NUMERICAL SOLUTION OF THE BOLTZMANN

NUMERICAL SOLUTION OF THE BOLTZMANN EQUATION BY TIME RELAXED MONTE CARLO (TRMC) METHODS LORENZO PARESCHI AND STEFANO TRAZZI Abstract. A new family of Monte Carlo schemes has been recently introduced for the numerical solution of the Boltzmann equation of rarefied gas dynamics[14]. After a splitting of the equation the time discretization of the collision step is obtained from the Wild sum expansion of the solution by replacing high order terms in the expansion with the equilibrium Maxwellian distribution. The corresponding time relaxed Monte Carlo (TRMC) schemes allow the use of time steps larger than those required by direct simulation Monte Carlo (DSMC) and guarantee consistency in the fluid-limit with the compressible Euler equations. Conservation of mass, momentum, and energy are also preserved by the schemes. Applications to a two-dimensional gas dynamic flow around an obstacle are presented which show the improvement in terms of computational efficiency of TRMC schemes over standard DSMC for regimes close to the fluid-limit.

Keywords. Boltzmann equation, Monte Carlo methods, Time Relaxed schemes, fluid-dynamic limit, Euler equations, stiff systems. AMS subject classifications. 65C05, 65L05, 76P05, 82C80 1. Introduction The mathematical model we consider is the kinetic Boltzmann equation for a rarefied gas dynamics (RGD) which characterizes the temporal evolution of a particle density function f (x, v, t) accordingly to ∂f 1 + v · ∇x f = Q(f, f ), ∂t ² supplemented with the initial condition

(1)

(2)

x ∈ Ω ⊂ R3 , v ∈ R3 ,

f (x, v, t = 0) = f0 (x, v).

The nonnegative density function f depends on position x ∈ R3 , velocity v ∈ R3 and time t > 0 and can depend on other independent variables like an internal energy [6]. In (1) the parameter ² > 0 is called Knudsen number and it is proportional to the mean free path between collisions. The bilinear collisional operator Q(f, f ) which describes the binary collisions between particles in a mono-atomic gas is given by Z Z (3) Q(f, f )(v) = σ(| v − v∗ |, ω)(f (v 0 )f (v∗0 ) − f (v)f (v∗ ))dωdv∗ , R3

S2

where for simplicity the dependence of f on x and t has been omitted. In the previous expression ω is a vector of the unitary sphere S 2 ⊂ R3 . The collisional velocities (v 0 , v∗0 ) are associated to the velocities (v, v∗ ) and to the parameter 1

2

LORENZO PARESCHI AND STEFANO TRAZZI

ω by the relations (4)

v0 =

1 (v + v∗ + |q|ω), 2

v∗0 =

1 (v + v∗ + |q|ω), 2

where q = v − v∗ is the relative velocity. The kernel σ is a non negative function which characterizes the details of the binary interaction between particles. In the case of a k-th power forces inversely proportional to the distance between particles, the kernel is (5)

σ(|q|, θ) = bα (θ)|q|α ,

where α = (k − 5)/(k − 1) and θ is the scattering angle between q and |q|ω. The Variable Hard Sphere (VHS) model is often used in numerical simulation of rarefied gases, and consists in choosing bα (θ) = Cα with Cα a positive constant. The case α = 0 corresponds to a Maxwellian gas, while α = 1 is called a Hard Sphere Gas. In this paper we consider the multidimensional extension of the Time Relaxed Monte Carlo (TRMC) methods recently introduced in[14] and present a numerical comparison with a classical Direct Simulation Monte Carlo (DSMC). Since TRMC methods are based on a suitably time discretized Boltzmann equation we use as a DSMC comparison the time discrete Nanbu-Babovsky[12, 1] method instead of Bird’s method[3]. A modified version of TRMC which avoids the introduction of a time discretization error and which is based on a recursive strategy has been recently presented in[17]. A comparison of the recursive TRMC method with Bird’s algorithm is actually under development[18] and will be presented elsewhere. The goal of TRMC methods is to construct simple and efficient numerical methods for the solution of the Boltzmann equation in regions with a large variation of the mean free path[15, 13, 14]. These methods are based on a splitting of the Boltzmann equation and on a time discretization of the collision step[8, 14], which is robust in the fluid-limit. As a consequence the resulting TRMC methods have the following features • For large Knudsen numbers, the TRMC method behaves as a classical DSMC method. In particular the sampling of new colliding particles is the same as in classical DSMC methods. • In the limit of the very small Knudsen number, the collision step replaces the distribution function by a local Maxwellian with the same moments. The TRMC method will behave as a stochastic kinetic scheme for the underlying Euler equations of gas dynamics[19]. • Mass, momentum, and energy are preserved. Previous techniques for acceleration of DSMC close to fluid regimes, such as [10, 9, 4, 11], have divided the spatial region into two subregions: a fluid region in which the Euler or Navier-Stokes equations are solved and an RGD region in which DSMC is used. The focus of these methods has been to determine how best to choose the two domains and on design of boundary conditions to couple the fluid and RGD regions. These coupling methods have proved to be very useful, but they are still limited by their performance close to fluid regions, where DSMC is slow but the fluid equations are inaccurate. We see this approach of coupling the fluid and RGD equations as a complementary strategy to the unified TRMC approach; i.e., it may make sense to use a TRMC-like method in a near continuum region between the pure fluid and pure DSMC regions.

NUMERICAL SOLUTION OF THE BOLTZMANN EQUATION BY TRMC METHODS

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After this introduction we will recall some basic facts about the Boltzmann equation and its fluid-dynamic limit. Next in section 3 we will discuss the problem of the time discretization of the Boltzmann equation. Section 4 is devoted to a short description of the different Monte Carlo methods. Finally in the last section we present the detailed numerical results of the simulation of a two-dimensional flow around an ellipse for several values of the mean free path. The results show a marked improvement in the efficiency of computations given by TRMC for small Knudsen numbers. 2. Physical background 2.1. Collisional invariants and the Euler equations. Let f and φ be such that R Q(f, f )φ(v)dv exists. Then it is possible to show the identity 3 R Z Z Z Z 1 Q(f, f )φ(v)dv = (6) f 0 f∗0 (φ0 + φ0∗ − φ − φ∗ )σ(|q|, ω)dvdv∗ dω, 2 R3 R3 S 2 R3 where f 0 = f (x, v 0 , t), f∗0 = f (x, v∗0 , t). If φ is such that φ + φ∗ = φ0 + φ0∗ then the previous integral vanishes independently from the function f . Such special functions φ are called collisional invariants. Assuming that φ is twice differentiable, the general form of the collisional invariant is [6] φ(v) = A + B · v + C|ξ|. More precisely φ(v) is a linear combination of the elementary collisional invariants 1, v, |v|2 , which correspond to the conservation of mass, moment and energy. If f is a mass density in the phase space then to obtain the density ρ = ρ(x, t) in the ordinary space we have to integrate f in v Z (7) ρ= f dv. R3

Similarly the gas velocity u is determined by Z (8) ρu = f vdv, R3

and the gas temperature by (9)

T =

1 3ρ

Z (v − u)2 f dv. R3

Finally we define (10)

E=

1 2

Z |v|2 f dv R3

the energy per volume density. Now, if we consider the Boltzmann equation ∂f 1 + v · ∇x f = Q(f, f ) ∂t ε and multiply it for the elementary collisional invariants 1, v, |v|2 and integrate in v we obtain Z Q(f, f )φ(v)dv = 0, φ(v) = 1, v, |v|2 , R3

4


and, if it is possible to invert the order of derivation - integration we get Z Z 3 X ∂ ∂ (11) φ(v)dv + vi φf dv = 0. ∂t ∂xi i=1 If we replace φ by the functions 1, v, |v|2 we have 3

(12)

∂ρ X ∂ + (ρui ) = 0, ∂t i=1 ∂xi

(13)

3 X ∂ ∂ (ρuj ) + (ρui uj + pij ) = 0, ∂t ∂x i i=1

(14)

j = 1, 2, 3

3 3 X X ∂ 1 ∂ 1 2 2 ( ρ|u| + ρe) + [ρui ( |u| + e) + ui pij + qi ] = 0. ∂t 2 ∂xi 2 i=1 j=1

These equation are the corresponding conservation laws for mass, momentum and energy. Unfortunately the differential equations system is not closed, since it involves higher order moments of the density function f . Taking φ = log(f ) in (6) it is easy to show that the collisional operator is such that the H-Theorem holds Z (15) Q(f, f ) log(f )dv ≤ 0. R3

This condition implies that each function f in equilibrium (i.e. Q(f, f ) = 0) has locally the form of a Maxwellian distribution µ ¶ ρ |u − v|2 (16) M (ρ, u, T )(v) = exp − , 2T (2πT )3/2 where ρ, u, T are the density, the mean velocity and the gas temperature. Formally as ε → 0 the function f is locally replaced by a Maxwellian. In this case it is possible to compute f from its moments thus obtaining the closed Euler system of compressible gas dynamics ∂ρ + ∇x · (ρu) = 0, ∂t ∂(ρu + ∇x · (ρu ⊗ u + p) = 0, ∂t ∂E + ∇x · (Eu + pu) = 0, ∂t 3 1 p = ρT, E = ρT + ρ|u|2 2 2 More in general using for small values of ε the Chapman-Enskog expansion

(17)

f = M + εf1 + ε2 f2 + · · · + εn fn it is possible to obtain the compressible Navier-Stokes equations (at the order n = 1) and going further the Burnett system (at the order n = 2). We refer to Cercignani[6] for more details.


5

2.2. Boundary conditions. Typically the equation (1) is completed with boundary conditions for x ∈ ∂Ω. The boundary conditions for mono-atomic gases are described by Z (18) |v · n(x)|f (x, v, t) = |v 0 · n(x)|K(x, v 0 −→ v, t)f (x, v 0 , t)dv 0 v 0 ·n(x) 0 is a constant, we have Pσ¯ (f, f ) = Qσ¯ (f, f ) + µf ≥ 0, for µ = 4π¯ σ ρ. Under this assumption, which is numerically essential, the solution provided by TR schemes is also nonnegative. Remark 3.1. The method described can be generalized by using different weight function, with the goal of finding a best approximation for the highest order coefficients of Wild sum expansion (31). In general this schemes can be written in the form m X Ak fk + Am+1 M, (36) f n+1 = k=0

8


where the coefficient fk are the determined by (30). The weights Ak = Ak (t) are not negative functions witch satisfy (i) Consistency: (37)

lim A1 (τ )/τ = 1, lim Ak (τ )/τ = 0, k = 2, . . . , m + 1

τ →0

τ →0

(ii) Conservations: m+1 X

(38)

Ak = 1τ ∈ [0, 1],

k=0

(iii) Asymptotic preservation (AP): (39)

lim Ak (τ ) = 0, k = 0, . . . , m.

τ →1

A possible choice of function satisfying the above conditions are (40)

Ak = (1 − τ )τ k , k = 0, . . . , m, Am+1 = τ m+1 ,

which correspond to the scheme (32). A better choice of parameter has been presented in [15]:

(41)

Ak = (1 − τ )τ k , k = 0, . . . , m + 1, m X Am = 1 − Ak − Am+1 , k=0

Am+1 = τ m+2 , that correspond to consider fm+1 = fm , fk = M, k ≥ m + 2 in the (31). We emphasize that other choice of parameters are possible, and the individuation of the optimal ones is an open problem. 4. Time Relaxed Monte Carlo (TRMC) methods In this section, we describe the TRMC method for the evolution of the density function f . We shall include the description of the classical DSMC no-time counter algorithm, so that it will be easier to make a comparison between the new formulation and the previous one. 4.1. The Nanbu–Babovsky DSMC method. Here we will describe the classical DSMC method in the general framework we have introduced in the previous sections. More specifically, we consider the Nanbu–Babovsky algorithm[12, 1]. The convergence of this scheme has been proved by Babovsky and Illner[2]. In the sequel we will implicitly assume that the collision kernel in the Boltzmann equation has been replaced by the bounded one. For simplicity of notations we will omit the subscript σ ¯ when referring to Qσ¯ (f, f ) and Pσ¯ (f, f ). The Boltzmann equation can be written in the form ∂f 1 = [P (f, f ) − µf ]. ∂t ² We assume that f is a probability density; i.e., Z ρ= f (v, t) dv = 1.

(42)

R3


9

Let us discretize time and denote by f n (v) an approximation of f (v, n∆t). The forward Euler scheme applied to (42) writes µ ¶ µ∆t µ∆t P (f n , f n ) n+1 (43) f = 1− fn + . ² ² µ Note that for bounded collision kernels P (f, f )/µ ≥ 0 if µ = 4π¯ σ and thus by mass conservation is a probability density. Note also that P (f, f )/µ = f1 that is the first term in the Wild expansion. Thus the equation has the following probabilistic interpretation. To sample one particle from f n+1 , with probability (1 − µ∆t/²) we sample the particle from f n and with probability µ∆t/² we sample the particle from P (f n , f n )(v)/µ. We remark here that this probabilistic interpretation of (43) breaks down if ∆t/² is too large because the coefficient of f n on the right-hand side may become negative. This implies that the time step becomes extremely small when approaching the fluid-dynamic limit. Therefore Nanbu-Babowski method becomes almost unusable near the fluid regime. The essential feature of the Monte Carlo method is the way particles are sampled from P (f, f )/µ. This is done with the aid of an acceptance-rejection technique combined with a collision strategy of the samples. In its first version, Nanbu’s algorithm was not conservative; i.e., energy and momentum were conserved only in the mean but not at each collision. A conservative version of the algorithm was introduced by Babovsky [1]. Instead of selecting single particles, independent particle pairs are selected, and conservation is maintained at each collision. The algorithm for evolving the density up to time t = ntot ∆t is the following. Algorithm 4.1 (DSMC for VHS molecules). • compute the initial velocity of the particles, {vi0 , i = 1, . . . , N }, by sampling them from the initial density f0 (v) • for n = 1 to ntot given {vin , i = 1, . . . , N }, ◦ compute an upper bound σ ¯ for the cross section ◦ set µ = 4π¯ σ ◦ set Nc = Iround(µN ∆t/(2²)) ◦ select Nc dummy collision pairs (i, j) uniformly among all possible pairs, and for those ◦ compute the relative cross section σij = σ(|vi − vj |) ◦ if σ ¯ Rand < σij ◦ perform the collision between i and j, and compute vi0 and vj0 according to the collisional law ◦ set vin+1 = vi0 , vjn+1 = vj0 else ◦ set vin+1 = vin , vjn+1 = vjn n+1 ◦ set vi = vin for the N − 2Nc particles that have not been selected end for

During each step, all the other N − 2Nc particle velocities remain unchanged. Here, by Iround(x), we denote a suitable integer rounding of a positive real number x. In our algorithm, we choose ½ [x] with probability [x] + 1 − x, Iround(x) = [x] + 1 with probability x − [x],

10


where [x] denotes the integer part of x. The post-collisional velocities are computed through relations |vi − vj | vi + vj |vi − vj | vi + vj + ω, vj0 = − ω, (44) vi0 = 2 2 2 2 where ω is chosen uniformly in the unit sphere, according to  cos φ sin θ ω =  sin φ sin θ  , cos θ 

(45)

θ = arccos(2ξ1 − 1),

φ = 2πξ2 ,

and ξ1 , ξ2 are uniformly distributed random variables in [0, 1]. The upper bound σ ¯ should be chosen as small as possible, to avoid inefficient rejection, and it should be computed fast. An upper bound can be derived taking σ ¯ as (46)

σ ¯ = max σ(|vi − vj |). vi ,vj

However this computation would require O(N 2 ) operations. Thus it is preferable to use an upper bound of (46) given by X (47) σ ¯ = σ(2∆v), ∆v = max |vi − v¯|, v¯ = vi /N. i

i

Remark 4.1. Note that if the time step is small enough, only a small fraction of particles, let us say Nc , will collide. The computational cost of the collisions is therefore O(Nc ). On the other hand, the cost of the computation of the upper bound σ ¯ is O(N ), which may be much larger than O(Nc ). A possible way to overcome this difficulty is to update at each time step the value of the upper bound σ ¯ only if it increases. This may be done as follows. During the computation of the collision between particles i and j, let v˜i and v˜j denote the new particle velocities. Then the quantity ∆v is updated according to (48)

∆v = max(∆v, |˜ vi − v¯|, |˜ vj − v¯|).

At the end of the collision loop, the upper bound on the cross section is computed as σ ¯ = σ(2∆v). In space nonhomogeneous calculations, assuming that there are several collisional time steps during a convection time step, the bound can be computed according to (47) the first time step and then updated as described above. 4.2. First and second order TRMC methods. The first order TRMC algorithm is based on the TR schemes (49)

f n+1 = A0 f n + A1 f1 + A2 M.

The probabilistic interpretation of the above equation is the following. To sample a particle form f n+1 , with probability A0 we sample the particle from f n , with probability A1 we sample the particle from f1 = P (f, f )/µ (as in standard DSMC method) and with probability A2 we sample the particle from a Maxwellian. In this formulation, the probabilistic interpretation holds uniformly in µ∆t/², at variance with standard DSMC, which requires µ∆t/² < 1. Furthermore, as µ∆t/² → ∞, the distribution at time n + 1 is sampled from a Maxwellian. In this limit, the density f n+1 relaxes immediately to its equilibrium distribution. In a


11

space nonhomogeneous case, this would be equivalent to the particle method for Euler equations proposed by Pullin [19]. The Monte Carlo schemes described above are conservative in the mean. It is possible to make it exactly conservative by selecting collision pairs uniformly, rather than individual particles, and by using a suitable algorithm for sampling a set of particles with prescribed momentum and energy from a Maxwellian. To this aim one can adopt the original algorithm proposed by Pullin[20]. An alternative and simpler method consists in rescaling the particles sampled from the Maxwellian in order to have exact preservation of the moments. Suppose that we want to sample NM particles by a Maxwellian with mean velocity u and energy E. First we have sample the particles vi , i = 1, ..., NM from a Gaussian density. To sample two of this particles vi , vj we can use the Box-Muller method vi = u + Eρ cos(θi ), vj = u + Eρ sin(θi ), with p ρ = − log(ξ1 ), θi = 2πξ2 where ξ1 , ξ2 are uniformly distributed random variables in [0, 1]. Let now n n 1 X 1 X 2 0 v= vi , E = v N i=1 2N i=1 i be the mean velocity and energy of the sampled particles (we fix for simplicity the mass of a single particle m = 1). We want to rescale the samples using a scalar τ and a vector λ such that N 1 X vi − λ (50) = u, N i=1 τ and N 1 X (vi − λ)2 = E. 2N i=1 τ2

(51) Solving the system we obtain (52)

τ 2 = (E 0 −

v2 u2 )/(E − ) 2 2

and (53)

λ = v − τ u.

Note that the right hand side of equation (52) is non negative and thus the system admits always a solution. The conservative version of the methods can be formalized in the following algorithm. Algorithm 4.2 (first order TRMC for VHS molecules). • compute the initial velocity of the particles, {vi0 , i = 1, . . . , N }, by sampling them from the initial density f0 (v) • for n = 1 to ntot given {vin , i = 1, . . . , N }, ◦ compute an upper bound σ ¯ of the cross section ◦ set τ = 1 − exp(−ρ¯ σ ∆t/²) ◦ compute A1 (τ ), A2 (τ )

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◦ ◦ ◦ ◦

set Nc = Iround(N A1 /2) perform Nc dummy collisions, as in Algorithm 4.1 set NM = Iround(N A2 ) select NM particles among those that have not collided, and compute their mean momentum and energy ◦ sample NM particles from the Maxwellian with the above momentum and energy, and replace the NM selected particles with the sampled ones ◦ set vin+1 = vin for all the N − 2Nc − NM particles that have not been selected end for

A second order Monte Carlo scheme is obtained by the TR scheme (54)

f n+1 = A0 f n + A1 f1 + A2 f2 + A3 M

with

P (f n , f n ) P (f n , f1 ) , f2 = . µ µ Given N particles distributed according to f n , the probabilistic interpretation of scheme (54) is the following: N A0 particles will not collide; N A1 will be sampled from f1 (as in the first order scheme); N A2 will be sampled from f2 ; i.e., N A2 /2 particles sampled from f n will undergo dummy collisions with N A2 /2 particles sampled from f1 ; and N A3 particles will be sampled from a Maxwellian. Once again, the methods can be made conservative using the same techniques adopted in the first order scheme. The various steps of the method can be summarized in the following algorithm. Algorithm 4.3 (second order TRMC for VHS molecules). f1 =

• compute the initial velocity of the particles, {vi0 , i = 1, . . . , N }, by sampling them from the initial density f0 (v) • for n = 1 to ntot given {vin , i = 1, . . . , N }, ◦ compute an upper bound σ ¯ of the cross section ◦ set τ = 1 − exp(−ρ¯ σ ∆t/²) ◦ compute A1 (τ ), A2 (τ ),A3 (τ ) ◦ set N1 = Iround(N A1 /2) , N2 = Iround(N A2 /4) ◦ select N1 + N2 dummy collision pairs (i, j) uniformly among all possible pairs ◦ for N1 pairs ◦ compute the relative cross section σij = σ(|vi − vj |) ◦ if σ ¯ Rand < σij ◦ perform the collision between i and j, and compute vi0 and vj0 according to the collisional law ◦ set vin+1 = vi0 , vjn+1 = vj0 ◦ for N2 pairs ◦ compute the relative cross section σij = σ(|vi − vj |) ◦ if σ ¯ Rand < σij ◦ perform the collision between i and j, compute vi0 and vj0 according to the collisional law and store them ◦ select 2N2 particles from f n ◦ perform the collision of these selected particles with the second set of 2N2 particles that have collided once


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◦ update the velocity of the 4N2 particles with the outcome of the 2N2 collisions (of particles that have never collided before with particles that collided once) ◦ set NM = Iround(N A3 ) ◦ replace NM particles with samples from Maxwellian, as in Algorithm 4.2 ◦ set vin+1 = vin for all the N − 2N1 − 4N2 − NM particles that have not been selected end for

Similarly, higher order TRMC methods can be constructed. For example, a third order scheme is obtained from f n+1 = A0 f n + A1 f1 + A2 f2 + A3 f3 + A4 M

(55) with f1 =

P (f n , f n ) , µ

f2 =

P (f n , f1 ) , µ

f3 =

1 [2P (f n , f2 ) + P (f1 , f1 )]. 3µ

We omit for brevity the details of the resulting Monte Carlo algorithm. We remark here that the TR scheme is a direct consequence of the time discretization of the Boltzmann equation. In this respect, the choice of a particular Monte Carlo scheme used to model the collision is not crucial. Here we considered a commonly used Monte Carlo scheme. If a different technique is used to model the collision, then the same technique could be used in conjunction with the TR discretization. 5. Numerical results In this section we compare DSMC and TRMC methods on a two dimensional space non homogeneous test problem. The physical problem we will study is represented by a rarefied gas flow around an elliptical section of an object. The rectangular domain Ω has been divided into a regular grid of cells. If we assume that in each cell the density is constant during the collision step, we can apply the algorithms previously seen. This is physically equivalent to say that each particle can collide only with particles of the same cell. 5.1. Free transport. In a spatial homogeneous case we supposed that the density function to be a probability function, that is the with unitary mass. In spatially non homogeneous cases the integral in the velocity determines the gas density number Z ρ (56) f (x, v, t)dv = , m R3 where m is a gas molecule mass. The algorithms seen in the previous sections can be modified considering (57)

µ = 4π¯ σ ρ/m

The density ρ is proportional to the number of particles in each cell j (58)

ρj = N j m ∗ ,

where m∗ is the mass of the particles used in the simulation. The transport step is made easily applying to each particle i (59)

x0i = xi + vi ∆t.

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5.2. Particle-surface interaction. At the surface of the obstacle the particles are absorbed and remitted in according with the Maxwellian distribution of a gas at the same temperature of the object. To compute the new ingoing velocity we must sample an ingoing particle from a Maxwellian with zero mean velocity and at the temperature of the wall. We describe here the method we used in the case of a two-dimensional elliptic object. Let ∆t∗ be the time of free flow of the particle before colliding the object (which can be computed for example using a suitable iterative method). The contact point particle-object will then be x∗i = xi + vi ∆t∗ . There are several ways to calculate the new position and velocity of a particle y x z after a collision with the boundary. For example, if vM = (vM , vM , vM ) is the velocity sampled by a Maxwellian corresponding to the thermal equilibrium gasobstacle and {k, n, t} is the orthogonal set of unitary vectors that characterize respectively the z-axes, the normal and the tangent to the surface at the contact point, then the new outgoing velocity of the particle will be y x z vi0 = |vM |n + vM k, t + vM

(60)

and its position will be determined by x0i = x∗i + vi0 (∆t − ∆t∗ )

(61)

where x∗i is the contact point and ∆t is the time step. 5.3. Two dimensional flow past an ellipse. In our calculations, the downstream state is characterized by ρ = 0.01,

T = 100.0,

Tobj = 100.0

M = 11.0,

where M is the Mach number of the shock, T and Tobj are respectively the gas and the object temperature. The downstream mean velocity is then given by p ux = −M (γT R), uy = 0, uz = 0, with γ = 5/3 and R is the perfect gas constant since we have considered a threedimensional monatomic gas in velocity space. The infinite physical space is truncated to the finite region [0, 15] × [0, 12]. We report the result obtained with the different schemes using 50 × 40 space cells and 30 particles ingoing in each downstream cell. Since we are computing a stationary solution after a fixed initial time, we can strongly improve the accuracy of the Monte Carlo solution by averaging in time the solution itself. In all our simulations we report the results obtained after averaging the solution over 400 time steps after a sufficiently large computational time has been reached. As a reference solution to estimate the relative L2 -error norms we used 50 particles per cell and 1200 averages with DSMC. The collisional time step in DSMC has been computed adaptively cell by cell to satisfy the stability condition. At variance in TRMC the collisional time step has been taken as C∆t, where C ≤ 1 is a constant and ∆t is the free flow time step. In the graphs reporting the number of collisions we have considered the sampling from a Maxwellian in TRMC like a collision. We studied the dynamic of the gas for different ranges of the Knudsen number and different values of the Mach number. The first example shows the case of a very rarefied gas (i.e. ² = 0.1). In this condition we cannot expect any gain of TRMC


15

with respect to DSMC. The numerical results obtained for C = 1 in TRMC with the different schemes are very similar as well as the efficiency (see Figure 12). As expected the cross sections along the lines x = 5 and y = 6 emphasize the higher accuracy obtained with TRMC II. For the intermediate regime (² = 0.01) we have considered the solutions obtained by TRMC with a collisional time of one third of the free flow time (C = 1/3). Again the solutions are comparable, with an improvement obtained by TRMC II, but with a better computational efficiency in TRMC methods (TRMC I is almost two times faster then DSMC, see Figure 12). Finally, the last example is near the equilibrium state (i.e. ² = 0.001). The improvement of TRMC with C = 1/5 over DSMC is clearly evident. The same degree of accuracy is reached with a speed up of computing time of approximatively 10 times for TRMC I and approximatively 5 times for TRMC II. For lower values of ² the DSMC method becomes unusable due to the increasing number of collisions, while the computational cost of TRMC reaches a constant value which depends only from the number of particles used in the simulation. 6. Conclusion We have presented a Monte Carlo methods which is suitable for the numerical simulation of the Boltzmann equation close to fluid regimes. In such fluid limit the methods become a kinetic particle scheme for the Euler equations and result in a greater efficiency if compared to standard DSMC schemes. The methods presented in this paper have limited accuracy in time and for such reason have been compared to Nambu version of DSMC. A recursive TRMC method that does not contain time discretization error is currently under development[18]. Numerical comparison of this last method with Bird’s DSMC algorithm will be presented elsewhere. Acknowledgments We would like to thank Giovanni Russo for the many stimulating discussions and suggestions. The support by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282, is gratefully acknowledged. References 1. H. Babovsky, On a simulation scheme for the Boltzmann equation, Math. Methods Appl. Sci., 8 (1986), pp. 223–233. 2. H. Babovsky and R. Illner, A convergence proof for Nanbu’s simulation method for the full Boltzmann equation, SIAM J. Numer. Anal., 26 (1989), pp. 45–65. 3. G. A. Bird, Molecular Gas Dynamics, Oxford University Press, London, 1976. 4. J. F. Bourgat, P. LeTallec, B. Perthame, and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 377–398. 5. E. A. Carlen, M. C. Carvalho, and E. Gabetta, Central limit theorem for Maxwellian molecules and truncation of the Wild expansion, Comm. Pure Appl. Math., 53 (2000), pp. 370– 397. 6. C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. 7. S. Deshpande. A second order accurate kinetic theory based method for inviscid compressible flow. Journal of Computational Physics, 1979. 8. E. Gabetta, L. Pareschi, and G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal., 34 (1997), pp. 2168–2194.

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9. A.L. Garcia, J.B. Bell, W.Y. Crutchfield and B.J. Alder Adaptive mesh and algorithm refinement using Direct Simulation Monte Carlo Journal of Computational Physics, 154, 134-55, 1999. 10. D.B. Hash and H.A. Hassan Assessment of schemes for coupling Monte Carlo and NavierStokes solution methods J. Thermophys. Heat Transf., 10, 242-249, 1996. 11. P. LeTallec and F. Mallinger Coupling Boltzmann and Navier-Stokes by half fluxes Journal of Computational Physics, 136, 51-67, 1997. 12. K. Nanbu, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), pp. 2042–2049. 13. L. Pareschi and R. E. Caflisch, Implicit Monte Carlo methods for rarefied gas dynamics I: The space homogeneous case, J. Comput. Phys., 154 (1999), pp. 90–116. 14. L. Pareschi and G. Russo, Time Relaxed Monet Carlo methods for the Boltzmann equation, SIAM J. Sci. Comput. 23 (2001), pp. 1253–1273. 15. L. Pareschi and G. Russo, Asymptotic preserving Monte Carlo methods for the Boltzmann equation, Transport Theory Statist. Phys., 29 (2000), pp. 415–430. 16. L. Pareschi, G. Russo, An Introduction to Monte Carlo Methods for the Boltzmann Equation, EDP Sciences, SMAI (1999, pp.1-38). 17. L. Pareschi and B. Wennberg, A recursive Monte Carlo method for the Boltzmann equation in the Maxwellian case, Monte Carlo Methods Appl., 7 (2001), pp. 349–358. 18. L. Pareschi, S. Trazzi and B. Wennberg, work in progress. 19. D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow, J. Comput. Phys., 34 (1980), pp. 231–244. 20. D. I. Pullin, Generation of normal variates with given sample, J. Statist. Comput. Simulation, 9 (1979), pp. 303–309. 21. G. Russo and R. E. Caflisch, Implicit methods for kinetic equations, in Rarefied Gas Dynamics: Theory and Simulations, Progress in Aeronautics and Astronautics 159, AIAA, New York, pp. 344–352. 22. E. Wild, On Boltzmann’s equation in the kinetic theory of gases, Proc. Cambridge Philos. Soc., 47 (1951), pp. 602–609. 23. C. Truesdell and R. G. Muncaster, Fundamentals of Maxwell Kinetic Theory of a Simple Monatomic Gas, Academic Press, New York, 1980.

[Figure 1 about here.] [Figure 2 about here.] [Figure 3 about here.] [Figure 4 about here.] [Figure 5 about here.] [Figure 6 about here.] [Figure 7 about here.] [Figure 8 about here.] [Figure 9 about here.] [Figure 10 about here.] [Figure 11 about here.] [Figure 12 about here.] [Figure 13 about here.] [Figure 14 about here.] [Figure 15 about here.] [Figure 16 about here.] [Figure 17 about here.] [Figure 18 about here.] [Figure 19 about here.]


17

University of Ferrara, Department of Mathematics, Via Machiavelli 35, 44100 Ferrara Italy. E-mail: [email protected] University of Ferrara, Department of Mathematics, Via Machiavelli 35, 44100 Ferrara Italy. E-mail: [email protected]

18


List of Figures 1 Iso-values of the density ρ (left) and mean velocity u (right) for ² = 0.1 and M = 1; DSMC (top), TRMC I (middle), TRMC II (bottom). 20 2 Longitudinal and transversal sections of density and velocity (left) and relative errors (right) at y = 6 and x = 5 respectively for ² = 0.1 and M = 1; DSMC (◦), TRMC I (+), TRMC II (×).

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3 Iso-values of the density ρ (left) and mean velocity u (right) for ² = 0.01 and M = 1; DSMC (top), TRMC I (middle), TRMC II (bottom).

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4 Longitudinal and transversal sections of density and velocity (left) and relative errors (right) at y = 6 and x = 5 respectively for ² = 0.01 and M = 1; DSMC (◦), TRMC I (+), TRMC II (×).

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7 Iso-values of the density ρ (left) and mean velocity u (right) for ² = 0.1 and M = 10; DSMC (top), TRMC I (middle), TRMC II (bottom). 26 8 Longitudinal and transversal sections of density and velocity (left) and relative errors (right) at y = 6 and x = 5 respectively for ² = 0.1 and M = 10; DSMC (◦), TRMC I (+), TRMC II (×).

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13 Iso-values of the density ρ (left) and mean velocity u (right) for ² = 0.1 and M = 20; DSMC (top), TRMC I (middle), TRMC II (bottom). 32 14 Longitudinal and transversal sections of density and velocity (left) and relative errors (right) at y = 6 and x = 5 respectively for ² = 0.1 and M = 20; DSMC (◦), TRMC I (+), TRMC II (×).

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18 Longitudinal and transversal sections of density and velocity (left) and relative errors (right) at y = 6 and x = 5 respectively for ² = 0.001 and M = 20; DSMC (◦), TRMC I (+), TRMC II (×). 37 19 Number of “collisions” respectively for M = 1 (left), M = 10 (center), M = 20 (right). From top to bottom ² = 0.1, 0.01, 0.001; DSMC (◦), TRMC I (+), TRMC II (×). 38

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Figure 1. Iso-values of the density ρ (left) and mean velocity u (right) for ² = 0.1 and M = 1; DSMC (top), TRMC I (middle), TRMC II (bottom).

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5

4

4

3

3

2

2

1

1 2

4

6

8

10

12

14

2

4

TRMC I Method

6

8

10

12

14

10

12

14

10

12

14

TRMC I Method

11

11

10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1 2

4

6

8

10

12

14

2

4

TRMC II Method

6

8

TRMC II Method

11

11

10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1 2

4

6

8

10

12

14

2

4

6

8


Figures

37

0.06

0.05

0.25

0.04

0.2

0.03

0.15

0.02

0.1

0.01

0.05

0 0

5

10

15

0 0

5

10

15

0.04

0.035 0.25

0.03 0.2

0.025

0.15

0.02

0.015 0.1

0.01 0.05

0.005

0 0

2

4

6

8

10

12

0 0

2

4

6

8

10

12

5000 4500 0.25

4000 3500 0.2

3000 0.15

2500 2000

0.1

1500 1000 0.05

500 0 0

5

10

15

0 0

5

10

15

5000 4500 0.25

4000 3500 0.2

3000 0.15

2500 2000

0.1

1500 1000 0.05

500 0 0

2

4

6

8

10

12

0 0

2

4

6

8

10


12

38

Figures

5000

5000

5000

4500

4500

4500

4000

4000

4000

3500

3500

3500

3000

3000

3000

2500

2500

2500

2000

2000

2000

1500

1500

1500

1000

1000

1000

500

500

0 0

50

100

150

200

250

300

350

400

4

4

500

0 0

50

100

150

200

250

300

350

0 0

400

4

x 10

4

4

3.5

3

3

3

2.5

2.5

2.5

2

2

2

1.5

1.5

1

1

1

0.5

0.5

0.5

100

150

200

250

300

350

400

0 0

5

4

50

100

150

200

250

300

350

400

4

4

3.5

3

3

3

2.5

2.5

2.5

2

2

2

1.5

1.5

1

1

1

0.5

0.5

0.5

100

150

200

250

300

350

400

300

350

400

0 0

50

100

150

200

250

300

350

400

50

100

150

200

250

300

350

400

x 10

3.5

1.5

50

250

5

x 10

3.5

0 0

200

x 10

0 0

5

x 10

150

3.5

1.5

50

100

4

x 10

3.5

0 0

50

50

100

150

200

250

300

350

400

0 0

Figure 19. Number of “collisions” respectively for M = 1 (left), M = 10 (center), M = 20 (right). From top to bottom ² = 0.1, 0.01, 0.001; DSMC (◦), TRMC I (+), TRMC II (×).