On the expedient solution of the Boltzmann equation by Modified Time Relaxed Monte Carlo (MTRMC) method

Transcription

1 On the expedient solution of the Boltzmann equation by Modified Time Relaxed Monte Carlo (M) method M. Eskandari 1 and S.S. Nourazar 1 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran Corresponding Author icp@aut.ac.ir (Received ; accepted ) ABSTRACT In the present study, a modified time relaxed Monte Carlo (M) method is developed for numerical solution of the Boltzmann equation in rarefied regimes. Taylor series expansion is employed to obtain a generalized form of the Wild sum expansion and consequently the modified collision functions with fewer inter-molecular interactions are obtained. The proposed algorithm is applied on the lid-driven micro cavity flow with different lid velocities and the results for velocity and shear stress distributions are compared with those from the standard and methods. The comparisons of the results of the method with those of the M and methods show reasonable agreement. The present study illustrates appreciable improvement in the computational expense of the M method compared to those from standard and methods. The improvement is more pronounced compared to the standard method. It is observed that up to 56% reduction in CPU time is obtained in the studied cases. Keywords: Boltzmann equation, Time relaxed Monte Carlo, modified time relaxed Monte Carlo, direct simulation Monte Carlo, Taylor series 1. INTRODUCTION The Navier-Stokes equations lose the accuracy of simulating the flows when the characteristic length of the flow becomes comparable to the mean free path. Under this circumstance, the governing equation is the Boltzmann equation of kinetic theory (Cercignani 1988; Bird 1994). In the past few years, the direct simulation Monte Carlo () method is widely used for simulation of the Boltzmann equation (Cercignani 1988; Bird 1994; Pareschi and Trazzi 25). It is noticeable that the main objective in those simulations, is to decrease the CPU time of the simulation (Pareschi and Trazzi 25; Oran, E.S., Oh, C.K., and Cybyk 1998; Gabetta, Pareschi, and Toscani 1997; Pan, Liu, Khoo, and Song 2; Filbet and Russo 23). The truncated Wild (Wild 1951) sum expansion is employed to calculate time discretization for a rarefied gas flow, where a model is presented for the collisional terms (Pareschi and Caflisch 1999). Later, Pareschi and Russo (Pareschi and Russo 2) did stability analysis on the method and showed the existence of the stability of the method. They performed the method to study the Kac equation and obtained reasonable results compared to the results of the standard method. Furthermore, Pareschi et al. (Pareschi and Wennberg 21; Pareschi and Russo 21a; Pareschi and Russo 21b) developed a method using the VHS model for simulation of method. Pareschi and Trazzi (Pareschi and Trazzi 25) used a new method to simulate the conservation of mass, momentum, and energy, simultaneously using the method. The third order collisions are performed by Pareschi et al. (Russo, Pareschi, Trazzi, Shevyrin, Bondar, and Ivanov 25) on the method with various Knudsen numbers for the flow under consideration. Moreover, Ganjaei and Nourazar (Ganjaei and Nourazar 29) study the binary mixture flow inside a cylinder using the and methods. Trazzi et al. (Trazzi, Pareschi, and Wennberg 29) used a new method with excellent improvement in the simulation of shock problems. Most recently, Eskandari and Nourazar (Eskandari and Nourazar 217) performed the first, second and third orders of the scheme to study the liddriven micro cavity flow with different lid velocities and Knudsen numbers. Furthermore, they employed the 1, 2 and 3 schemes to investigate the flow over the nano plate with different free stream velocities and Knudsen numbers (Eskandari and Nourazar 218). The investigations showed excellent agreement between the results obtained from the third order scheme with the ones from the standard method (Eskandari and Nourazar 217; Eskandari and Nourazar 218). 1.1 The purpose of the present work In the present work, it is intended to develop a modified method, called the M method,

2 to lessen the CPU time of simulation when compared to the CPU time of the standard and methods. To reach this aim, Taylor series expansion is employed in derivatives to reform the Wild sum expansion. Simulation of a lid-driven micro cavity flow is considered as the benchmark problem to investigate the accuracy and computational expense of the proposed algorithm. The present work intends to investigate the lid-driven micro cavity flow from the following points of view: Investigating the accuracy of proposed algorithm for the M method Comparison of demanded CPU time, demanded for the, and M methods studied cases Investigating the influence of lid velocity on the flow properties 2. THE GOVERNING EQUATION 2.1 The equation of Boltzmann The governing equation for the present study is written as (Cercignani 1988; Bird 1994): g t + υ x = 1 Q(g,g). (1) Kn Where Q(g, g) indicates the binary collisions of molecules. The above equation may be rewritten as the following, where Q(g,g) = P(g,g) g (Gabetta, Pareschi, and Toscani 1997; Wild 1951; Carlen, Carvalho, and Gabetta 2): g t + υ x = 1 (P(g,g) g). (2) Kn P(g,g) is a new binary collisions of molecules and the parameter is the frequency of collision. The above equation (2) may be split into the convective (i.e. Q ) (3) and collisional steps (i.e. υ x ) (4) (Bird 1994; Pareschi and Trazzi 25; Pareschi and Caflisch 1999; Trazzi, Pareschi, and Wennberg 29; Jahangiri, Nejat, Samadi, and Aboutalebi 212; Yanenko 1971). g t + υ x =. (3) g t = 1 (P(g,g) g). Kn (4) In the above equations (equations (3) and (4)) the equation of convection step (equation (3)) may be directly solved using the Lagrangian coordinates by particle tracing method, however the equation of collision step (equation (4)) needs a model to simulate collision of the molecules. 2.2 Approach of the method The collision term in the method of is solved by discretization of equation (4) using the first order Euler upwind scheme as: g n+1 g n = P(g, g) g. t Kn Kn (5) g n+1 = (1 t Kn )gn + t P(g, g). Kn (6) The probabilistic interpretation of the scheme (6) may be stated as the following: to sample a particle from g n+1, a particle is sampled from g n with probability of 1 ( t/kn) and a particle is sampled from P(g, g)/ with probability of t/kn. 2.3 Approach of the method The relaxation time τ and the distribution function of probability G(v, τ) are expressed as: (Pareschi and Trazzi 25; Pareschi and Caflisch 1999): τ = (1 e t/kn ). (7) G(v,τ) = g(v,τ)e t/kn. (8) Equation (4) may be rewritten by employing variables (7) and (8) as the followings: G τ = 1 P(G,G). G(v,τ = ) = g(v,). (9) The solution of Cauchy equation (9) is written as a power series solution (Pareschi and Trazzi 25) as: G(v,τ) = k= τ k g k (v). g(v,t) = e t/kn ( ( 1 e t/kn) k gk (v)). k= (1) By arrangement of the above equation (9) and using equation (1) we arrive at the following (Pareschi and Trazzi 25): G τ = kτ k 1 g k (v) = k= P(G,G) = P ( k= k= τ k g k (v), (k + 1)τ k g k+1 (v). k= τ k g k (v) = P(g,g ) + 2τP(g,g 1 )+ ) (11) + τ 2 (2P(g,g 2 ) + P(g 1,g 1 )) +. (12) 2

3 By using the above equation, the following recursive equations for the coefficients g k are obtained: g k+1 = 1 k ( ) 1 k + 1 P(g k,g k h ). (13) h= The scheme is written in the following way: g n+1 (v) =e t/kn m τ k g k (v) k= ( + 1 e t/kn) m+1 M(v). (14) The equation of the method can generally be obtained as (Pareschi and Caflisch 1999; Pareschi and Russo 2; Pareschi and Russo 21a; Pareschi and Russo 21b; Pareschi and Trazzi 25; Pareschi and Wennberg 21; Trazzi, Pareschi, and Wennberg 29): g n+1 (v) = m k= (A k g k (v)) + A m+1 M(v). (15) Where the coefficients g k are calculated by the recursive formula (13). In the present simulation, the weight functions are depicted as the ones introduced by Pareschi (Pareschi and Trazzi 25; Pareschi and Russo 2) as the followings: A k = (1 τ)τ k A m = 1 m k= A k A m+1 A m+1 = τ m+2. (16) In the present work the third order and the most accurate (3) method is used in all simulations (Eskandari and Nourazar 217). Therefore the third order (3) with the corresponding weight functions (16) may be written as: g n+1 = (1 τ)g + ( τ τ 2) P(g,g ) + ( τ 2 τ 3) P(g,g 1 ) ( + τ 3 τ 5) 2P(g,g 2 ) + P(g 1,g 1 ) + τ 5 M(v) 3 = A g n + A 1 g 1 + A 2 g 2 + A 3 g 3 + A 4 M(v). (17) From probability point of view the equation (17) may be explained in the following way (Eskandari and Nourazar 217): A chosen particle at the n th time step is sampled from g 1 with probability of A 1, sampled from g 2 with probability of A 2, sampled from g 3 with probability of A 3 and sampled from Maxwellian distribution with probability of A 4. The rest of the particles do not collide. 2.4 Approach of the M method In the M method, Taylor series expansion is employed to obtain derivatives with higher accuracies: G τ = Gn+1 G n. (18) τ G n+1 = m k= (τ + τ) k g k (v). (19) τ = τ t t = t Kn e t/kn. (2) Introducing equations (18) to (2), to equation (9) yields: m k= (τ + τ)k g k m k= τk g k τ = P(G,G). (21) Applying equations (12) to equation (21) gives: ((τ + τ)g 1 τg 1 ) + ( (τ + τ) 2 g 2 τ 2 g 2 ) + τ = 1 (P(g,g ) + 2τP(g,g 1 ) + ). (22) Coefficients g k can be obtained by rearranging the right and left hand sides of equation (22) for the same corresponding powers of the relaxation time: τ g 1 = (τ + τ) τ g 2 = g 3 = P(g,g ) 2τ τ P(g,g 1 ) (τ + τ) 2 τ 2 3τ 2 τ (τ + τ) 3 τ 3 ( ) 2P(g,g 2 ) + P(g 1,g 1 ) 3. (23) The M scheme can be achieved by combining the Maxwellian form of equation (1) with coefficients introduced in equation (23) as: g n+1 = m k= B k g k + B m+1 M(v). (24) Comparison of equation (24) with equation (6) indicates that for higher values of Knudsen numbers the M method has the same ability as the method. Considering the first three terms of the M method (24) and equation (16), will 3

4 result in the third order of M method: g n+1 = g n (1 τ + τ4 τ(3τ + 2 τ)(1 τ 2 ) (2τ + τ)(3τ 2 + 3τ τ + τ 2 ) + τ2 τ(3τ 2 + 3τ τ + τ 2 )(1 τ 3 ) (2τ + τ)(3τ 2 + 3τ τ + τ 2 ) + ( τ τ 2) P(g,g ) (( τ 2 τ 3) ) 2τ P(g,g 1 ) + 2τ + τ ( (τ 3 τ 5 ) 3τ 2 ) 2P(g,g 2 ) + P(g 1,g 1 ) + 3τ 2 + 3τ τ + τ τ 5 M(v) = B g n + B 1 P(g,g ) ) + B 2 P(g,g 1 ) + B 3 M. (25) The privilege of using the M method to the method is the freedom of selecting higher values of time steps, when compared to the method, where the freedom of choosing is very limited ( t/kn 1). It is noticeable that the mentioned limitation on the time step is resolved by defining relaxed time and transformed probability distribution function in the and M methods. On the other hand, comparison of equations (17) and (25) indicates fewer collisions in the M method compared to the standard method. The fewer collisions may result in lower computational expense for the M method, compared to that of the standard method. The third order of the M method may be formalized as the following: Algorithm 1 Third order M scheme with VHS model 1: evaluate the initial velocity of particles; 2: for t = to n tot t do 3: mark all particles with the -collision label; 4: estimate upper bound for the cross-section σ; 5: set τ = (1 e t/kn ); 6: compute B 1 (τ),b 2 (τ),b 3 (τ),b 4 (τ); 7: set N 1 = [NB 1 /2], N 2 = [NB 2 /4], N 3 = [NB 3 /3], N 4 = [NB 4 ]; 8: select (N 1 + N 2 + N 3 ) dummy collision pairs among all particles; 9: for (N 1 + N 2 + N 3 ) pairs do 1: compute σ i j = σ( v i v j ); 11: generate random number ε 1; 12: if ε < (σ i j /σ) then 13: collide particle i with particle j; 14: evaluate velocities of post collision; 15: update labels of participant particles to 1-collision ; 16: end if 17: end for 18: select 2N 2 particles among -collision and 2N 2 particles among 1-collision particles; 19: for 2N 2 pairs do 2: perform the collision between pairs; 21: compute the post collision velocities; 22: update the labels of participant particles to 1-collision and 2-collision ; 23: end for 24: select N 3 /3 pairs among 1-collision particles; 25: for N 3 /3 pairs do 26: perform the collision between pairs; 27: compute the post collision velocities; 28: update the labels of participant particles to 2-collision ; 29: end for 3: select N 3 /3 particles among -collision and N 3 /3 particles among 2-collision particles; 31: for N 3 /3 pairs do 32: perform the collision between pairs; 33: compute the post collision velocities; 34: update labels of participant particles to 1- collision and 3-collision ; 35: end for 36: for N 4 particles do 37: replace particles with samples from the Maxwellian, with the same total energy and moment; 38: end for 39: for (N 2N 1 4N 2 3N 3 N 4 ) particles do 4: the energy and momentum wll not change for remained particles; 41: end for 42: end for 3. THE GEOMETRY AND CALCULATION CONDITION For validating the proposed modified time relaxed Monte Carlo (M) method, a lid-driven micro cavity flow is considered. Figure 1 presents the geometry and the boundary conditions for the benchmark problem. The geometry consists of a wall with length of 1m, where the diffuse reflection is used for the walls with temperatures of 3K. The upper lid moves in the x direction while the remaining ones are kept stationary. The cavity consists initially of Argon gas, where the temperature, number density and Knudsen number of the Argon gas are 3K, molecules per cubic meter and 4

5 1 moving diffuse reflector lid 1 y (microns) xstationary diffuse reflector wall Horizontal axis Vertical axis stationary diffuse reflector wall y stationary diffuse reflector wall x (microns) Fig. 1. The geometry and the boundary conditions for lid-driven micro cavity..5, respectively. The velocities for the lid are considered to be 1m/s, 1m/s and 1m/s for cases I, II and III, respectively. These values for velocities are selected according to data of published articles (Amiri-Jaghargh, Roohi, Niazmand, and Stefanov 212; Amiri-Jaghargh, Roohi, Niazmand, and Stefanov 213; Mohammadzadeh, Roohi, and Niazmand 213; John, Gu, and Emerson 211; John, Gu, and Emerson 21; Jiang, Fan, and Shen 23; Sheremet and Pop 215; Safdari and Kim 215; Gutt and Groan 215; Rana, Torrilhon, and Struchtrup 213). The model for the molecular collision is chosen to be variable hard sphere (VHS) scheme. The diameter, viscosity and mass of the molecules are chosen to be m,.81 and kg, respectively (Bird 1994). The selection of time step for the method is confined to t/kn 1, while larger time steps comparable with the free flow time step, are considered for the and M methods. 3.1 Mesh refinement test The finer mesh size, the more accurate are the results. However, the more finer mesh size results in larger amount of CPU time. Hence, a mesh independency test is performed in the present study with four mesh resolutions. The uniformity of the mesh size is kept consistent in both x and y directions. Our present results of simulation for the M method are compared with those obtained for and methods using two models for the collisional sampling. Figures 2(a), 2(b) and 2(c) show that the mesh refinement test is satisfactory for the fine mesh size of 4 4. Moreover, the average number of particles per each cell is 2 particles where the total number of 3,3, particles are considered in the mean. L U x / y/l U y / (a) U x distribution along the vertical axis τ wall /τ (b) U y distribution along the horizontal axis. moving lid (c) Wall shear stress distribution. Fig. 2. Mesg refinement tests. 5

6 4. THE RESULTS OF SIMULATION FOR THE M METHOD Figure 3 illustrates the non-dimension wall shear stress and velocity distributions for case I U x / Jaghargh et al. M y/l (a) U x distribution along the vertical axis. ( = 1m/s). The reference shear stress is defined as τ = /L. The results obtained from the M method are compared with those obtained from the standard and methods, and the ones published by Jaghargh et al. (Amiri-Jaghargh, Roohi, Niazmand, and Stefanov 212; Amiri-Jaghargh, Roohi, Niazmand, and Stefanov 213). Figure 3 illustrates a tremendous agreement in the comparison of the M results with the ones from and methods. Figure 4 shows the non-dimensional wall shear stress and velocity distributions for the cavity flow for case II ( = 1m/s). The distributions illustrate excellent agreement between the results of the M method with those obtained from the standard and methods, and the ones published by Jaghargh et al. (Amiri-Jaghargh, Roohi, Niazmand, and Stefanov 212; Amiri- Jaghargh, Roohi, Niazmand, and Stefanov 213). Moreover, figure 4(c) indicates that the maximum U y / Jaghargh et al. M U x / 1 Jaghargh et al. M.5 τ wall /τ (b) U y distribution along the horizontal axis. moving lid M (c) Wall shear stress distribution. Fig. 3. The velocity and shear stress distributions for case I ( = 1m/s). 6 U y / y/l (a) U x distribution along the vertical axis M Jaghargh et al (b) U y distribution along the horizontal axis.

7 1 5 Jaghargh et al. M.2.1 M τ wall /τ -5 moving lid U y / (c) Wall shear stress distribution (b) U y distribution along the horizontal axis. Fig. 4. The velocity and wall shear stress distributions for case II ( = 1m/s). 1 M wall shear stress occurs at both ends of the moving lid. This may be attributed to higher velocity gradients at both ends of the moving lid. Figure 5 presents the distributions of U x /, U y / and τ wall /τ for case III ( = 1m/s). The results of the M method shows a very good agreement with the results of the simulation of the and methods. Figures 3 to 5 demonstrate that the results of the M method show excellent agreement with their counterparts from the standard and method for a wide range of lid velocities including low subsonic (i.e. = 1m/s) to supersonic (i.e. = 1m/s) lid velocities. Figure 6 presents the CPU time against the physical time at different lid velocities for the, τ wall /τ -1-2 moving lid (c) Wall shear stress distribution. Fig. 5. The velocity and wall shear stress distributions for case II ( = 1m/s). U x / M y/l (a) U x distribution along the cavity axes. CPU time( 1 3 sec) M = 1m/s Physical time( 1-9 Sec) (a) Case I ( = 1m/s ) 7

8 CPU time( 1 3 sec) CPU time( 1 3 sec) 3 M = 1m/s Physical time( 1-9 Sec) (b) Case II ( = 1m/s ) M = 1 m/s Physical time( 1-9 Sec) (c) Case III ( = 1m/s ) Fig. 6. The comparison of physical time and CPU time of different schemes and different lid velocities and M methods. Figure 6 indicates that for the same physical time, the demanded CPU time for the and methods is greater than that for the M method. The enormous difference between the computational costs of the M and methods originates from replacing the time consuming intermolecular collision with the local Maxwellian distribution (see equations (6) and (25)) and using the relaxed time which allows employing higher time steps, while the improvement in comparison to the method, can be attributed to the modified collision functions which yield fewer intermolecular collisions in the M method compared to the standard method (see equations (17) and (25)). Table 1 presents the normalized consumed CPU time for, and M methods in the present work, at different lid velocities. The Table 1. The normalized CPU time for the, and M schemes at different lid velocities. (m/s) M comparison of the consumed CPU times indicate that for the same physical time, the CPU time required for the M method to simulate the liddriven micro cavity flow accurately, is comparably less than those of the and methods. The improvement in the demanded CPU time of M method compared to the method, is more pronounced as the lid velocities increase. CPU time reduction(%) (m/s) Fig. 7. The reduction of CPU time in the M method compared to the standard method. Figure 7 presents the reduction in the demanded CPU time for the M method in comparison to the ones from the standard method. Figure 7 indicates that in the M method, the required CPU time of simulation is significantly less than those required for the and methods. This is more pronounced for higher lid velocities where, for values of lid velocities higher than 1m/s the reduction in the CPU time approaches an asymptotic value of about 58%. Figure 7 and table 1 showed that the difference between the M and the schemes about the computational costs becomes significant for the lid velocities 5(m/s). 5. THE INFLUENCE OF THE LID VELOC- ITY ON THE RESULTS In this section, the validated results obtained from the M method for different lid velocities, are compared. It should be indicated that the Knudsen 8

9 number, based on the wall length is considered to be Kn =.5 (see section 3.). Figure 8 presents U x / = 1 m/s = 1 m/s = 1 m/s y/l (a) U x distribution along the cavity axes. = 1 m/s = 1 m/s = 1 m/s the normalized velocity (U x,u y ) and shear stress (τ wall ) variations, varying from low to very high supersonic velocities. It indicates that for lid velocities up to 1m/s the distribution remains nearly intact, while for = 1m/s distribution patterns change, where the steeper velocity gradients are obtained for higher lid velocities. Moreover, the same results at different velocities are obtained for the shear stress distributions. Figure 9 shows the streamline and temperature patterns at different lid velocities obtained from the simulations, performed by the M method. The contours illustrate that as the velocity increases, the maximum temperature increases significantly. The increment of the temperature for cases I and II is not appreciable, however the increment of temperature for case III is appreciable and a hot spot close to the moving lid is recognized. The appearance of hot spot may be interpreted to the occurrence of the shock wave in case III = 1m/s..5 U y / (b) U y distribution along the horizontal axis. τ wall /τ moving lid =1 m/s =1 m/s =1 m/s (a) Case I ( = 1m/s) (c) Wall shear stress distribution. Fig. 8. The velocity and shear stress distributions for different lid velocities and Kn =.5. (b) Case II ( = 1m/s) 9

10 indicates that our present results of simulation are accurate. Figure 1 shows the contours of Mach number in the lid driven micro cavity flow for case III ( = 1m/s) and Kn =.5, where Mach = 1 iso-values are determined in bold black lines. 6. CONCLUSION (c) Case III ( = 1m/s) Fig. 9. Streamlines and temperature contours at various lid velocities and Kn =.5. Table 2. The primary vortex center location for different lid velocities. Location of vortex centers Location of vortex centers Re (m/sec) (present study) (x c /L,y c /L) (Jaghargh et al.) (x c /L,y c /L) (.51,.76) (.51,.76) (.61,.75) (.61,.75) (.55,.59) - The modified time relaxed Monte Carlo (M) method has been introduced in present work. Taylor series expansion is employed in the Wild sum expansion to obtain the modified collision functions with fewer intermolecular collision. It is observed that the proposed algorithm for the M method is capable of accurately simulating the cavity flow at various lid velocities, covering low subsonic to supersonic lid velocities. Moreover, it is illustrated that for the same physical time, the computational CPU time of the M method is appreciably less that that for the standard and methods. The reduction in the computational CPU time is enhanced at higher lid velocities. The improvement in the CPU time is more pronounced when the comparison is made with the results of method. It is shown that up to 56% reduction in the computational CPU time can be obtained from M method compared to the standard method. REFERENCES Table 2 shows the locations of the created vortex centers for different lid velocities, that is achieved from the M method. The results obtained for the M method when compared with the results obtained by the previous researches (Amiri- Jaghargh, Roohi, Niazmand, and Stefanov 213), Fig. 1. The contours of local Mach number for case III ( = 1m/s) and Kn =.5. Amiri-Jaghargh, A., E. Roohi, H. Niazmand, and S. Stefanov (212, may). Low speed/low rarefaction flow simulation in micro/nano cavity using method with small number of particles per cell. Journal of Physics: Conference Series 362, 127. Amiri-Jaghargh, A., E. Roohi, H. Niazmand, and S. Stefanov (213). Simulation of Low Knudsen Micro/Nanoflows Using Small Number of Particles per Cells. Journal of Heat Transfer 135(1), 118. Bird, G. A. (1994). Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford,: Clarendon Press. Carlen, E. A., M. C. Carvalho, and E. Gabetta (2, mar). Central limit theorem for Maxwellian molecules and truncation of the wild expansion. Communications on Pure and Applied Mathematics 53(3), Cercignani, C. (1988). The Boltzmann Equation and Its Applications, Volume 67 of Applied Mathematical Sciences. New York, NY: Springer New York. Eskandari, M. and S. Nourazar (217, aug). On 1

11 the time relaxed Monte Carlo computations for the lid-driven micro cavity flow. Journal of Computational Physics 343, Eskandari, M. and S. Nourazar (218, jan). On the time relaxed Monte Carlo computations for the flow over a flat nano-plate. Computers & Fluids 16, Filbet, F. and G. Russo (23, apr). High order numerical methods for the space nonhomogeneous Boltzmann equation. Journal of Computational Physics 186(2), Gabetta, E., L. Pareschi, and G. Toscani (1997, dec). Relaxation Schemes for Nonlinear Kinetic Equations. SIAM Journal on Numerical Analysis 34(6), Ganjaei, A. A. and S. S. Nourazar (29, oct). Numerical simulation of a binary gas flow inside a rotating cylinder. Journal of Mechanical Science and Technology 23(1), Gutt, R. and T. Groan (215, sep). On the liddriven problem in a porous cavity. A theoretical and numerical approach. Applied Mathematics and Computation 266, Jahangiri, P., A. Nejat, J. Samadi, and A. Aboutalebi (212, may). A high-order Monte Carlo algorithm for the direct simulation of Boltzmann equation. Journal of Computational Physics 231(14), Jiang, J.-Z., J. Fan, and C. Shen (23). Statistical Simulation of Micro-Cavity Flows. In AIP Conference Proceedings, Volume 663, pp AIP. John, B., X.-J. Gu, and D. Emerson (21). Investigation of Heat and Mass Transfer in a Lid-Driven Cavity Under Nonequilibrium Flow Conditions. Numerical Heat Transfer, Part B: Fundamentals 58(5), John, B., X.-J. Gu, and D. R. Emerson (211). Effects of incomplete surface accommodation on non-equilibrium heat transfer in cavity flow: A parallel study. Computers & Fluids 45(1), Mohammadzadeh, A., E. Roohi, and H. Niazmand (213). A Parallel Investigation of Monatomic/Diatomic Gas Flows in a Micro/Nano Cavity. Numerical Heat Transfer, Part A: Applications 63(4), Oran, E.S., Oh, C.K., and B. Z. Cybyk (1998, jan). DIRECT SIMULATION MONTE CARLO: Recent Advances and Applications. Annual Review of Fluid Mechanics 3(1), Pan, L. S., G. R. Liu, B. C. Khoo, and B. Song (2, mar). A modified direct simulation Monte Carlo method for low-speed microflows. Journal of Micromechanics and Microengineering 1(1), Pareschi, L. and R. E. Caflisch (1999, sep). An Implicit Monte Carlo Method for Rarefied Gas Dynamics. Journal of Computational Physics 154(1), Pareschi, L. and G. Russo (2, apr). Asymptotic preserving Monte Carlo methods for the Boltzmann equation. Transport Theory and Statistical Physics 29(3-5), Pareschi, L. and G. Russo (21a). An introduction to Monte Carlo method for the Boltzmann equation. ESAIM: Proceedings 1, Pareschi, L. and G. Russo (21b, jan). Time Relaxed Monte Carlo Methods for the Boltzmann Equation. SIAM Journal on Scientific Computing 23(4), Pareschi, L. and S. Trazzi (25, jul). Numerical solution of the Boltzmann equation by time relaxed Monte Carlo () methods. International Journal for Numerical Methods in Fluids 48(9), Pareschi, L. and B. Wennberg (21). A recursive Monte Carlo method for the Boltzmann equation in the Maxwellian case. Monte Carlo Methods and Applications 7(3-4). Rana, A., M. Torrilhon, and H. Struchtrup (213, mar). A robust numerical method for the R13 equations of rarefied gas dynamics: Application to lid driven cavity. Journal of Computational Physics 236, Russo, G., L. Pareschi, S. Trazzi, A. A. Shevyrin, Y. A. Bondar, and M. S. Ivanov (25). Plane Couette Flow Computations by and MFS Methods. In AIP Conference Proceedings, Volume 762, pp AIP. Safdari, A. and K. C. Kim (215, aug). Lattice Boltzmann simulation of the threedimensional motions of particles with various density ratios in lid-driven cavity flow. Applied Mathematics and Computation 265, Sheremet, M. and I. Pop (215, sep). Mixed convection in a lid-driven square cavity filled by a nanofluid: Buongiorno s mathematical model. Applied Mathematics and Computation 266, Trazzi, S., L. Pareschi, and B. Wennberg (29, jan). Adaptive and Recursive Time Relaxed 11

12 Monte Carlo Methods for Rarefied Gas Dynamics. SIAM Journal on Scientific Computing 31(2), Wild, E. (1951, jul). On Boltzmann s equation in the kinetic theory of gases. Mathematical Proceedings of the Cambridge Philosophical Society 47(3), 62. Yanenko, N. N. (1971). The Method of Fractional Steps. Berlin, Heidelberg: Springer Berlin Heidelberg. 12