A micro-macro decomposition based asymptotic-preserving scheme for the multispecies Boltzmann equation

Transcription

1 A mcro-macro decomposton based asymptotc-preservng scheme for the multspeces Boltzmann equaton Sh Jn Yngzhe Sh Abstract In ths paper we extend the mcro-macro decomposton based asymptotc-preservng scheme developed n [3] for the sngle speces Boltzmann equaton to the multspeces problems. An asymptotcpreservng scheme for netc equaton s very effcent n the flud regme where the Knudsen number s small and the collson term becomes stff. It allows coarse ndependent of Knudsen number mesh sze and large tme step n the flud regme. The dffculty assocated wth multspeces problems s that there are no local conservaton laws for each speces, resultng n extra stff nonlnear source terms that need to be dscretzed properly n order to avod Newton type solvers for nonlnear algebrac systems and to be asymptotc-preservng. We show that these extra nonlnear source terms can be solved usng only lnear system solvers, and the scheme preserves the correct Euler and Naver-Stoes lmts. Numercal examples are used to demonstrate the effcency and applcablty of the schemes for both Euler and Naver-Stoes regmes. Keywords: multspeces Boltzmann equaton, BGK model, mcro-macro decomposton, asymptotcpreservng scheme, flud dynamc lmt AMS subject classfcatons.: 65M6, 8C4, 8C8, 4A6, 76P5 Introducton In netc theory, the Boltzmann equaton s a fundamental equaton to descrbe the evoluton of rarefed gases. In ths paper, we are nterested n the multspeces Boltzmann equaton for the gas mxture. Such equatons arse n many applcatons. One example s the atmosphere whch must be consdered at least as the mxture of Oxygen and Ntrogen. Other applcatons of the gas mxture nclude the problems n evaporaton-condensaton or n the nuclear engneerng. Typcal computatonal challenges for the Boltzmann equaton nclude ts hgh dmensonalty, and the exstence of multscale where the Knudsen number the rato of the mean free path over a typcal length scale such as the doman sze can have dfferent order of magntude n dfferent part of the doman. When the Knudsen number s small, the soluton to the Boltzmann equaton can be approxmated by the compressble Euler or Naver-Stoes equatons va the Chapman-Ensog expanson [9]. Ths s the so-called flud dynamc regme, whch s nown to be numercal stff due to the stff collson term. Our am s to develop numercal schemes for the multspeces Boltzmann equaton that are effcent n the flud dynamc regme, namely we are seeng numercal schemes that can allow macroscopc or flud dynamc mesh sze and tme step. The asymptotc-preservng AP methods are a general framewor for netc equatons wth dfferent scales of the Knudsen number. Accordng to Jn [3], a scheme for netc equaton s AP f t preserves the dscrete analogy of the Chapman-Ensog expanson, namely, t s a sutable scheme for the netc equaton, yet, when holdng the mesh sze and tme step fxed and lettng the Knudsen number go to zero, the scheme becomes a sutable scheme for the lmtng Euler equatons Ths wor was partally supported by NSF grant No. DMS-687 and NSF FRG grant DMS SJ was also supported by a Van Vlec Dstngushed Research Prze from Unversty of Wsconsn-Madson. Department of Mathematcs, Unversty of Wsconsn, Madson, WI 5376, USA jn@math.wsc.edu Department of Mathematcs, Unversty of Wsconsn, Madson, WI 5376, USA sh@math.wsc.edu

2 mplct collson terms can be mplemented explctly, or at least more effcently than usng the Newton type solvers for nonlnear algebrac systems. Comparng wth a multphyscs doman decomposton type method [4, 6, 7,, 8, 4], the AP schemes avod the couplng of physcal equatons of dfferent scales where the couplng condtons are dffcult to obtan, and nterface locatons hard to determne. The AP schemes are based on solvng one equaton the netc equaton, and they become a robust macroscopc flud solver automatcally when the Knudsen number goes to zero. An AP scheme mplyng a numercal convergence unformly n the Knudsen number was proved by Golse-Jn-Levermore for lnear transport equaton n the dffuson regme []. Ths result can be extended to essentally all AP schemes, although the specfc proof s problem dependent. For examples of AP schemes for netc equatons n the flud dynamc or dffusve regmes see for examples [5, 5, 6, 7, ]. An AP scheme for sngle speces Boltzmann equaton was ntroduced by Benoune, Lemou and Meussens [3] usng the mcro-macro decomposton of the Boltzmann equaton. The mcro-macro decomposton wrtes the densty dstrbuton functon as the sum of the local Maxwellan and the non-thermoequalbrum devator. A coupled system for the hydrodynamc moments densty, momentum and total energy and the devator can be formed whch recovers both Euler and Naver-Stoes equatons va the Chapman-Ensog expanson. It has found theoretcal success [], and has also been used for numercal purposes, see [8, 9]. It was shown n [8] that a AP scheme can be constructed by usng the mcro-macro decomposton of the Boltzmann equaton. The scheme s AP n the Euler lmt. By sutably resolvng the vscous term t s also consstent to the Naver-Stoes approxmaton. It s our goal to extend ths scheme to multspeces Boltzmann equaton. For multspeces Boltzmann equaton, each speces does not conserve the momentum and energy, although these quanttes for the entre systems are conserved. Ths feature brngs new dffculty for numercal approxmatons not encountered n the sngle speces case. The non-conservaton of each speces ntroduces stff nonlnear source terms that must be dscretzed wth care n order to be AP and to be solved effcently by avodng the teratve Newton solvers for nonlnear algebrac systems. Our dscretzatons are desgned to satsfy these two propertes, as wll be shown asymptotcally and demonstrated numercally. For convenence and clarty we wll mostly use the smpler multspeces BGK model ntroduced n [], yet for completeness and generalty we wll also present the framewor for the general Boltzmann equaton. The paper s organzed as follows. In Secton, we brefly revew the multspeces Boltzmann equaton. We then ntroduce the consstent BGK model for gas mxtures and ts hydrodynamc lmts of Euler and Naver-Stoes equatons. In Secton 3, we present the mcro-macro decomposton method, and use t to construct the equvalent netc/flud system to the Boltzmann equaton. Ths system s proved to have the Euler and Naver-Stoes lmts va the Chapman-Ensog expanson. Secton 4 gves the detaled numercal approxmatons based on the netc/flud couplng system. We show that t s AP to the Euler lmt, and s also consstent to the Naver-Stoes approxmaton for sutably small mesh sze and tme step. We also show how the mplct nonlnear source term can be solved va only lnear system solvers. In Secton 5, some numercal tests are conducted to valdate our model and the schemes. We mae some concludng remars n secton 6. Multspeces models. The multspeces Boltzmann equaton The Boltzmann equaton descrbes the densty dstrbuton of evoluton of rarefed gases. For the mxtures, the Boltzmann equaton refer to [9] s wrtten as t f + ξ x f = ǫ Q f, f, t, x, ξ R d R d. f = f t, x, ξ represents the densty dstrbuton functon of speces- partcles that have poston x and velocty ξ at tme t; ǫ s the dmensonless Knudsen number, the mean free path over a typcal length scale;

3 d s the space dmenson. The collson term Q s defned by Q f, f = = Q f, f, Q f, f = R d B + f f f f B Ω V, V dξ dω where B Ω V, V s the collson ernel; ξ and ξ are the molecular pre-collsonal veloctes; ξ and ξ are the post-collsonal veloctes; f = f t, x, ξ, f = f t, x, ξ, f = f t, x, ξ ; Ω s an unt vector, B + s the sem-sphere defned by Ω V =, V s the relatve velocty V = ξ ξ. We consder the elastc collsons of two partcles: one from speces and the other from speces. Thus, the post-collsonal veloctes are { ξ = ξ µ m Ω[ξ ξ Ω], ξ = ξ + µ m Ω[ξ ξ Ω]. the mass of speces s m and the reduced mass s µ = m m /m + m. These velocty relatons arse from the conservaton laws for the momentum and energy n the molecules collson If we defne the mcroscopc collson operator the collson s reversble and satsfes Ths property gves the followng denttes:. Macroscopc quanttes m ξ + m ξ = m ξ + m ξ m ξ + m ξ = m ξ + m ξ Υ : ξ, ξ ξ, ξ, Υ Υ = I dξdξ = dξ dξ ξ ξ Ω = ξ ξ Ω ξ ξ = ξ ξ. We ntroduce the notatons for macroscopc quanttes of each speces : n s the number densty, ρ the mass densty, u the average velocty, E the total energy, e the nternal energy per partcle, T the temperature, gven by n = f dξ, ρ = m f dξ, ρ u = m f ξdξ, E = ρ u + n e = m e = d T = m n f ξ dξ, f ξ u dξ. 3

4 We also defne global quanttes for the mxture: the total mass densty ρ, the number densty n, the mean velocty u, the total energy E, the nternal energy ne, and the mean temperature T = e d : ρ = ρ, n = n, ρu = ρ u, E = d nt + ρ u = E..3 Propertes of the Boltzmann equaton We use the notaton n the followng: ϕ = ϕdξ and H =, ξ, ξ. The macroscopc quanttes come from the moments of densty dstrbuton functon U = m Hf = ρ, ρ u, ρ u + dn T. The collson term Q satsfes the conservaton laws of mass, the total momentum and the total energy m Q =, m ξq =, m ξ Q =. Moreover, the H-theorem holds: m Q log f for any f >, the = s satsfed at the equlbrum, whch mples Q f, f = and f s the local Maxwellan where u = u = u, T = T = T for any and see [9]..4 The multspeces BGK model f = M U = n m πt d/ exp m ξ u 3 T Consder the consstent BGK model ntroduced n [] for the Maxwell molecules df dt = tf + ξ x f = ǫ Q := ν ǫ M f 4 ν s the collson frequency defned as ν = n χ ; χ s the nteracton coeffcent χ = cosω B ωdω B + wth the collson angle ω = Ω V/ V ; and M = n m π T d/ exp m ξ ũ T 4

5 s a Maxwellan dstrbuton wth ũ and T defned below. For the Maxwell molecule, the collson ernel B Ω V, V s only related wth the collson angle ω between the relatve velocty V and unt vector Ω B Ω V, V = B ω. Thus, the moments of the collson operator can be obtaned as m Q =, m ξq = ν ρ ũ u = µ χ n n [u u ], 6 m ξ Q = ν Ẽ E = u u µ χ n n [u u u + e ] e + m. 7 m + m Note that the momentum and energy are not conserved for each speces. From 5-7, the ũ and T n M have the expresson ẽ = d T : m ν ũ = m ν u + ν ẽ = ν e m ν ũ u + µ χ n u u, 8 µ χ n m + m e e + m u u Tang moments on the BGK model 4, 5-7 gves the macroscopc equatons: t ρ + x ρ u =, t ρ u + x ρ u u + P = ǫ µ χ n n [u u ], t E + x E u + P u + q = ǫ µ χ n n [u u u + m +m where P s a d d stress tensor; q s the heat flux of speces. It was shown n [] that the BGK model possesses the followng propertes: the non-negatvty of denstes e e + m u u the exchange relatons of momentum and energy comples wth those of the Maxwell partcles 5. 9 the ndfferentablty prncple holds: when all speces are the same the same mass and all χ χ,,, the model wll degenerate nto the sngle speces BGK model the equlbrum dstrbutons are local Maxwellans as 3 wth mean velocty u and mean temperature T the H theorem holds true..5 The flud dynamcs approxmatons of the multspeces BGK model At the zero Knudsen number lmt ǫ, all Q =. From 8-9, we have ũ = u = u and T = T = T see []. Thus, f = M = n m πt d/ exp m ξ u, T and the macroscopc equatons become the Euler system t ρ + x ρ u =, t ρu + x ρu u + ntī =, t E + x E + ntu =, ], 5

6 wth the pressure P = ntī, where Ī s the unt matrx, and the heat flux q =. When ǫ, the Chapman-Ensog expanson was used n [] to get the Naver-Stoes equaton as wth t ρ + x ρ u = x J, t ρu + x ρu u + P =, t E + x [E + Pu + q] =, J = ρ u u, P = ntī η x u + x u T d x uī, q = d + T n u u κ x T, where the vscosty coeffcent η = ǫt n ν, the thermal conductvty coeffcents κ = ǫ d+ T n m ν, and J s the dffuson velocty J = ǫ x n T L + Oǫ, ρ n whch L s a symmetrc matrx dependng only on the denstes. 3 A netc/flud formulaton 3. Mcro/Macro decomposton For each speces, as was done s [, 3], we decompose the dstrbuton functon f = f t, x, ξ nto the sum of ts Maxwellan M U and g = f M ǫ f = M U + ǫg, 3 where the Maxwellan s M U = n m πt d/ exp m ξ u T. 4 The Maxwellan for speces has the same moments as the densty dstrbuton functon f thus, Hf = HM 5 Hg =. One can use a projecton method { to separate the macroscopc and mcroscopc quanttes M and g. Consder the Hlbert space L } M = ϕ ϕ M n L R d, the scalar product ϕ, ψ M = ϕψ M n. Defne the space D M =Span { } M, ξm, ξ M, then the orthogonal bass of DM s { } M ξ u M ξ u B =,, d M, n T /m n T /m n and the orthogonal projecton n L M onto D M s Π M ϕ: Π M ϕ = [ ϕ + ξ u ξ u ϕ ξ u + d ξ u d ] ϕ M. n T /m T /m d T /m One can establsh the followng propertes of Π M ϕ as n [3]: 6

7 Lemma. As the defnton of M and g n 34, we have Proof. We now that t M = I Π M t M = Π M g = Π M t g =. [ t ρ + m ξ u m ξ u t u + d ρ T T clearly belongs to D M. Thus, Π M t M = t M. Snce Hg =, H t g = t Hg =. So t mples Π M g = Π M t g =. Now apply the operator I Π M to, t T T I Π M t M + ξ x M + ǫi Π M t g + ξ x g = ǫ I Π M Q. Usng Lemma one obtans: t g + I Π M ξ x g = ǫ [ ] ǫ I Π M Q I Π M ξ x M. 6 ] M Tang the moments of, t gves: t m HM + x m ξhm + ǫ x m ξhg = ǫ m HQ. The macroscopc quanttes U are defned as m HM. Let FU = m ξhm be the flux vector of U. Then, The equatons for macroscopc quanttes are t U + x FU + ǫ x m ξhg = ǫ m HQ. 7 The coupled system 6-7 gves a netc/flud formulaton of the multspeces Boltzmann equaton. Next we wll show that the system s equvalent to the Boltzmann equaton, whch essentally follows the proof n [3]. Proposton. Let f be a classcal soluton of the Boltzmann equaton wth ntal data f t =, x, ξ = f x, ξ, and M = M U ts assocated Maxwellan gven n 4. Then the par U, g, where U = m HM and g = f M ǫ, s a soluton to the coupled system 6-7 wth the correspondng ntal data U t = = U = m Hf and g t = = g = f M. 8 ǫ Conversely, f U, g satsfes system 6-7 wth the ntal data 8 such that m Hg =, then f = M +ǫg s a soluton to the Boltzmann equaton wth ntal data f = M U +ǫg and we have U = m Hf and m Hg =. Proof. s straghtforward due to the constructon of the coupled system 6-7. Consder, we have from 6 ǫ t g + ξ x M + ǫξ x g = ǫ I Π M Q + Π M ξ x M + ǫπ M ξ x g. Addng t M to the above equatons gves, t f + ξ x f = ǫ Q + [ t M + Π M ξ x f ǫ Π M Q ]. t M + Π M ξ x f ǫ Π M Q belongs to the space D M. On the other hand, 7 s equvalent to H t M + ξ x f ǫ Q =, whch mples t M + Π M ξ x f ǫ Π M Q s orthogonal to the space D M. Consequently, t M + Π M ξ x f ǫ Π M Q = and f satsfes the Boltzmann equaton. 7

8 Correspondng to 6-7, the netc/flud system from the multspeces BGK model 4 s: t g + I Π M ξ x g = ǫ [ ν ] ǫ I Π M M ν g I Π M ξ x M, 9 t U + x FU + ǫ x m ξhg = ǫ m HQ. The calculaton on the frst term of the rght sde 9 gves I Π M M = M M [ + ξ u ξ u ũ u + T /m T T + ũ u T T /m d T /m = M M + u M ũ u + T M T T + m ũ u. ] In the next two sectons we wll show that the netc/flud system 9 - recovers the flud approxmaton of the Euler and CNS equatons as the standard Chapman-Ensog expanson on the BGK model. 3. The Euler system As ǫ, Q = HQ =. The equatons n 5-7 mply that u = u = u, T = T = T. Under the common velocty and temperature, m d/ f = M U = n exp m ξ u, πt T and g =. Hence, the equaton automatcally turns nto the Euler equatons: t ρ + x ρ u =, t ρu + x ρu u + ntī =, 3 t E + x [E + ntu] =. 3.3 Chapman-Ensog expanson and the Naver-Stoes system When ǫ, from, the states of speces are close to the equlbrum after the collson process see []: u u Oǫ and T T Oǫ. 4 Consequently, I Π M M n s the dfference of M and ts lnear approxmaton at M wth a second order resdual, whch means I Π M M Oǫ. 5 Thus, 9 gves Through the calculatons as n [], we have [ I Π M ξ x M = B : g = ν I Π M ξ x M + Oǫ. 6 x u + x u T d x u Ī + A xt T ]M + Oǫ, where Ī s the dentty matrx, and ξ u A = d + ξ u, B = [ ξ u ξ u ξ u ] Ī. T /m T T /m dt /m 8

9 Pluggng the leadng term of 6 nto, Note t U + x FU = ǫ m HQ + ǫ x ǫ m ξh I Π M ξ x M ν m ξh I Π M ξ x M + Oǫ. 7 ν = σ σ u + q whch s nothng but the CNS system s vscosty and heat conducton parts for the speces. The rescaled vscosty tensor s σ = η x u + x u T d x u Ī wth the vscosty coeffcent η = ǫt n ν. The heat flux s wth the thermal conductvty coeffcent κ = ǫ d+ q = κ x T T n m ν. From 7, the CNS equatons of speces can be wrtten as, t ρ + x ρ u = x ρ u ρ u, 8a t ρ u + x ρ u u + n T Ī + σ = ǫ m ξq + Oǫ, 8b t E + x [E + n T u + σ u + q ] = m ǫ ξ Q + Oǫ. 8c 8a s exactly the same wth the frst equaton of CNS for the mxture -. To prove that the system 8 s the approxmaton to - n Oǫ, we need the followng lemma. Lemma 3. If u u Oǫ and T T Oǫ ǫ, then: ρ u u = ρu u + Oǫ, 9 Proof. We prove 9, whch s equvalent to n T = nt + Oǫ, 3 ρ u u = ρ u u + Oǫ. 3 ρ ρ u u = ρu ρu + Oǫ. 3 Recall the defnton of ρ, u and T: ρ = ρ, ρu = ρ u, and wrte the vector u = u j n d T + ρ u = j=,,...,d LHS j,s = E = n d T + ρ u, 33. When j s, the left hand sde n 3 s ρ ρ u j u s = 9 ρ ρ u j u s,

10 whle the rght hand sde n 3 s RHS j,s = ρ u s ρ u j = ρ ρ u j u s. Thus, LHS RHS j,s = The subscrpt and are exchangeable; then LHS RHS j,s = ρ ρ u j u s ρ ρ u j u s u s. us. Add up these two results, we can get LHS RHS j,s = ρ ρ u j u j u s u s Oǫ as n 9. The same approach wth the energy conservaton n 33 ndcates that n d T n d T = ρ u ρ u Oǫ as n 3, whch s only the case j = s n 3. Multplyng the equaton 3 by the total mass densty ρ, the left hand sde s LHS = ρ ρ u u = ρ j ρ u u, whle the rght hand sde We have RHS = ρ ρ u u = ρ j ρ u u + Oǫ = ρ j ρ u u j + Oǫ. LHS RHS = ρ j ρ u u u j. Exchange the subscrpt and j and we fnally prove the result as n 3. j LHS RHS = ρ j ρ u u u j Oǫ j Recallng the conservaton laws and summng up the second moments 8b for all speces, we obtan the momentum equatons of the compressble Naver-Stoes system for the mxture: The vscosty coeffcent: η = ǫt j t ρu + x ρu u + P =, 34 P = ntī η x u + x u T d x uī + Oǫ. n ν.

11 Summng up the energy equaton 8c for all speces and usng 3 lead to E + n T u = ρ u u + d + n T u = ρ u u + d + ntu + d + n T u u + Oǫ = ρ u u + d + ntu + d + T n u u + Oǫ note that the thrd term s Oǫ. Therefore, we obtan the thrd equaton of CNS : t E + x [Eu + Pu + q] =, 35 q = d + T n u u κ x T + Oǫ, n κ = ǫ d+ T m ν s the thermal conductvty coeffcent. Thus, the system 8a-8c s a second order n ǫ approxmaton of the CNS system arsng from the Chapman-Ensog expanson of the BGK system - derved n []. 4 Asymptotc-preservng numercal approxmatons In ths secton, we wll present an asymptotc-preservng numercal approxmaton of the couplng system 9-. Gven the fxed tme step and space mesh sze, we wll show that our numercal scheme approxmates the dscretzaton of the Euler system wth error Oǫ, thus asymptotcally preserves the Euler lmt. It s also consstent wth the CNS system f x, ǫ. Ths scheme s an extenson of the scheme [3] for sngle speces, and t recovers the method of [3] f there s only one speces. 4. Tme dscretzaton In the frst step, we gve the AP tme dscretzaton. Space and velocty dscretzatons wll be studed n the next subsecton. We denote a fxed tme step and a sequence of dscrete tme t l = l, l N. Thus, g lx, ξ = g t l, x, ξ, U lx = U t l, x. In the netc equaton 9, the term ν ǫ I Π M M s O from 5, thus s not a stff term; so t wll be treated explctly. For the stablty ndependent of ǫ, we treat g term n the collson part mplctly. Ths sem-mplct scheme on the netc equaton has the expresson g l+ g l + I Π M l ξ x g l = ǫ [ ] ν l ǫ I Π M l M l ν l g l+ I Π M l ξ x M l. 36 Now, we dscretze the second equaton. In, there are three scales: O ǫ collson term; O convecton term and Oǫ dffuson term. The latest two terms essentally come from the molecule convecton. We wll use splttng method to separately calculate the collson and convecton terms n U l+ U l + x FU l + ǫ x m ξhg l =, 37 U l+ U l+ = ǫ m HQ l+. 38

12 We order these steps n the followng way: U l+ g l+ U l = ǫ g l + [ ν l+ ǫ = ǫ U l+ U l+ m HQ l+ I Π M l+ I Π M l+, 39 ξ x g l l+ M ν l+ g l+ I Π l+ M ] ξ x M l+, 4 + x FU l+ + ǫ x m ξhg l+ =. 4 Although 39 s fully mplct, note the results n 5-7, one can wrte t as ρ l+ ρ l =, ρ l+ u l+ ρ l ul = ǫ E l+ E l = ǫ + m + m [ ] µ χ n l+ n l+ u l+ u l+, 39 [ µ χ n l+ n l+ u l+ u l+ u l+ e l+ e l+ + m u l+ ] u l+ /. Ths mplct scheme satsfes the conservaton laws as well as the lmt propertes see Remar 5. Smplfy the above and we wll obtan a lnear system u l+ E l+ n l+ = n l, u l = ǫ E l = ǫ 4 m n l+ [ ] χ u l+ u l+, 43 m + m 4m m m + m χ n l+ n l+ [ m m u l+ u l+ + El+ n l+ ] El+. 44 n l+ Suppose that there are K speces n the mxture, let u = u u... u K. Due to 4, 43 s a lnear system for u. Once u l+ s obtaned, 44 s a lnear system for E = E E... E K. Thus, although the rght hand sde of 39 s nonlnear, we only need to solve lnear systems. 4 s clearly lnear for g l+ whch can be obtaned easly. Proposton 4. The tme dscretzatons as 39-4 gve a scheme consstent wth the Euler system 3 when ǫ ; for small ǫ, the scheme 39-4 s asymptotcally the Oǫ approxmaton to an explct tme dscretzaton of CNS system -. Proof. When ǫ, the mplct collson scheme 39 leads to u l+ = u l+ and T l+ and 4 becomes the followng forward Euler dscretzaton n tme of the Euler system: Euler: U l+ U l+ = T l+ for, + x FU l+ =. 45 From, the frst term n the rght hand sde of 4 s O. The sem-mplct scheme n equaton 4 yelds g l+ = I Π ν l+/ l+ ξ M x M l+ + Oǫ.

13 Applyng t nto 4, we get U l+ U l+ + x FU l+ = ǫ x m ξh ν l+/ I Π M l+ ξ x M l+ + Oǫ, 46 whch s an Oǫ approxmaton to the forward Euler dscretzaton n tme of CNS for speces 7. The summaton over all speces s also consstent wth multspeces CNS as u u Oǫ and T T Oǫ, whch has been proved n Secton 3.3. Remar 5. In ths paper, we derve the mplct scheme 4-44 for the collson process of the consstent BGK model 4. Although the collson term s a nonlnear functon of the macroscopc quanttes, the Maxwell partcle relatons 5-7 provde a method to solve a lnear system. It s easy to chec that the scheme has the conservaton property durng the collson 39, namely ρ l+ = ρ l, ρl+ u l+ = ρl ul and El+ = El. It also satsfes that at the lmt ǫ = ul+ = u l+ and T l+ = T l+ for,. 4. Space dscretzaton The terms that need the spatal dscretzatons nclude: the fluxes on the left-hand sde of both 4 and 4; and the dffuson term n 4. The convecton terms wll be dscretzed by upwnd scheme. For the dffuson term, the central dfferences defned on two staggered grds s used. Snce n 4 g s nfluence on U s Oǫ as dffuson, the explct central dscretzaton for the leadng term of I Π M ξ x M n 4 wll be taen. The crtcal steps are shown n the remander of ths secton. [ ] x m, x m+. The mesh sze x = of speces are dscretzed at the grd center ponts We use a unform grd x m+ and defne x m the center of the cell x m+ x m = x m+ x m, m N. The macroscopc U l as U,m l = U t l, x m whle the mcroscopc g l are at the grd end ponts as gl,m+ = g t l, x m+, ξ. The velocty s dscretzed evenly n a bounded doman. The rectangular quadratures are appled to approxmate the ntegrals wth respect to the velocty. For a smplfed expresson, we wll not explctly express the dscrete velocty for the mcroscopc functons. The numercal scheme for the system 39-4 s MM: U l+,m Ul,m g l+,m+ = ǫ g l,m+ νl+,m+ ǫ U l+,m Ul+,m = ǫ m HQ l+,m, 47 + I Π l+/ M ξ x g l,m+,m+/ I Π l+/ M M l+ ν l+,m+ g l+ I Π,m+/,m+,m+ l+/ M,m+/ + F m+ Ul+ F m Ul+ g l+ g l+,m+ + ǫ m ξh,m x x ξ Ml+,m+ Ml+,m x 48, =. 49 where The flux term n 48 s approxmated by a frst order upwnd scheme ξ x g l,m+ Φ m+ gl Φ m gl, x Φ m+ gl = ξ+ g l,m+ + ξ g l,m+ 3 we defne ξ ± = ξ± ξ. The projecton operator Π M,m+/ s chosen by Π M,m+/ ΠM,m +ΠM,m+. The flud flux F m+ n 49 s dscretzed by the netc flux vector splttng scheme n []: F m+ Ul+ = m H ξ + M l,m + ξ M l,m+. 5 3

14 The asymptotc behavor of scheme when ǫ can be understood as followng. For ǫ, the leadng order n 48 gves g l+ I Π,m+ l+/ M ξ Ml+,m+ Ml+,m.,m+/ x Pluggng t nto 49 gves NS/MM: = ǫ m ξh x U,m l+ Ul+,m ν l+/,m+ ν l+/,m+/ + F m+ Ul+ F m Ul+ x I Π M l+/,m+/ ξ Ml+,m+ Ml+,m x ν l+/,m I Π M l+/,m / ξ Ml+,m Ml+ x whch s a second order approxmaton of the CNS equaton 7. Thus the scheme s asymptotc preservng AP for the Euler lmt x, = O, ǫ. 5 s also consstent to the CNS f x, Oǫ. To reduce numercal vscosty, we wll use the second order upwnd scheme n the sense of Van Leer n [5] to reduce the numercal vscosty. Such a scheme needs to use the slope lmter. In our numercal tests, the classcal mnmod slope lmter see [] s appled. For numercal comparsons, we also present the schemes for the BGK model and the CNS equatons n the followng two subsectons. 4.3 A numercal approxmaton of the BGK model To dscretze the -D BGK model: t f + ξ x f = ν M f, ǫ we use the splttng method: solve the collson process by mplct scheme and the convecton process by frst order explct upwnd scheme:,m 5, S : f l+,m fl,m f l+,m fl+,m = νl+,m ǫ Ml+,m fl+,m ǫ Ql+,m, 5 + Φ m+ fl+ Φ m fl+ =. 53 x Here f,m l = f t l, x m and Φ m+ fl+ = ξ + f l+,m +ξ f l+,m+. The conservaton of mass leads to νl+,m = νl,m ; M l+,m s a nonlnear functon of the moments of fl+,m, whch s Ul+,m. We tae the moments of 5: U l+,m Ul,m = ǫ m HQ l+,m The same process 4-44 wll solve U l+,m by nvertng lnear systems. Consequently, we get M l+,m, and solve 5 for f l+,m. 4.4 A numercal approxmaton of the CNS equatons In the -D CNS, the pressure P = nt and the equatons are: t ρ + x ρ u = x J,. t ρu + x ρu + nt =, 54 t E + x [E + ntu] = x κ x T 3 T J. m 4

15 Here the thermal conductvty coeffcents κ = ǫ 3 T n m ν, and the dffuson velocty J = ǫ L xn T ρ, of whch L s defned n [] as a functon of denstes. The left hand sde are the same as the Euler equatons, and we wll replace them by t U + x FU and mplement the netc flux vector splttng scheme 5 on the flux terms. The rght hand sde s the dffuson terms whch are dscretzed by the central dscretzaton NS = x U l+ m Ul m + F l U l F l m+ m U l x J l J l,m+,m κ l T l m+ T l m m+ x 3 T l J l,m+ m+ m κ l T l m T l m m x 3 T l J l,m m m, 55 where n l,m+ T l m+ nl,m T l m J l,m+ = ǫ L l,m+ ρ l,m+ x. The quanttes at the half grd T m+, κ m+, ρ,m+ and L,m+ nearby two grds as T m+ Tm++Tm. 5 Numercal results wll be estmated by the average of the Wthout loss of generalty, all numercal examples studed n ths paper wll be conducted for a two-speces mxture. We frst show the numercal results for the one-dmensonal BGK model 4, to nvestgate the behavor of the mxture. We then present several numercal solutons of the one-dmensonal couplng system usng correspondng to the soluton of the BGK model To llustrate the asymptotc preservng propertes, we wll chec that our scheme of s AP to the Euler system 45 to Oǫ, as well as captures the CNS asymptotcs 55 wth sutable mesh sze and tme step. For convenence, we wll denote our numercal schemes as followng: mcro/macro decomposton scheme as MM; the mplct scheme 5-53 for BGK model as S ; the numercal scheme 55 for CNS as NS; the netc scheme for Euler equatons 45 as Euler; and the approxmaton of MM to CNS 5 as NS/MM. 5. A space homogeneous problem We frst use the mplct scheme S for the BGK model n the space homogeneous case, to chec our scheme and the behavor of the mxture n the collson process. The ntal condton s { ma =,, n a =, u a =.5, T a = m b =.5, n b =., u b =., T b =.. There are velocty grd ponts n the range [, ] and the tme step = 5 4. In Fg, the veloctes and temperatures of speces a and b converge along wth the dfferent Knudsen number ǫ =.5 and.. It shows that the smaller the Knudsen number ǫ s, the faster veloctes and temperatures converge nto the equlbrum. 5. A statonary shoc problem In ths example, we test the couplng system for a statonary shoc. We gve the ntal macroscopc data on the left-hand sde, whle the rght-hand sde data are gven by the Ranne-Hugonot relatons: { m a =, m b =.5, n a = n b =, u a = u b =.5, T a = T b =.4, for x ; m a =, m b =.5, n a = n b =.49, u a = u b =.7, T a = T b =.865, for x >. 5

16 The computatonal doman n space s [.5,.7] dscretzed by a unform grds x =.5. We use a velocty grd of ponts n doman [, ]. To satsfy the CFL condton, the tme step s taen as = x. Frstly, n Fg the results by our scheme MM show that the densty dstrbuton functon converges to the Maxwellan wth dmnshng Knudsen number ǫ. The results match well wth the BGK model S for the statonary shoc, shown n Fg 3. We llustrate the asymptotc behavor of scheme for dfferent values of ǫ ǫ = θ, θ N, and plot the densty, velocty and temperature of the mxture as the functons of x n Fg 4, comparng them wth the exact soluton of the statonary shoc. There exts the oscllaton n the statonary shoc when ǫ, whch results from the numercal vscostes of the fnte dfference method see [4]. Then we compare the relatve numercal dfference n l norm wth the Euler lmt and nvestgate the convergence speed. Both mcro/macro decomposton scheme MM and mplct BGK scheme S converge to the Euler lmt n Oǫ. We observe that the slope n Fg 5 s about as expected, except when ǫ s O, where the Euler equatons are not accurate approxmatons to the Boltzmann equaton. 5.3 Sod problem We use the classcal Sod problem [3] wth the ntal condton: { n a =, n b =., u a = u b =, T a = T b =, for x ; n a =.5, n b =., u a = u b =, T a = T b =., for x >. We compare our mcro/macro decomposton scheme MM wth the mplct scheme on BGK S and the Euler system Euler. The computatonal doman n space s [.5,.5] dscretzed by the unform grds x =.5. We use a velocty grd of ponts n doman [, ]. To satsfy the CFL condton, the tme step s taen as = x. Fg 6 shows that MM and S are almost the same for the dfferent regmes: ǫ = netc regme; ǫ =. transton regme and ǫ =. 5 flud regme. Fg 7 llustrates the asymptotc behavor of MM for the densty, velocty and the temperature of the mxture under the dfferent Knudsen number ǫ, wth ts lmt Euler. We also compare the CNS system NS wth our schemes MM and ts asymptotcs NS/MM n Fg 8, whch ndcates that NS, NS/MM and MM all match well even for small ǫ. 6 Concluson In ths wor, we extend the mcro-macro decomposton based asymptotc-preservng scheme, developed n [3] for sngle speces Boltzmann equaton, to multspeces Boltzmann equaton. In addton to the essental propertes of the scheme [3], we overcome the addtonal stff source terms due to the nonconservatve momentum and total energy for each speces. The dscretzaton s asymptotc-preservng n the Euler lmt, and s also consstent to the Naver-Stoes lmt wth sutably small tme step and mesh sze. Moreover, the nonlnear source terms can be solved by only usng lnear system solvers. Numercal experments demonstrate the effcency and correct asymptotc behavor of ths scheme. References [] P. Andres, K. Ao and B. Perthame, A consstent BGK-type model for gas mxtures, J. Stat. Phys [] C. Bardos, F. Golse and D. Levermore, Flud dynamc lmts of netc equatons. I. Formal dervatons, J. Stat. Phys [3] M. Bennoune, M. Lemou, and L. Meussens, Unformly stable numercal schemes for the Bolztmann equaton preservng compressble Naver-Stoes asympotcs, J. Comput. Phys

17 [4] J.-F. Bourgat, P. Le Tallec, B. Perthame, and Y. Qu, Couplng Boltzmann and Euler equatons wthout overlappng, n doman decomposton methods n scence and engneerng Como, 99, , Contemp. Math. 57, Amer. Math. Soc., Provdence, RI, 994. [5] R. Caflsch, S. Jn and G. Russo, Unformly accurate schemes for hyperbolc systems wth relaxatons, SIAM J. Num. Anal [6] P. Degond and S. Jn, A smooth transton model between netc and dffuson equatons, SIAM J. Numer. Anal [7] P. Degond, S. Jn and L. Meussens, A smooth transton model between netc and hydrodynamc equatons, J. Comp. Phys [8] P. Degond, J.-G. Lu and L. Meussens, Macroscopc flud modes wth localzed netc upscalng effects Multscale Model. Smul [9] J. H. Ferzger and H. G. Kaper, Mathematcal theory of transport processes n gases, North-Holland Publshng Company 97 [] F. Golse, S. Jn and C.D. Levermore, The convergence of numercal transfer schemes n dffusve regmes I: the dscrete-ordnate method, SIAM J. Num. Anal [] L. Gosse and G. Toscan, An asymptotc-preservng well-balanced scheme for the hyperbolc heat equatons, C.R. Math. Acad. Sc. Pars [] M. Günther, P. Le Tallec, J.-P. Perlat, and J. Strucmeer, Numercal modelng of gas flows n the transton between rarefed and contnuum regmes. Numercal flow smulaton I, Marselle, 997, 4, Notes Numer. Flud Mech., 66, Veweg, Braunschweg, 998. [3] S. Jn, Effcent Asymptotc-Preservng AP schemes for some multscale netc equatons, SIAM J. Sc. Comp [4] S. Jn and J.G. Lu, The effects of numercal vscostes I: slowly movng shocs, J. Comput. Phys [5] S. Jn, L. Paresch and G. Toscan, Dffusve relaxaton schemes for dscrete-velocty netc equatons, SIAM J. Num. Anal [6] S. Jn, L. Paresch and G. Toscan, Unformly accurate dffusve relaxaton schemes for multscale transport equatons, SIAM J. Num. Anal [7] A. Klar, An asymptotc-nduced scheme for nonstatonary transport equatons n the dffusve lmt, SIAM J. Num. Anal [8] A. Klar, H. Neunzert, and J. Strucmeer, Transton from netc theory to macroscopc flud equatons: a problem for doman decomposton and a source for new algorthm, Transp. Theory and Stat. Phys [9] M. Lemou and L. Meussens, A new asymptotc preservng scheme based on mcro-macro formulaton for lnear netc equatons n the dffuson lmt. SIAM J. Sc. Comput. 3 8 no., [] R. J. Leveque, Fnte volume methods for hyperbolc problems, Cambrdge Unversty Press [] T.P. Lu and S.H. Yu, Boltzmann equaton: mcro-macro decompostons and postvty of shoc profles. Comm. Math. Phys no., [] J. C. Mandal and S. M. Deshpande, Knetc flux vector splttng for Euler equatons, Comput. & fluds [3] G. A. Sod, A survey of several fnte dfference methods for systems of nonlnear hyperbolc conservaton laws, J. Comput. Phys

18 [4] P. Le Tallec, and F. Mallnger, Couplng Boltzmann and Naver-Stoes equatons by half fluxes, J. Comput. Phys [5] B. van Leer, Towards the ultmate conservatve dfference scheme V. A second order sequel to Godunov s method, J. Comput. Phys

19 .9.8 ε=.5 u a u b T a T b tme t.9.8 ε=. u a u b T a T b tme t Fgure : Space homogeneous soluton by S : the convergence of the veloctes and temperatures between speces a and b, wth dfferent Knudsen numbers ǫ =.5 on the top, and ǫ =. on the bottom. 9

20 ε=3 θ θ= θ= θ=3 θ=4 θ=5 densty f b x= velocty v Fgure : Statonary shoc: densty dstrbuton functon f b as a functon of velocty v at pont x =, gven by the scheme MM wth dfferent Knudsen numbers ǫ = 3 θ, at t =...8 ε=.7 ε=..7 S MM.6 S MM.6.5 densty f b x= densty f b x= velocty v velocty v Fgure 3: Statonary shoc: densty dstrbuton functon f b as a functon of velocty v at pont x =, gven by the scheme MM wth dfferent Knudsen numbers ǫ = and ǫ =., comparng wth the mplct scheme for BGK model S at t =..

21 densty ρ exact θ= θ= θ= θ=3 θ=4 θ= x velocty u exact θ= θ= θ= θ=3 θ=4 θ= x temperature T exact θ= θ= θ= θ=3 θ=4 θ= x Fgure 4: Statonary shoc: the asymptotc propertes of the scheme MM wth dfferent ǫ = θ at t =.. Profles of mass densty ρ on the top, mean velocty u n the mddle, and mean temperature T on the bottom, as a functon of space x.

22 5 densty ρ S /Euler MM/Euler logrelatve dfference logε 5 velocty u S /Euler MM/Euler logrelatve dfference logε temperature T S /Euler MM/Euler 5 logrelatve dfference logε Fgure 5: Statonary shoc: the relatve error between MM and Euler, and between S and Euler for dfferent ǫ = θ, θ =,,, 3,..., 5 at t =..

23 .7 ε= S.6 MM.5 velocty u x ε=. S.6 MM.5 velocty u x ε=. S.6 MM.5 velocty u x Fgure 6: Sod problem: compare the mean velocty u gven by the scheme MM wth the mplct scheme S for the BGK model under dfferent regmes: netc regme ǫ =, transton regme ǫ = and flud regme ǫ = 5 at t =.. 3

24 .5 Euler θ= θ= θ= θ=3 θ=4 θ=5 densty ρ x Euler θ= θ= θ= θ=3 θ=4 θ=5 velocty u x temperature T Euler θ= θ= θ= θ=3 θ=4 θ= x Fgure 7: Sod problem: the asymptotc propertes of the scheme MM wth dfferent ǫ = θ at t =.. Profles of mass densty ρ on the top, mean velocty u n the mddle, and mean temperature T on the bottom, as a functon of space x. 4

25 3.5 MM NS/MM NS densty ρ x.7.6 MM NS/MM NS.5 velocty u x.9.8 MM NS/MM NS.7 temperature x Fgure 8: Sod problem: compare the schemes MM, NS and NS/MM at ǫ =., x =.5, x = and t =.. 5