A new algorithm for the simulation of the boltzmann equation using the direct simulation monte-carlo method

Transcription

1 Joural of Mechacal Scece ad echology 3 (009) 86~870 Joural of Mechacal Scece ad echology wwwsprgerlkcom/cotet/ x DOI 0007/s A ew algorthm for the smulato of the boltzma equato usg the drect smulato mote-carlo method A A Gajae ad S S Nourazar Mechacal Egeerg Departmet, Amrkabr Uversty of echology, ehra, IRAN (Mauscrpt Receved December, 008; Revsed Jue 9, 009; Accepted July 5, 009) Abstract A ew algorthm, the modfed drect smulato Mote-Carlo (MDSMC) method, for the smulato of Couette- aylor gas flow problem s developed he aylor seres expaso s used to obta the modfed equato of the frstorder tme dscretzato of the collso equato ad the ew algorthm, MDSMC, s mplemeted to smulate the collso equato the Boltzma equato I the ew algorthm (MDSMC) there exsts a ew extra term whch takes to accout the effect of the secod order collso hs ew extra term has the effect of ehacg the appearace of the frst aylor stabltes of vortces streamles I the ew algorthm (MDSMC) there also exsts a secod order term tme step the probablstc coeffcets whch has the effect of smulato wth hgher accuracy tha the prevous DSMC algorthm he appearace of the frst aylor stabltes of vortces streamles usg the MDSMC algorthm at dfferet ratos of ω ν (expermetal data of aylor []) occurred at less tme-step tha usg the DSMC algorthm he results of the torque developed o the statoary cylder usg the MDSMC algorthm show better agreemet comparso wth the expermetal data of Kuhlthau [] tha the results of the torque developed o the statoary cylder usg the DSMC algorthm Keywords: Boltzma equato; DSMC; Rotatg cylder; Hgh order DSMC; Modfed DSMC Itroducto I the gas flow problems where the legth scale of the system s comparable to the mea free path for molecules the gas flow the cocept of the cotuum s o more vald, Kudse umber greater tha 0 [3] I ths case the smulato s doe usg the Drect Smulato Mote-Carlo (DSMC) or the Collsoal Boltzma Equato (CBE) methods I most cases the drect soluto of the CBE s mpractcable due to the huge umber of molecules, however most of the tme; the mplemetato of the DSMC s more practcable So far the rarefed gas flow problems are smulated hs paper was recommeded for publcato revsed form by Assocate Edtor Ghu So Correspodg author el: , Fax: E-mal address: cp@autacr KSME & Sprger 009 usg the DSMC algorthm by Vogetz et al [4], Stefaov et al [5], Brd G A [3] & [6], Shagawa et al [7] ad Myog [8] he DSMC algorthm s used the flow smulato of the prevous researchers ad the results of the smulatos show some dscrepaces whe compared wth the expermetal data For example Vogetz et al [4], studed the theoretcal ad expermetal aspects of the rarefed supersoc flow about several smple shapes (sphere, cylder, coe ad wedge) her results of smulato show less dscrepacy at hgh Kudse umber tha low Kudse umber whe compared wth measuremets he Couette-aylor gas flow s smulated usg the DSMC algorthm by Stefaov ad Cercga [5] he formato of the aylor stabltes of vortces s clearly exhbted he usteady axally symmetrc ad three-dmesoal Couette-aylor flow s smulated by Brd [6] I ths work, the curret status of the

2 86 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~870 DSMC algorthm s revewed wth partcular emphass o ts rage of valdty, the extet of ts valdato agast expermet, ad the DSMC applcatos to the study of flow stabltes are dscussed he smulato of the rarefed gas flow through crcular tube of fte legth the trastoal regme at both low Kudse umber ad hgh Kudse umber are doe usg the DSMC algorthm by Shagawa et al [7] ad Myog [8] I ths study we would lke to develop a ew algorthm to smulate the collso equato the Boltzma equato wth hgher accuracy tha the prevous algorthms avalable the lterature Purpose of the preset work So far the rarefed gas dyamc problems, whe the Kudse umber s large eough, are smulated usg the frst-order tme dscretzatos of the Boltzma equato [3] he Boltzma equato s splt tme to purely covectve equato (collso term s zero) ad purely collso equato (covectve term s zero) he collso equato s dscretzed tme by the frst-order Euler scheme ad the probablstc terpretato of the dscretzed equato breaks dow whe the rato µ t ε s large eough, [9-4] However the preset work our goal s to develop a ew algorthm whch cosders the effects of the trucato errors the tme dscretzato of the frst-order Euler scheme for the collso equato order to acheve more accurate probablstc terpretatos o acheve ths goal, we wrte the modfed equato of the frst-order Euler scheme usg the aylor expaso seres ad the we capture the hgher order trucated terms I the preset work due to the lmtato of the computg tme we are lmted to choose oly the frst two terms the aylor expaso seres he detal of the dervato of our ew algorthm whch we call that modfed drect smulato Mote- Carlo (MDSMC) algorthm s preseted the secto b the MDSMC algorthm Mathematcal formulatos he boltzma equato he Boltzma equato s wrtte as, [5]: f + v r f = σ ( v v, )( f ( v ) f ( v ) f ( v) f ( v )) d dv ε Ω Ω 3 R S = Q( f, f ) ε () I Eq (), f ( v) s the oegatve desty probablty dstrbuto fucto of molecule of class havg the velocty of v, f ( v ) s the oegatve desty probablty dstrbuto fucto of the molecule of class havg the velocty of v, f ( v ) s the post-collso oegatve desty probablty dstrbuto fucto of the molecule of class havg the velocty of v, ad f ( v ) s the post-collso oegatve desty probablty dstrbuto fucto of the molecule of class havg the velocty of v ad Ω s the agle the sphercal coordates he Q( f, f ) s the tegral collso whch descrbes the bary collsos of the molecules he kerel σ s a o egatve fucto whch s descrbed, [-4]: ( ) bα ( θ) α σ v v, Ω = v v () Where, θ s the scatterg agle betwee v v ad v v Ω he varable hard sphere (VHS), [3], model s ofte used umercal smulato of rarefed gases, where, bα ( θ ) = C wth C a postve costat ad α = he value of C s equal to (Brd, 994), C = σ,where σ s the collso cross secto ad s equal π d 4 he MDSMC algorthm Splttg equato, the Boltzma equato, [6], to equato for the effect of collso, v r f 0, ad equato for the effect of covecto, Q( f, f ) 0 he equato for the effect of covecto s wrtte as, [-4]: f + v r f = 0 (3) Ad the equato for the effect of collso s wrtte as, [-4]: f = Q( f, f ), ε (4)

3 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~ ad the collso term s wrtte as: Q f f ε (, ) ( ( ) ( ) ( ) ( )) σ v v f v f v f v f v dωdv ( ) ( ) = σ v v f v f v dωdv ( ) ( ) σ v v f v f v dωdv = = P( f, f ) µ ( v) f ε + Where, ( ) ( ) 4π 0 f ( v ) (5) µ v = σ v v f v dω dv = σ v v dω d v = κρ m s the mea collso frequecy for the molecules havg velocty v, ρ s the desty of the gas, m s the mass of a molecule of the 4π gas, κ s a molecular costat κ = σ v v dω + ad ρ ( v ) m= f dv [7] I specal case whch σ v v s depedet of v v (Maxwella molecules), we have: κρ µ ( v ) = µ = (6) m Substtutg Eq (6) to Eq (5) ad the to Eq (4): f = Q( f, f ) = P( f, f ) µ f ε ε 0 (7) he frst order tme dscretzato of Eq (7) s wrtte as: (, f ) P f + µ t µ t f = f + ε ε µ (8) he probablstc terpretato of Eq (8) s the followg I order a partcle s sampled from f +, a partcle s sampled from f wth probablty of ( µ t ε ) ad a partcle s sampled from ( P f, f ) µ wth probablty of µ t ε It s to be oted that the above probablstc terpretato fals f the rato of µ t ε s too large because the coeffcet of f o the rght had sde may become egatve, [9-4] + I our algorthm, we wrte for f aylor seres expaso as: ( ) + f t f 3 ( ) f = f + t + + O t! he secod order dervatve, 9 s wrtte as: f (9) f t, equato f = = P( f, f ) µ f = ε P( f, f ) µ f = ε ε P( f, f ) µ P( f, f ) µ f ε ε ε P( f, f ) µ µ = P ( f, f ) + f ε ε ε (0) Substtutg equato0 for the value of f t ad equato 7 for the value of f t to Eq (9): + t f = f + P( f, f ) µ f ε + µ P f! ε ε µ 3 + f O + ( t) ε ( t) P( f, f ) σ (, f ) he (, ) ( ) ( ) () P f f = v v f v f v dω dv s the blear operator descrbg the collso effect of two molecules he tme dervatve of ( P f, f ) s wrtte as: (, f ) P f = f ( v ) σ v v f ( v ) dω dv he tme dscretzato of (, ) P f f s wrtte as: ()

4 864 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~870 (, f ) P f = f ( v ) f ( v ) σ v v v v t f ( ) dωd = σ v f ( v ) f ( v ) dωdv t v σ v v f ( v ) f ( v ) dωdv σ (3) Substtutg for P( f, f ) = v v f ( v ) ( ) f v dω dv to Eq (3): ( f ) P f, = { P( f, f ) P( f, f )} t Substtutg Eq (4) to Eq (): (, f ) P f + µ t f = f + f + ε µ! ε t µ µ ( t) µ P( f, f ) P( f, f ) (, f ) µ P f µ + + ε µ ε where (, ) ( ) f O t 3, (4) (5) f = P f f µ Rearragg ad trucatg terms hgher tha the secod order Eq (5): f + ( t) µ t µ = + ε ε µ t µ + ( t) P( f, f ) ε ε µ (, f ) µ t P f + ε µ f (6) he probablstc terpretato of Eq (6) s the followg I order a partcle s sampled from f a partcle s sampled from f wth probablty of ( µ t ε + µ ( ) ) t ε, a partcle s sampled from ( P f, f ) µ wth probablty of ( µ t ε µ ( ) ) t ε ad a partcle s sampled from (, P f f ) µ wth probablty of µ t ε Comparg equato 6, the ew algorthm +, (MDSMC), wth equato 8 (the DSMC algorthm) reveals two facts as follows: ) Eq (7), the ew algorthm (MDSMC), cossts of three terms that are sampled probablstcally, however Eq (8) (the DSMC algorthm) cossts of two terms that are sampled probablstcally, the thrd extra term Eq (6), (, P f f ) µ, s terpreted as the collso betwee the partcles sampled from f ad the partcles sampled from ( P f, f ) µ ) the probablstc coeffcets Eq (7), the ew algorthm (MDSMC), cosst of the secod order terms tme step however the probablstc coeffcets Eq (8) (the DSMC algorthm) cosst of the frst order terms tme step 3 Aalytcal solutos he eergy E of a partcle a axally symmetrc gas flow sde a rotatg cylder s gve as, [8], E( r) = Iω = mr ω (7) he rotatoal effect s the same of addtoal exteral feld actg o the system ad may be wrtte as: U( r) = Iω = mr ω (8) Usg the Boltzma dstrbuto for the partcle umber desty ad substtutg for U( r ) from Eq (8): ( ) mr ω U r ( r) = Aexp = Aexp, k k (9) the ormalzato factor A Eq 9, s determed N = r dv : by ( ) R π mr ω N = Aexp 0 rdrdϕ 0 k π Ak mrω = exp mω k he the ormalzato factor mr ω exp k A s (0) Nmω π k

5 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~ Substtutg for A to Eq (0): mω r exp Nmω k ( r) = π kl mω R exp k () Where, N s the total umber of molecules, m s the mass of a molecule of gas, ω s the agular velocty, k s the Boltzma costat, s the absolute temperature, L s the legth of the cylder, R s the cylder radus ad r s the radal dstace 4 he umercal procedures 4 he DSMC Algorthm (the VHS collso model molecules): for all partcles - Compute a upper boud σ= max ( πd v v j ) 4 for the cross secto, σ s updated each collso - Set µ = 4πσ - Set Nc = Iroud ( µ N t /( ε )) - Select N c dummy collso pars (, j) uformly amog all possble pars, ad for those - Compute the relatve cross secto σ j = π d v v j 4 - Geerate uform radom umbers Rad - If Rad < σ j σ Perform the collso betwee ad j, ad compute the post-collso veloctes v ad v j Set v + = v, v + j = vj else Set v + = v, v + j = v j Set v + = v for the N Nc partcles that have ot bee selected Ed for Durg each step, all the other N Nc partcle veloctes rema uchaged 4 he MDSMC Algorthm (the VHS collso model molecules): for t = to tot - Compute a upper boud, σ - Set µ = 4πσ - Compute ( t) N µ t µ N µ t N = Iroud + + c ε ε 4 ε - Select N c dummy collso pars (, j ) uformly amog all possble pars - Compute the relatve cross secto σ j - Geerate uform radom umbers ( Rad ) - If Rad < σ j σ Perform the collso betwee ad j, ad compute the post-collso veloctes v ad v j N µ t - Set N = Iroud c ε - Select Nc partcles amog those that have ot collded ad select N c partcles amog those that have collded - Compute the relatve cross secto σ j - Geerate uform radom umbers ( Rad ) - If Rad < σ j σ Perform the collso betwee ad j, ad compute the post-collso veloctes v ad v j Set v + = v for all the N N c N c partcles that have ot bee selected Ed for 5 Dscussos of results 5 Descrpto of case study problems I order to valdate our ew algorthm we cosder three dfferet case study problems, the frst case study problem the umber desty of argo a rotatg cylder s smulated usg the MDSMC ad DSMC algorthms, wth real molecules, 0000 model molecules ad (000 50) total umber of cells, ad the results of the smulato are compared wth the aalytcal soluto he radus of the cylder s 00m ad ts legth s 0m ad rotates wth agular velocty of5900rev s he gas sde the cylder s Argo (Ar) wth the tal temperature of 300K ad the tal pressure of 30Pa absolute I the secod ad thrd case study problems the Couette-aylor flow s smulated usg the MDSMC ad DSMC algorthms, wth real molecules, 0000 model molecules ad 5000 (50 0) total umber of cells, ad the results of the smulato are compared wth the expermetal data of aylor, G I, [] ad Kuhlthau, A R, [] For the comparso purposes we choose the same case study problems depcted by aylor, G I, [] ad Kuhlthau, A, [] respectvely

6 866 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~870 5 Comparsos of the umber desty results wth the aalytcal soluto Fg shows the comparso of the aalytcal soluto of the umber desty of the argo gas sde a rotatg cylder wth the results of the MDSMC ad the DSMC smulatos he comparso of the results of the umber desty usg the MDSMC algorthm wth the aalytcal soluto shows closer agreemet tha the results of the umber desty usg the DSMC algorthm he agreemet betwee our results of the umber desty usg the MDSMC algorthm s more proouced tha the results of the umber desty usg the DSMC algorthm the rego closer to the ceter of the cylder Furthermore, the fluctuatos of the umber desty results usg the MDSMC algorthm are less tha the fluctuatos of the umber desty results usg the DSMC algorthm the rego closer to the ceter of the cylder Fgs (a), (b), (c) ad (d) show the results of the streamles ad the desty cotours usg the MDSMC ad the DSMC algorthms ad the results of the streamles ad the temperatures cotours usg the MDSMC ad the DSMC algorthms respectvely he results of the streamles, the desty cotours ad the temperatures usg the MDSMC have less fluctuato tha the results of the DSMC smulatos hs s agreemet wth the results of the smulato usg the MDSMC ad the DSMC algorthms for the umber desty Fg 53 Comparsos of the results of Couette-aylor flow smulatos wth expermets of aylor he frst aylor stabltes of vortces streamles Fg Comparso of the aalytcal soluto of the umber desty wth the results of the MDSMC ad the DSMC algorthms Fg the streamles, the costat desty cotours ad the costat temperature cotours: (a) the streamles ad the costat desty cotours usg the MDSMC algorthm, (b) the streamles ad the costat desty cotours usg the DSMC algorthm, (c) the streamles ad the costat temperature cotours usg the MDSMC algorthm, (d) the streamles ad the costat temperature cotours usg the DSMC algorthm

7 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~ Fg 3 Streamles: (a) usg the MDSMC algorthm at a= after 6000 teratos, (b) usg the DSMC algorthm at a= after 4000 teratos the aylor, [] expermet appear at the ratos of ων= 303rad m, ων= 707rad m ad ω ν = 89 rad m, where these values of the rato ω ν correspod to three geometres the aylor, [] expermet as follows; the frst geometry cossts of R = 30 cm, R = 4035cm ad L=03cm, the secod geometry cossts of R = 355 cm, R = 4035cm ad L=03cm ad the thrd geometry cossts of R = 38cm, R = 4035cm ad L=03cm, where R s the outer radus of the er cylder, R s the er radus of the outer cylder ad L s the heght of the cylder Fgs 3(a) ad 3(b) show the results of the aylor stabltes of vortces streamles at the rato of ων= 300rad m usg the MDSMC ad the DSMC algorthms respectvely he frst aylor stabltes of vortces streamles usg the MDSMC ad the DSMC algorthms appear at the rato of ων= 300rad m, whereas the expermet of aylor, [] the frst aylor stabltes of vortces streamles appear at the rato of ων= 303rad m he results of the aylor stabltes of vortces streamles usg the MDSMC algorthm appear after 6000 teratos, whereas the Fg 4 Streamles: (a) usg the MDSMC algorthm at a= after 0000 teratos, (b) usg the DSMC algorthm at a= after 4000 teratos Fg 5 Streamles: (a) usg the MDSMC algorthm at a= after 0000 teratos, (b) usg the DSMC algorthm at a= after 3000 teratos

8 868 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~870 results of aylor stabltes of vortces streamles usg the DSMC appear after 4000 teratos Fgs 4(a) ad 4(b) show the results of the aylor stabltes of vortces streamles at the rato of ων= 707 rad m usg the MDSMC ad the DSMC algorthms respectvely he frst aylor stabltes of vortces streamles usg the MDSMC ad the DSMC algorthms appear at the rato of ων= 700rad m, whereas the expermet of aylor, [] the frst aylor stabltes of vortces streamles appear at the rato of ων= 700rad m he frst aylor stabltes of vortces streamles usg the MDSMC algorthm appear after 0000 teratos, whereas the frst aylor stabltes of vortces streamles usg the DSMC appear after 4000 teratos Fgs 5(a) ad 5(b) show the results of the aylor stabltes of vortces streamles at the rato of ων= 890rad m usg the MDSMC ad the DSMC algorthms respectvely he frst aylor stabltes of vortces streamles usg the MDSMC ad the DSMC algorthms appear at the rato of ων= 890rad m, whereas the expermet of aylor the frst aylor stabltes of vortces streamles appear at the rato of ων= 89rad m he frst aylor stabltes of vortces streamles usg the MDSMC algorthm appear after 0000 teratos, whereas the frst aylor stabltes of vortces streamles usg the DSMC appear after 3000 teratos herefore, the effect of the ew algorthm (MDSMC) the smulato s to ehacg the appearace of the aylor stabltes of vortces streamles 54 Comparsos of the results of Couette-aylor flow smulatos wth expermets of Kuhlthau he torque developed o the outer cylder the Couette-aylor flow s measured by Kuhlthau, [] he geometry of Couette-aylor flow Kuhlthau, [] expermet cossts of R = 508cm, R = 635cm ad L=38cm where R s the outer radus of the er cylder, R s the er radus of the outer cylder ad L s the heght of the cylder he pressure of the gas the space betwee the two cylders s 00 µ m Hg ad the er cylder rotates at sx dfferet agular veloctes of 400, 800, 000, 00, 400 ad 600 Rev Sec he results of the torque developed o the outer cylder usg the MDSMC ad the DSMC algorthms are compared wth the expermetal data of Kuhlthau, [] Fg 6 shows the able shows the comparso of the results of the torque developed o the statoary cylder usg the MDSMC ad DSMC algorthms wth the expermetal data of Kuhlthau (960) Rotatoal Velocty (R/S) Calculated orque of the MDSMC (Nm) 0 4 Calculated orque of the DSMC (Nm) 0 4 Measured orque by Kuhlthau (960) (Nm) 0 4 Error Error Calculato Calculato DSMC MDSMC wth wth Measuremet Measuremet % 44309% % 58% % 78% % 99% % 7% % 86% Fg 6 Comparso of expermetal data of the developed torque o the statoary cylder usg the MDSMC ad DSMC algorthms comparso of the expermetal data of the developed torque o the statoary cylder wth the results of smulato usg the MDSMC ad DSMC algorthms at sx dfferet agular veloctes of 400, , 00, 400 ad 600 Rev Sec he comparso of the preset results of smulato wth the expermetal data of Kuhlthau, [] show that the results of smulato usg the MDSMC are closer agreemet tha the results of smulato usg the DSMC algorthm able shows the error assocated wth the torque developed o the statoary cylder usg the MDSMC ad DSMC algorthms at sx dfferet agular veloctes of 400, 800, 000, 00, 400 ad 600 Rev Sec whe compared wth the expermetal data he errors assocated wth the results of the torque usg the MDSMC are less tha the errors of the results of the torque usg the DSMC whe com-

9 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~ pared wth the expermetal data 6 Coclusos Comparg the results of the smulato wth the expermetal data shows that the ew algorthm developed the preset work (the MDSMC algorthm) has the capablty of smulatg the Couette-aylor gas flow problem wth hgher accuracy tha the prevous DSMC algorthm he appearace of the frst aylor stabltes of the vortces streamles usg the MDSMC algorthm s ehaced whe compared wth the appearace of the frst aylor stabltes of the vortces streamles usg the DSMC algorthm hese are due to the facts that, the ew algorthm (MDSMC) there exsts a ew extra term (, P f f ) µ whch takes to accout the effect of the collso betwee the partcles sampled from f ad the partcles sampled from ( P f, f ) µ whch we call that the secod order collso term hs ew extra term has the effect of ehacg the appearace of the frst aylor stabltes of vortces streamles Fally the ew algorthm (MDSMC) there exsts a secod order term tme step the probablstc coeffcets whch has the effect of smulato wth hgher accuracy tha the prevous DSMC algorthm Refereces [] G I aylor, Stablty of a Vscous Lqud Cotaed betwee wo Rotatg Cylders, Phl ras, Ser A, 3 (93) [] A R Kuhlthau, Recet Low Desty Expermets usg Rotatg Cylder echques, I Proceedg of the frst teratoal symposum o rarefed gas dyamcs, (ed FM Deuee), (960), 9-00, Pergamo, Lodo [3] G A Brd, Molecular Gas Dyamcs ad the Drect Smulato of Gas Flows, Oxford Uv Press, Lodo (994) [4] F W Vogetz, G A Brd, J E Broadwel ad H Rugalder, heoretcal ad Expermetal Study of Rarefed Supersoc Flows about Several Smple Shapes, AIAA Joural, 6 () (968) [5] S Stefaov ad C Cercga, Mote Carlo Smu- lato of the aylor-couette Flow of a Rarefed Gas, Joural of Flud Mechacs, 56 (993) 99-3 [6] G A Brd, Recet Advaces ad Curret Challeges for DSMC, Computers ad Mathematcs wth Applcatos, 35 (-) (998) -4 [7] H Shagawa, H Setyawa, Asa, Y Sugyama ad K Okuyama, A expermetal ad theoretcal vestgato of rarfed gas flow through crcular tube of fte legth, Pergamo, Chemcal Egeerg Scece, 56 (00) [8] R S Myog, A geeralzed hydrodyamc computatoal model for rarfed ad mcro scale gas flows, Joural of Computatoal Physcs, 95, (004), [9] K Nabu, Drect Smulato Scheme Derved From the Boltzma Equato, Joural of the Physcal Socety of Japa, 49 (980) [0] H Babovsky, O a Smulato Scheme for the Boltzma Equato, Mathematcal Method the Appled Sceces, 8 (986) 3-33 [] L Paresch ad R E Calfsch, A Implct Mote- Carlo Method for Rarefed Gas Dyamcs, J Comput Phys, (999), 54, 90 [] L Paresch ad G Russo, me Relaxed Mote- Carlo Methods for the Boltzma Equato, SIAM J Sc Comput, 3, (00), [3] L Paresch ad S razz, Asymptotc Preservg Mote Carlo Methods for the Boltzma Equato, rasport heory Statst Phys, 9, (005), pp [4] L Paresch ad S razz, Numercal Soluto of the Boltzma Equato by me Relaxed Mote- Carlo (RMC) Method, Iteratoal Joural of Numercal Method Fluds, 48 (005) [5] C Cercga, he Boltzma Equato ad Its Applcatos, Lectures Seres Mathematcs, 68, Sprger-Verlag, Berl, New York, (988) [6] E Gabetta, L Paresch ad G osca, Relaxato Schemes for Nolear Ketc Equatos, SIAM J, Number Aal, 34 (997) [7] E Wld, O Boltzma s Equato the Ketc heory of Gases, Proc Camb Phl Soc, 47 (95) [8] L Y Kuo, Problems ad solutos o hermodyamcs ad Statstcal Mechacs, World Scetfc Publcato, (990)

10 870 A A Gajae ad S S Nourazar / Joural of Mechacal Scece ad echology 3 (009) 86~870 Dr S S Nourazar receved hs BSc degree from Amrkabr Uversty of echology ehra, Ira he he proceeded hs graduate studes Caada ad receved the M Sc ad PhD degrees Mechacal egeerg from Ottawa Uversty Caada Dr Nourazar s actg ow as assstat professor Mechacal Egeerg Departmet of Amrkabr Uversty of echology he research terests of Dr Nourazar are the CFD compressble ad compressble turbulet oreactve flow as well as rarefed gas dyamcs A A Gajae receved hs BSc degree from Scece & echology of Ira Uversty ehra, Ira he he proceeded hs graduate studes Amrkabr Uversty of echology ad receved the M Sc degrees Mechacal egeerg A A Gajae s studyg ow as PhD studet Mechacal Egeerg Departmet of Amrkabr Uversty of echology he research terests of Gajae are the CFD compressble ad compressble turbulet oreactve flow as well as rarefed gas dyamcs