigured as described above so that each new cell l0 (l0 = 0

0

0

1, ..., M + 1) (with volume V (l ) = |dx(l ) | = hα and reference point x(l ) = x(l) − ε(l) ) is shifted at distance ε(l) in each coordinate direction with respect to the corresponding cell of the initial grid. For an orthogonal uniform grid ε(l) = h/2. All sub-steps of the first step are repeated on the shifted grid. Within the second step the pairs of particles at a separation distance less than h/2, which have belonged to neighboring cells in the first step, are checked for a collision. It can be seen that the number of cells in the second step is increased by one in each co-ordinate direction so that the second grid overlaps 0

the first one. The volume V (l ) of the cells intersecting the boundaries of the computational domain (or the corresponding subcell area) should be reduced to a fractional volume equal to the part, which is inside the computational domain (or the considered subcell area). Only particles belonging to this volume should be counted for collisions in this cell. For an orthogonal uniform 0

α

grid V (l ) = (h/2) . The second stage (free molecular motion) of the general DSMC algorithm remains the same as specified in the standard scheme. The quadratic dependance of the efficiency of algorithm BT-A on the particle number can be compensated when the number of particles in cell is small N (l) → 1. We suggest another way for reduction of the required number of operations. The following simplified algorithm BT-B realizes a number of computations proportional to the number of particles (N (l) − 1) and can be used in many cases instead the algorithm BT-A in the general DSMC scheme. Its description is given as follows:

Algorithm BT-B: • a sequence of pairs i = 1, . . . , (N (l) − 1) is chosen from N (l) particles in cell l as follows:

DSMC WITH SMALL NUMBER OF PARTICLES

19

- the first particle i is the particle with index i in the particle list created for cell l; £ ¤ - the second particle j ∈ i + 1, N (l) is chosen with probability p1 = 1/k from k = (N (l) − i) particles taking place in the list after particle i. • particle pair (i, j) is checked for collision with probability ˆ ij = k4 σij gij τ /2 , W V (l)

(4.3)

ˆ ij must satisfy the condition where W ˆ ij ≥ 1} → 0. Prob{W

(4.4)

Thus, the total probability for a collision of pair (i, j) is equal to PA = ˆ ij = Wij , i.e. it is equal to the collision probability (4.1) in algop1 W rithm BT-A ; • if collision is accepted then velocities (ξi , ξj ) are changed to the postcollision values (ξ 0 i , ξ 0 j ), otherwise they remain unchanged. Remark. For an appropriate non-dimensional form of the variables the scaling factor 4 persisting in both algorithms becomes equal to 4 = 1. In dimensional form 4 is equal to the number of real gas molecules represented by each simulated particle. The better efficiency of algorithm BT-B with respect to the number of particles in cell requires a stronger limitation of the time step, i.e. a smaller time step (see condition (4.4)). At the same time both algorithms, BT-A and BT-B, avoid the realization of repeated collisions within the considered step. In the framework of the general Strang splitting scheme algorithms BT-A or BT-B are applied twice with step (τ /2,h)

τ /2 realizing the action of operator SQ

. Within a full time step τ a particle pair

can realize a repeated collison only in the case if both particles are situated in the 0

volume (V (l) ∩ V (l ) ) shaped by the intersection of cells of standard and shifted grid, or in other words, when two particles belonging to a cell within the first half-time step are found to be in one cell again within the second half-time step. τ,h For more details on the approximation SQ of the Boltzman collision operator

by the Bernoulli trials scheme (algorithm BT-A) the reader is referred to Yanitskiy’s paper [28] and Appendix B in Sone’s book [21]. A rigorous derivation of the simplified

20

S. STEFANOV

version (algorithm BT-B) is presented in [24] . Here we shall restrict ourselves to the consideration of the mean collision frequency determined using the Bernoulli trials scheme (algorithm BT-A) showing that it is equal to the Boltzmann collision frequency for an arbitrary number of particles in a cell if the particle number varies over time in accordance with the Poisson distribution (4.5)

P (N, λ) =

λN e−λ , N!

where N = N (l) and λ is the mean value of N (l) . Let h.i be an averaging operator and let hσgi be the mean value (time or ensemble averaged) of (σij gij ) of all particles in a given cell. It can be shown that the average number of collisions per cell realized within a time step is equal to ¿ À 2 hN i hσgi τ N (N − 1) N (4.6) Nth = W = 4 (l) = n hσgi τ, 2 2 2 V where n = 4(N/V (l) ) is the local number density and the equality hN (N − 1)i = 2

hN i , which holds for the Poisson distribution (4.5), is taken into account. The last term in (4.6) coincides with the theoretical expression derived from the Boltzmann equation. Condition (4.2) is important to hold in order to keep Nth ≤ hN (N − 1)/2i. It can also be shown that the Poisson distribution of the particle number in a cell follows from the assumption for statistical independence of the local particle velocities (see Eq. (2.2)). Practically, we have observed a Poisson distribution of the particle number in cell in many rarefied gas flows. However, it should be noted that the complete treatment of the matter is a difficult task beyond the aim of the present paper. The splitting method with the new two-step collision algorithm employs the following operator: (τ /2,h)

τ,h τ SQ+D ≈ SD SQ

(4.7) (τ /2,h)

where SQ

0

(τ /2,h)

SQ

, 0

(τ /2,h)

acts on the basic grid within the first half-time step and SQ

acts on the shifted grid within the second half-time step. If the sampling procedure is (τ /2,h)

executed between SQ

0

(τ /2,h)

and SQ

the splitting formula (4.7) takes the symmetric

form (4.8)

(τ /2,h)

τ,h SQ+D ≈ SQ

0

(τ /2,h)

τ SD SQ

.

21

DSMC WITH SMALL NUMBER OF PARTICLES

The splitting formula (4.8), applied to Eq. (2.3) on a dual grid, is the analog of the scheme proposed by Strang [25], [6], which is second-order accurate in time. Thus, the proposed modification is twofold: first, it improves the stochastic properties of the collision algorithm in order to be run accurately with small number of particles per cell, and second, the new splitting scheme is second-order accurate in time. A disadvantage is that the efficiency of the “Bernoulli trials” collision scheme (algorithm BT-A)is proportional to (1/N 2 ) , while the efficiency of NTC is proportional to (1/N ). The simplified version (algorithm BT-B) reduces the quadratic dependency of the algorithm BT-A to linear one (its efficiency is proportional to 1/(N − 1)). Since each of the collision algorithms requires a number of operations proportional to the number of checked particle pairs for collision within a time step it can be shown that when the condition N − 1 < Nc (Nc is calculated from (2.11)) is fulfilled the algorithm BT-B is more efficient than the standard NTC. However, in practice for a larger number of particles in cell N ≥ 2 the computational requirements for time step and cell size lead to N − 1 > Nc and the standard NTC becomes a bit more efficient. There are indications that the modified algorithm has some influence upon the spatial approximation of the simulated problem and, respectively, it should also be analyzed along with the time approximation. A detailed analysis of the modified DSMC scheme requires additional mathematical work. This work is in progress and the results will be published in another paper. Here we refrain from doing this in detail and satisfy ourselves by giving an illustration of the main idea along this line of analysis. To this aim, consider the operator action on a group of adjacent orthogonal uniform cells within a time step τ . Let us decompose the two-step collision operator into two parts as follows: (4.9)

(τ /2,h)

SQ

0

(τ /2,h)

SQ

(τ /2,h/2)

= SQ

0

(τ /2,h/2)

SQ

(τ /2,h− /2,h/2)

+ SQ

0

(τ /2,h/2,h+ /2)

SQ

,

where formally (h/2) denotes the size of the overlapped volume of the cells belonging to the dual grid, whose particle pairs are checked for collisions during both half-time steps τ /2 of the two-step collision algorithm, (h− /2) is the complimentary volume of the cells used within the first half-time step and (h+ /2) is the complimentary volume of the cells used within the second half-time step only. Thus, the first term on the right-hand side of Eq. (4.9) represents a two-step collision operator applied

22

S. STEFANOV

to the overlapped domain having a size of order (h/2) i.e. half the basic spatial step size. The second term represents the two-step collision operator that accounts for the collisions of particles from the overlapped domain with particles belonging to the complementary domain surrounding the overlapped domain. Within the first halftime step, these are collisions with particles from domain (h− /2), within the second half-time step - with particles from (h+ /2). It is worth noting that in the second term, the separation distance of collided particles in both steps does not exceed the space step h. Taking into account Eq. (4.7), one comes to the following interpretation of the splitting scheme with included two-step collision operator: (4.10)

τ,h/2

(τ /2,h/2,h∓ /2)

τ,h SQ+D ≈ SQ+D + SQ+D

τ,h/2

(τ /2,h)

= SQ+D + SQ+D .

The splitting scheme (Eq.(4.10)) consists of two terms: the first represents the contribution of the splitting operator with a collision operator acting within each time step τ on each cell of a double refined grid; the second represents the contribution of the splitting operator with a collision operator acting within each half-time step (τ /2) on each pair of adjacent cells (collisions are allowed only between particles belonging to different cells). As seen from Eqs. (4.9) and (4.10), one could find some space refinement effect on the collision process that should give an improvement of the space approximation of the simulated process. Practically, the test calculations (see section ??) showed that the approximation effect on sufficiently fine grids was negligible. 5. Numerical validation of the modified scheme. Before proceeding with the numerical examples we need to settle on the calculations performed by using algorithms BT-A and BT-B. Further, in all considered simulations the time step is small enough so that the condition (4.4) is fulfilled almost always; the results obtained by using both algorithms are almost identical and we will relate them to a general modified algorithm (where needed the differences will be distinguished). As a first numerical example, we consider the one-dimensional unsteady-state problem of the formation of a strong shock wave by a piston that impulsively starts to move with a constant velocity of 2285.5 m/s in the x-direction. The gas in front of the piston uses the variable hard sphere model of argon at a temperature of 273o K and a number density of 1020 m−3 . The piston can travel within an interval of 1 m

DSMC WITH SMALL NUMBER OF PARTICLES

23

i.e. approximately 100 mfps in the undisturbed gas. This is exactly the same case as described in Bird’s book [5](§13.2). Bird’s original program DSMC1U has been used to obtain the results by the standard DSMC method. We have used the same source code to obtain the modified scheme, substituting the ”No Time Counter” collision procedure by the modified two-step one using one of algorithms BT-A or BT-B. The indexing procedure has also been reorganized slightly in order to meet the requirements of the modified two-step algorithm. Both codes, standard and modified, have been run on a fine uniform grid without dividing the basic cells into subcells, an important condition for a correct comparison of both schemes. It was found that the numerical solutions obtained from the standard scheme on grids with more than 400 cells containing an average number of 100 particles per cell practically converged with each other. The time step in all runs was chosen to be very small, τ = 1.0−7 s, in order to reduce the effect of the time discretization. Thus, only the average number of particles per cell is different in each run of the corresponding scheme. An ensemble average over a large number of repeated runs (from 1000 to 50000 depending on hN i) of the unsteady flow problem was taken for evaluation of the shock wave profiles of density, mean velocity and temperature. Figure 5.1 presents the profiles of x-velocity (a) and temperature (b) along the x-direction, computed by both the standard and modified schemes on a grid with 400 uniform cells with different number of particles per cell. Each marker of the plotted results represents every second point of the averaged value over a group of 4 neighboring cells. The reference profiles (shown as thick solid lines) are computed by the standard ”No Time Counter” NTC collision scheme with hN i = 100.0 particles per cell. The profiles computed by the NTC scheme with hN i = 10.0 and hN i = 2.0 (shown in stars) increasingly deviate from the reference profile in the lower part of the shock wave front, where the average number of particles per cell is smaller. The characteristic swelling in the lower part of the shock front is an effect of the insufficient number of effective collisions owing to an increase of repeated collisions of particle pairs with high relative velocity. Unlike these results, all profiles, obtained by the modified scheme with hN i = 10.0, hN i = 2.0 and hN i = 1.0 (shown in circles), are in a good agreement with the reference profiles obtained by the standard scheme with

24

S. STEFANOV

2500

9000 Standard, N=2 Standard, N=10 Standard, N=100 Modified, N=1 Modified, N=2 Modified, N=10

2000

7000 6000

T oK

1500 U [m/s]

Standard, N=2 Standard, N=10 Standard, N=100 Modified, N=1 Modified, N=2 Modified, N=10

8000

1000

t=2.0e−4 s

t=1.0e−4 s 4000

t=2.0e−4

t=1.0e−4

5000

500

3000 2000

0 1000 −500 0.2

0.3

0.4

0.5

0.6 x [m]

0.7

0.8

(a)

0.9

1

0 0.2

0.3

0.4

0.5

0.6 x [m]

0.7

0.8

0.9

1

(b)

Fig. 5.1. Formation of a strong shock wave in front of a piston moving with a constant velocity of 2285.5 m/s in the x-direction; (a) comparison of velocity profiles at times t = 1.0 × 10−4 s and t = 2.0 × 10−4 s; (b) comparison of temperature profiles at the same times. The reference profiles, computed by the standard DSMC scheme with a mean number of particles hN i = 100.0 in cell , are given by the thick solid line (blue in color). The other profiles, obtained by the standard scheme with hN i = 10.0 and hN i = 2.0, are shown as stars (blue in color); the profiles, obtained by the modified two-step collision scheme with hN i = 10.0, hN i = 2.0 and hN i = 1.0, are shown as circles (red in color).

hN i = 100.0. The comparison analysis of both schemes on the shock wave formation problem demonstrates that, under certain conditions, the modified DSMC scheme can be used successfully for simulation of high-speed flows with very small numbers of particles per cell compared with the standard DSMC method. The second example is also taken from the Bird’s book [5](§12.11). It concerns the one-dimensional steady-state problem of normal shock wave. Bird’s original program DSMC1S has been used to obtain the results by the standard DSMC method. Again we have used the same source code to obtain the modified scheme, however this time we have illustrated only the effect of the replacement of the standard collision procedure by one of the collision algorithms BT-A or BT-B in the standard first order splitting scheme (see section 2 and (2.10)). In the case the two-step collision algorithm applied on a dual grid was not used. An uniform grid of 600 cells without subcells was found fine enough to produce an accurate numerical solution when the other requirements are fulfilled. The shock wave has been considered in the steady flow frame of reference, and

DSMC WITH SMALL NUMBER OF PARTICLES

25

the case of shock Mach number (M a)s = 8.0 ( defined as the ratio of the speed of wave to the speed of sound relevant to the upstream flow) was simulated. Reference results for this case might be found in figure (12.31) in the Bird’s book [5](§12.11). A variable soft sphere (VSS) model of argon is used with parameters for argon at reference temperature 293o K as follows: the molecular diameter d = 4.092 × 10−10 m m; the molecular mass m = 6.64 × 10−26 kg, the viscosity-temperature index ω = 0.81 and the VSS scattering parameter α = 1.6625. The time step τ = 0.75 × 10−6 s is taken small enough to produce an accurate solution by using the first order splitting scheme. The upstream flow quantities are chosen equal to: velocity U1 = 2549.19 m/s, number density n1 = 1.0 × 1020 and temperature T1 = 293o K. The relevant mean free path and speed of sound are equal respectively to λ1 = 1.3442 × 10−2 m and a1 = 318.65 m/s. The downstream macroscopic characteristics are obtained from the Rankine-Hugoniot relations: velocity U2 = 667.17 m/s , number density n1 = 3.821 × 1020 and temperature T1 = 6115.52o K. The obtained results for density and temperature are normilized as follows: (5.1)

ρˆ =

ρ − ρ1 , ρ2 − ρ1

T − T1 Tˆ = . T2 − T1

To reproduce the reference density and temperature profiles of the strong shock wave, which are given in figure(12.31) [5], we have run the standard (original) DSMC code employing approximately 30000 simulator molecules (i.e. the average number was 50 particles per cell ). The time average was taken over the time for flow to move several thousand shock widths as pointed out in the Bird’s book. As a result the obtained profiles coincided exactly with the profiles shown in the book. Then we have performed runs with a total number of molecules approximately equal to 1200 (2 particle per cell) successively applying the standard NTC collision algorithm (original code) and the modified code with collision algorithms BT-A and BT-B. The time average was taken over 25 times larger interval in order to obtain a sample size equivalent to the one in the reference case with 50 particles per cell. In order to prevent the time-averaged shock profiles of smearing by the consequent movement of the wave in all runs the stabilizing procedure STABIL has been added as described in the Bird’s book. The results for the density and temperature profiles are compared in figure 5.2.

26

S. STEFANOV

1.2

1 Standard, N=50 Modified, N=2 Standard, N=2

0.8

ρˆ, Tˆ

0.6

Tˆ

ρˆ

0.4

0.2

0

−25

−20

−15

−10

−5

0

5

10

15

20

25

x/λ1

Fig. 5.2. Comparison of density and temperature profiles of a strong shock of Mach number (M a)s = 8.0 in argon. The reference profiles, computed by the standard DSMC scheme with a mean number of particles hN i = 50.0 in cell , are given by solid lines. The profiles, obtained by the standard scheme with hN i = 2.0 are given by dashed lines; the profiles, obtained by the modified scheme with BT-B collision algorithm with hN i = 2.0 are shown in circles.

When the time step is small enough and the condition (4.4) was fulfilled almost always the results obtained by the modified code with both collision algorithms BT-A and BT-B are almost indistinguishable. In figure 5.2 only the profiles obtained by algorithm BT-B are presented by circles. Each marker represents the sampled macroscopic value in every second cell of the computational grid. The profiles computed by the standard NTC collision algorithm are shown as solid lines (for hN i = 50.0 particles in cell) and dashed lines (for hN i = 2.0 particles per cell). As one can see from figure 5.2 the profiles obtained by the modified scheme with hN i = 2.0 (shown in circles) are in an excellent agreement with the reference profiles obtained by the standard scheme with hN i = 50.0. Under the same conditions, the profiles calculated by the standard NTC scheme with hN i = 2.0 deviate notably from the reference ones. The position of the profiles is shifted right and the density

DSMC WITH SMALL NUMBER OF PARTICLES

27

overpredicts the downstream value obtained from the Rankine-Hugoniot relations. An observable distinction in the calculation process showed that the stabilizing procedure STABIL applied to the modified scheme continued working well for small number of simulator particles in cells and the fluctuations of the total number of particles around the user-defined value in the steady regime was within the expected limits. Under the same conditions the fluctuations of the total number of particles in the calculations, carried out by using the standard NTC algorithm, were unacceptably larger and the procedure STABIL failed to stabilize the shock to the correct density values. The excellent agreement between the reference profiles, obtained by the standard NTC collision algorithm with 50 particles per cell, and those, obtained by algorithms BT-A and BT-B with 2 particles per cell, indicates that the improved accuracy, demonstrated in the runs with algorithms BT-A and BT-B, is a result of the replacement of the standard NTC collision algorithm with one of these algorithms in the first order splitting DSMC scheme. The meaning of this fact is that in the considered onedimensional steady-state problem a major effect can be achieved by applying only the first element of the modified scheme - the replacement of the NTC collision algorithm with the Bernoulli trials one. The second element, related to the splitting of the collision process into two steps performed consecutively over the cells of a dual grid, adds a little to the accurate estimation of the basic macroscopic profiles such as density, bulk velocity and temperature. More clearly, the modified two-collision step splitting scheme demonstrates an improved accuracy with small number of particles in cells in steady-state two- and three-dimensional calculations, where the number of cell edges and surfaces, separating closely situated particles, increases progressively with the increase of space dimensionality. The relative computational times of the runs performed by the standard NTC, Bernoulli trials BT-A, and simplified BT-B collision algorithms are shown in table 5.1 for different average number of particles per cell. Since the computational time depends on the computer platform characteristics the presented times are normalized by the time required for obtaining of a reference NTC solution with an average number of 50 particles per cell. The computational times are given for calculations performed with an equal sample size, which means that the product of number of samplings Ns

28

S. STEFANOV

and average number of particles per cell is Ns hN i = const. The number of operations, required by each collision algorithm, depends mainly on the number of checked particle pairs for collision. For the fixed time step and cell size the number of checked pairs (2.11) in the standard NTC algorithm in all cases is less than the number of pairs Np = N (N − 1)/2 checked in the BT-A algorithm and less than N − 1 pairs checked in algorithm BT-B. Accordingly, the efficiency of the standard collision algorithm is better. However, for a number of particles per cell N ≤ 2 the difference in the computational times of all algorithms is small and the efficiency of algorithms BT-A and BT-B is comparable to the standard NTC one. For a larger number of particles in cell the quadratic dependency makes BT-A the most time consuming algorithm, while the linear dependency on the number of particles in cell in the algorithm BT-B keeps the consumed computational time close to the consumed time by the standard NTC algorithm. Table 5.1 Relative computing time for 1D normal shock wave

hN i

algorithm NTC

algorithm BT-A

algorithm BT-B

2

0.91

1.77

1.23

5

0.92

3.54

1.38

10

0.94

7.57

1.51

20

0.97

12.13

1.66

50

1.00

30.06

1.76

The third example considers a two-dimensional steady-state lid-driven cavity flow. This is often used as a validation test for numerical schemes because of its sensitivity to choice of grid dimension and time step [15]. We consider a rarefied gas flow in a square cavity for Kn = 0.01. The cavity is confined by four diffusely reflecting walls at temperature T0 . A hard sphere gas model is employed and all variables are normalized by using the following scales: for density, ρ0 = mn0 (m is the molecular mass, n0 is the √ initial number density in the cavity); for velocity, Vth = 2RT0 ; for length, the cavity size L; for time, t0 = L/Vth ; for temperature, the wall temperature T0 ; the upper boundary is given a velocity of U0 = 2.0 in the x-direction. The macroscopic variables

DSMC WITH SMALL NUMBER OF PARTICLES

29

1 U0

0.9 0.8

A

0.7

y

0.6 B

0.5 0.4 0.3

C

0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

x

Fig. 5.3. The lid-driven vortex flow in a square cavity; the (x, y)-velocity vector field is calculated for Kn = 0.01 and U0 = 2.0. The macroscopic characteristic profiles, shown in the subsequent figures, are calculated on lines either A, B or C.

are calculated by using time averaged sampling on a single run omitting the transition period. The simulations are performed by the DSMC program that has been used for calculations of a low-speed cavity flow [15]. In that study [15] the results, obtained by the standard NTC scheme for Kn = 0.05, are in an excellent agreement with the slip-velocity continuum model calculations. In the present consideration the same algorithm is employed for the standard DSMC runs. The modified DSMC algorithm is built on the same source code by substituting the standard collision procedure with the Yanitskiy’s Bernoulli trials collision scheme BT-A or its simplified version, algorithm BT-B, and rearranging the particle indexing in accordance with the two-step collision algorithm. The macroscopic variables are sampled between the collision half-steps according to the splitting formula (4.8). Figure 5.3 shows the velocity vector field of the cavity flow which is established in the form of a single vortex. The macroscopic characteristic profiles, presented in the subsequent figures, are calculated on either lines A, B or C. Most of the calculations for this example have been performed by using the standard NTC and the Bernoulli trails algorithm BT-A. Several computations with included simplified algorithm BT-B and small enough time step gave identical results with these obtained by using algorithm BT-A. Thus, for brevity, we will discuss the following results referring to the standard and modified schemes without distinguishing which one of algorithms BT-A and BT-B has been used in the modified scheme.

30

S. STEFANOV

Initially, we have tested the accuracy of both schemes with regard to grid dimensions. To eliminate the effects of particle number in each cell and the time step, we have run the simulations from both schemes with a large average number of particle per cell hN i = 20.0 and a small time step τ = 0.1. An extra test calculation has been also performed with a larger average number of particle per cell hN i = 50.0. The obtained results coincided with those computed with hN i = 20.0. These numerical calculations showed that the total number of particles in the entire domain was enough for both cases. The results are shown in Fig. 5.4. Density (a) and temperature (b) profiles ρ(y) and T (y) are calculated along the vertical line A at x = 0.5. The profiles of y-velocity v(x) (c)and x-velocity u(x) (d) are calculated along the horizontal lines B and C at y = 0.5 and y = 0.25, respectively. The results obtained by the standard scheme are shown in the figures as stars and those obtained by the modified scheme as circles. Both schemes are run on three different uniform grids with resolutions (100 × 100), (200 × 200) and (400 × 400) cells, respectively, and the profiles are represented by smaller number of values averaged over groups of neighboring cells. As shown in Fig. 5.4, the profiles computed by both schemes are located very closely to each other for each grid resolution. This means that for the considered range of grid resolution, both the standard and modified schemes exhibit the same spacial accuracy, which depends on the grid coarseness. Consequently, an accurate solution can be obtained on a grid with fine enough resolution. This is particularly important for the simulation of complex vortex or unstable flows. In these cases the use of a fine grid is unavoidable. However, when run on a coarser grid with small number of particles per cell the two schemes exhibit different properties that should be investigated in detail ( see the final remarks in section 4). For the purpose of the present work, it is sufficient to show that for both schemes, the cavity flow simulation is sensitive with respect to the grid dimension . All profiles calculated on grids with (200 × 200) and (400 × 400) cells are almost identical (a tangible difference is observed for the x-velocity profiles at y = 0.25 shown in Fig. 5.4 (d)), while the profiles, calculated on the grid with (100 × 100) cells have considerable deviations from the reference solution calculated on the grid with (400 × 400) cells. The next results, presented in figures 5.5 and 5.6, are calculated with different numbers of particles per cell and time steps on the finest

31

DSMC WITH SMALL NUMBER OF PARTICLES 1.4

1.7 Standard, N =(100,100) c Standard, N =(200,200) c Standard, N =(400,400) c Modified, N =(100,100) c Modified, Nc=(200,200) Modified, Nc=(400,400)

1.3

1.2

1.6

1.5

1.4

ρ

T

1.1

1

1.3

0.9

1.2

0.8

1.1

Standard, N =(100,100) c Standard, Nc=(200,200) Standard, N =(400,400) c Modified, Nc=(100,100) Modified, N =(200,200) c Modified, N =(400,400) c

0.7 0

0.2

0.4

0.6

0.8

1 0

1

0.2

0.4

y

(a)

0.8

1

0.8

1

(b)

0.4

0 Standard, N =(100,100) c Standard, N =(200,200) c Standard, N =(400,400) c Modified, N =(100,100) c Modified, N =(200,200) c Modified, Nc=(400,400)

0.3 0.2 0.1

Standard, N =(100,100) c Standard, Nc=(200,200) Standard, Nc=(400,400) Modified, Nc=(100,100) Modified, N =(200,200) c Modified, N =(400,400)

−0.05 −0.1

c

−0.15 U

0 V

0.6 y

−0.1 −0.2

−0.2 −0.25

−0.3 −0.3 −0.4 −0.35

−0.5 −0.6 0

0.2

0.4

0.6 x

(c)

0.8

1

−0.4 0

0.2

0.4

0.6 x

(d)

Fig. 5.4. Comparison of standard (shown as stars) and modified (shown as circles) DSMC solutions for Kn = 0.01 on grids with resolutions (100 × 100), (200 × 200) and (400 × 400) cells. The time step and mean number of particles per cell are fixed at τ = 0.1 and hN i = 20.0, respectively. Density and temperature profiles (a) ρ(y) and (b) T (y) are calculated at x = 0.5 (vertical line A, Fig. 5.3); vertical y-component of velocity (c) v(x) is calculated at y = 0.5 (horizontal line B, Fig. 5.3); horizontal u-component of velocity (d) u(x) calculated at y = 0.25 (horizontal line C, Fig. 5.3).

grid with (400 × 400) cells. Figures 5.5 (a) and (b) present the profiles of the vertical component of velocity v(x) at y = 0.5, and figures 5.5 (c) and (d) - the profiles of the horizontal component of velocity u(x) at y = 0.25, calculated by both standard ((a), (c)) and modified ((b), (d)) schemes with a fixed small time step τ = 0.1 and different numbers of particles in cell, hN i = 2.0, hN i = 1.0 and hN i = 0.5. It is remarkable that for both schemes, the simulation of the considered flow with a small number of particles hN i = 2.0 per

32

S. STEFANOV

cell gives results identical to the reference solution, calculated with hN i = 20.0. This means that the computational requirements to the DSMC method with regard to the number of particles per cell are not so strong when the gas flow, which is not so far from local equilibrium, is considered on a fine enough grid. However, it should be noted that this conclusion is not valid for the case of a strong non-equilibrium shock wave (see Fig 5.1). In that particular case, the standard DSMC scheme requires a larger number of particles per cell hN i > 20 for an accurate simulation. The second case (Fig. 5.5) illustrates the fact that for both schemes, the simulation of the lid-driven flow becomes sensitive to the number of simulated particles per cell when hN i < 2. The velocity profiles, calculated by the standard scheme (Fig. 5.5 (a) and (c)) with hN i = 1.0 and hN i = 0.5, deviate increasingly from the reference profile (the solid line), while the profiles, calculated by the modified scheme (Fig. 5.5 (b) and (d)) with the same number of particles per cell, remain close to the reference solution compared to the standard DSMC data. Figure 5.6 illustrates how the accuracy of both schemes depends on the time step. The number of particles per cell is fixed to hN i = 2.0 in all runs. Figures 5.6(a) and 5.6(b) show the density and temperature profiles, calculated along the vertical line A (x = 0.5); while figures 5.6(c) and 5.6(d) show the profiles of y-velocity v(x) and x-velocity u(x), calculated along the horizontal lines B and C at y = 0.5 and y = 0.25, respectively. The results obtained by the standard scheme are shown as stars and those obtained by the modified scheme as circles. Both schemes have been run for three different time steps: τ = 0.1, τ = 0.25, and τ = 0.5. The simulations performed by both schemes with time step τ = 0.1 give almost identical results and the calculated profiles coincide with the reference solution (shown with a solid line), calculated with hN i = 20.0 and τ = 0.1. The profiles, calculated by the standard scheme (shown in stars), deviate increasingly from the reference profile (the solid line) for τ = 0.25, and τ = 0.5, while the profiles, calculated by the modified scheme (shown in circles) with the same time steps, remain much closer to the reference solution than the standard DSMC data. The differences are most clearly seen in Fig. 5.6(d) for the profiles u(x) at y = 0.25. It is worth noting that the improved accuracy of the modified DSMC scheme with regard to time step confirms completely the generalized

33

DSMC WITH SMALL NUMBER OF PARTICLES 0.4

0.4 N=20 N=2 N=1 N=0.5

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5

−0.5

−0.6 0

0.2

0.4

0.6

0.8

N=20 N=2 N=1 N=0.5

0.3

V

V

0.3

−0.6 0

1

0.2

0.4

x

(a) 0 N=20 N=2 N=1 N=0.5

−0.1

−0.1

−0.15

−0.15

−0.2

−0.2

−0.25

−0.25

−0.3

−0.3

−0.35

−0.35

0.4

1

0.6

N=20 N=2 N=1 N=0.5

−0.05

U

U

−0.05

0.2

0.8

(b)

0

−0.4 0

0.6 x

0.8

1

−0.4 0

x

(c)

0.2

0.4

0.6

0.8

1

x

(d)

Fig. 5.5. Comparison of standard DSMC solution (graphics (a),(c), shown as stars, blue in color) and modified ((b),(d), shown as stars, red in color) obtained with different mean numbers of particles per cell hN i = 2.0, hN i = 1.0 and hN i = 0.5 and a fixed time step τ = 0.1 for Kn = 0.01 on a fine grid with a resolution (400 × 400) cells. The reference profiles, obtained by the standard scheme with mean number of particles in cell hN i = 20.0, are given with the thick solid line (blue in color). Profiles of vertical y-component of velocity v(x) (graphics (a),(b)) are calculated at y = 0.5 (horizontal line B, Fig. 5.3); profiles of horizontal u-component of velocity u(x) ( graphics (c),(d)) are calculated at y = 0.25 (horizontal line C, Fig. 5.3).

theoretical analysis [6] concerning the accuracy of Strang’s type splitting methods. Consequently, an important benefit of the modified scheme is that it can be run with a larger time step for the same accuracy compared to the required by the standard scheme. The modified two-step collision procedure increases the computational time of the simulation. For example, the “Bernoulli trials” collision algorithm BT-A requires

34

S. STEFANOV 1.4

1.5

Standard, N=20, τ=0.1 Standard, N=2, τ=0.1 Standard, N=2, τ=0.25 Standard, N=2, τ=0.5 Modified, N=2, τ=0.1 Modified, N=2, τ=0.25 Modified, N=2, τ=0.5

1.3

1.2

1.45 1.4 1.35 1.3

ρ

T

1.1

1

1.25 Standard, N=20, τ=0.1 Standard, N=2, τ=0.1 Standard, N=2, τ=0.25 Standard, N=2, τ=0.5 Modified, N=2, τ=0.1 Modified, N=2, τ=0.25 Modified, N=2, τ=0.5

1.2 1.15

0.9

1.1 0.8 1.05 0.7 0

0.2

0.4

0.6

0.8

1 0

1

0.2

0.4

y

(a)

0.8

1

0.8

1

(b)

0.4

0

Standard, N=20, ∆t=0.1 Standard, N=2, ∆t=0.1 Standard, N=2, ∆t=0.25 Standard, N=2, ∆t=0.5 Modified, N=2, ∆t=0.1 Modified, N=2, ∆t=0.25 Modified, N=2, ∆t=0.5

0.3 0.2 0.1

Standard, N=20, ∆t=0.1 Standard, N=2, ∆t=0.1 Standard, N=2, ∆t=0.25 Standard, N=2, ∆t=0.5 Modified, N=2, ∆t=0.1 Modified, N=2, ∆t=0.25 Modified, N=2, ∆t=0.5

−0.05 −0.1 −0.15 U

0 V

0.6 y

−0.1 −0.2

−0.2 −0.25

−0.3 −0.3 −0.4 −0.35

−0.5 −0.6 0

0.2

0.4

0.6 x

(c)

0.8

1

−0.4 0

0.2

0.4

0.6 x

(d)

Fig. 5.6. Comparison of standard (shown in stars, blue in color) and modified (shown in circles, red in color) DSMC solutions obtained with a fixed mean number of particles per cell hN i = 2.0 and different time steps τ = 0.1, τ = 0.25 and τ = 0.5 for Kn = 0.01 on a fine grid with resolution (400 × 400) cells. The reference profiles, obtained by the standard scheme with mean number of particles in cell hN i = 20.0 and time stepτ = 0.1, are given with thick solid line (blue in color). Density and temperature profiles (a) ρ(y) and (b) T (y) are calculated at x = 0.5 (vertical line A, Fig. 5.3); vertical y-component of velocity (c) v(x) is calculated at y = 0.5 (horizontal line B, Fig. 5.3); horizontal u-component of velocity (d) u(x) calculated at y = 0.25 (horizontal line C, Fig. 5.3).

a number of operations proportional to (N (N − 1)/2), the simplified Bernoulli trials algorithm BT-B requires a number of operations proportional to (N − 1), and for the standard NTC the required number of operations is proportional to N by a factor, which depends on the chosen computational parameters and might be above or below unity. In the considered cases this factor was always les than 1. However, for both

DSMC WITH SMALL NUMBER OF PARTICLES

35

algorithms BT-A and BT-B of the modified scheme this is compensated by the use of a smaller average number of particles per cell for obtaining results with the same accuracy. To compare the real computational time required by each scheme, a number of simulations have been run by all three collision schemes with different average number of particles per cell at a fixed time step τ = 0.1 accumulating an equivalent sample-size on a grid with (400 × 400) cells . The relative computational times of the runs performed by the standard NTC, Bernoulli trials BT-A, and simplified BT-B collision algorithms are shown in table 5.2 for different average number of particles per cell. The presented times are normalized by the time required for obtaining of a reference NTC solution with an average number of 20 particles per cell. The estimates are indicative and approximately illustrate the relative computational efficiency of the standard and modified schemes achieved for the two-dimensional cavity flow problem. Table 5.2 Relative computing time for 2D cavity flow

hN i

algorithm NTC

algorithm BT-A

algorithm BT-B

1

0.983

1.26

1.23

2

0.997

1.65

1.37

20

1.00

4.65

1.96

Thus, from these results it can be seen that for large hN i the standard scheme is vastly more efficient than the modified with collision algorithm BT-A and relatively more efficient than the modified with collision algorithm BT-B. With the decrease of hN i the standard scheme keeps approximately the same efficiency. However, when the modified scheme (with BT-A or BT-B) is run with a small average number of particles per cell hN i ∼ 1, 2, its efficiency becomes much better and closely matches the efficiency of the standard scheme. Although the modified scheme with two-step collision algorithm requires more computational operations than the first order splitting scheme with one-step collision algorithm the analysis of the relative computing times given in the corresponding tables 5.2 and 5.1 shows an improved efficiency of the modified scheme in the two-dimensional calculations in comparison with one-dimensional ones.

36

S. STEFANOV

An important benefit of the modified scheme is that it considerably reduces the memory requirements. This becomes evident for multidimensional simulations or for use with local refined meshes and is illustrated in our recent paper [22], where the modified scheme has been applied successfully for the simulation of three-dimensional Rayleigh-B´enard convection of a rarefied gas. In that particular study we used a combined approach: the standard scheme was applied in subcell areas with hN i ≥ 5 particles per cell and the modified algorithm was applied in areas with hN i < 5 particles per cell. 6. Concluding remarks. This paper presents a modified DSMC algorithm for the simulation of rarefied gas flows that improves the stochastic modeling and allows the use of a smaller average number of particles per cell in comparison with the standard DSMC method. The proposed modification is twofold: first, it improves the stochastic properties of the collision algorithm, and second, the new splitting scheme is second-order accurate in time. In the new algorithm, the standard ”No Time Counter” collision scheme is replaced by a two-step collision procedure based on Yanitskiy’s ”Bernoulli trials” algorithm BT-A or a simplified Bernoulli trials algorithm BT-B, which is applied twice to the cells (or subcells) of a dual grid. The modified scheme is less efficient compared to the standard scheme when a large average number of particles per cell is used. However, when the number of particles per cells is small this disadvantage is avoided and the efficiency of the new algorithm closely approaches the efficiency of the standard DSMC method. An important advantage of the new computational scheme is the reduced amount of memory required during a simulation. This is particularly useful in DSMC calculations of complex vortex and unstable gas flows on two- or three-dimensional fine meshes when the use of the standard scheme requires a huge total number of particles (for DSMC computational requirements see ref. [18]). Another advantage is that the Strang’s type splitting scheme used in the new algorithm improves the simulation accuracy with respect to time step making the modified DSMC method an effective numerical tool for unsteady flow calculations. For simulation of high-speed rarefied gas flows with large density gradients [23]

DSMC WITH SMALL NUMBER OF PARTICLES

37

the following combined approach may be more efficient: the standard DSMC scheme is applied in areas of the computational domain with hN i ≥ Np particles per cell and the modified algorithm in areas with hN i < Np particles per cell, where Np is a small integer that can be chosen by numerical experimentation to give an optimal effect. 7. Acknowledgments. The research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement ITN GASMEMS - No 215504 and from the NSF of Bulgaria under Grant No DID 02/20-2009. The author is very grateful to Professor Carlo Cercignani and Dr. Robert Barber for many helpful discussions.

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