BGK models for inert mixtures: comparison and applications
Sebastiano Boscarino, Seung Yeon Cho, Maria Groppi, Giovanni Russo
BBGK MODELS FOR INERT MIXTURES: COMPARISON ANDAPPLICATIONS
SEBASTIANO BOSCARINO, SEUNG YEON CHO, MARIA GROPPI, AND GIOVANNI RUSSO
Abstract.
Consistent BGK models for inert mixtures are compared, first in their ki-netic behavior and then versus the hydrodynamic limits that can be derived in differentcollision-dominated regimes. The comparison is carried out both analytically and nu-merically, for the latter using an asymptotic preserving semi-Lagrangian scheme for theBGK models. Application to the plane shock wave in a binary mixture of noble gases isalso presented. Introduction
Since the seminal paper of Bhatnagar, Gross and Krook [10], BGK models of the Boltz-mann equation [14, 28] play the role of simpler and effective modeling tools to describe thedynamics of rarefied gases. Their importance is much more evident when gas mixtures aretaken into account. However, the extension to an arbitrary mixture of monoatomic gasesis not trivial, since interactions and exchanges between different components, as well asconservation of total momentum and energy, must be properly considered.The first mathematically rigorous contribution appeared in [2], where the authors proposeda BGK model characterized by a single global operator for each species, able to reproducethe correct exchange rates for momemtum and kinetic energy among species of the Boltz-mann equations for mixtures, assuming intermolecular potentials of Maxwell molecules type.Later, several BGK models have been introduced, characterized by different structures ofthe BGK operators (see for instance [24, 26, 27] and the reference therein). In [22] theauthors consider a BGK model for inert mixtures whose single global collision operator foreach species allows to correctly reproduce the conservation of total momentum and kineticenergy; moreover, in the same paper a comparison among this BGK model and the oneproposed in [2] is presented and discussed. For both BGK models in [2] and [22] the pos-sibility to include simple chemical reactions has been considered, and their extension tobimolecular chemical reactions in mixtures of monoatomic gases can be found in [23] and[6, 19], respectively.Recently, in [11] a further different consistent BGK-type model is built up, that mimics thestructure of the Boltzmann equations for mixtures, namely in which the collision operatorfor each species is a sum of bi-species BGK operators. This last model is consistent andwell–posed as the one in [2], but in addition allows to deal with general intermolecular po-tentials. The exchange rates for momentum and energy of each BGK operator coincide byconstruction with the corresponding exchange rates of each Boltzmann binary integral op-erator, getting thus exact conservations. The structure of the collision operator in this caseallows to consistently derive evolution equations for the main macroscopic fields in differenthydrodynamic regimes, according to the dominant collisional phenomenon [3, 4].This paper is aiming at comparing the behaviors of the above mentioned BGK models forinert mixtures of monoatomic gases in various regimes, from kinetic to hydrodynamic, and a r X i v : . [ m a t h - ph ] F e b S. BOSCARINO, S. Y. CHO, M. GROPPI, AND G. RUSSO to highlight their analogies and discrepancies. In addition, two different Navier-Stokes hy-drodynamic limits, characterized by global velocity and temperature or multi-velocity andmulti-temperature, respectively, and obtained from the consistent BGK model in [11], arenumerically tested with respect to their capability to reproduce the correct dynamics of real-istic mixtures of noble gases, with particular attention to the case of mixtures of heavy andlight particles (the so called ε -mixtures [18]). For the numerical solutions to BGK models,semi-Lagrangian methods proposed in [15, 22] will be adopted together with a conserva-tive reconstruction technique [12, 13], which enables us to capture the correct behaviors ofhydrodynamic limit models. As regards details and properties of semi-Lagrangian schemes(for single gas BGK model), we refer to papers regarding the construction of high ordersemi-Lagrangian schemes [8, 20, 33], treatment of boundary problems [21, 32] and conver-gence analysis [9, 34, 35].The paper is organized as follows. In section 2, we briefly recall the classical Boltzmannequation for inert gas mixtures. Section 3 is devoted to the description of the three BGKmodels considered in this paper. Next, in section 4, we study analytically the discrepancybetween BGK models at the level of collision operators. Then, in section 5, we present hy-drodynamic limits at Navier-Stokes level that can be obtained from the kinetic BGK models.In section 6, we perform numerical experiments in which we investigate the discrepancy ofthe three BGK models, compare them with their two corresponding Navier-Stokes limits,and study the Riemann problem and the steady shock structure in binary mixtures of noblegases, with different mass ratios.2. Kinetic Boltzmann-Type Equations
Let us consider a mixture of L monoatomic inert gases. Under the assumption that s -thgas has a mass m s >
0, its dynamics can be described through the distribution functions f s ( x , v , t ), s = 1 , · · · , L defined on phase space ( x , v ) ∈ R × R at time t >
0, whoseevolution is governed by the Boltzmann-type equations: ∂f s ∂t + v · ∇ x f s = Q s , (2.1)where Q s is the collision term of the s -th species, which collects bi-species collision operatorsbetween s -th and other k -th gases: Q s = N (cid:88) k =1 Q sk ( f s , f k ) . The binary collision operator Q sk can be cast as Q sk ( f s , f k ) = (cid:90) R × S d w d ω g sk ( | y | , ˆy · ω ) (cid:20) f s ( v (cid:48) ) f k ( w (cid:48) ) − f s ( v ) f k ( w ) (cid:21) (2.2)with a non-negative scattering kernel g sk depending on intermolecular potentials. Here weuse the integration variable w ∈ R , a unit vector on a sphere ω ∈ S , the relative velocity y := v − w and its unit vector ˆ y := y / | y | . The other two variables v (cid:48) and w (cid:48) are postcollisional velocities of s -th and k -th gases whose masses are m s , m k and pre-collisionalvelocities are v , w , respectively: v (cid:48) = m s v + m k w m s + m k + m k m s + m k | y | ω , w (cid:48) = m s v + m k w m s + m k − m s m s + m k | y | ω . (for more details, see for instance [14] and references therein). GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 3
The distribution function f s , s = 1 , . . . , L , can be used to reproduce s -species macroscopicquantities such as number density n s , average velocity u s , absolute temperature T s : n s = (cid:104) f s , (cid:105) , n s u s = (cid:104) f s , v (cid:105) , n s K B T s = m s (cid:104) f s , | v − u s | (cid:105) where (cid:104) f, h (cid:105) := (cid:90) R dvf ( v ) h ( v ) . Similarly, global macroscopic variables such as number density n , mass density ρ , velocity u and temperature T of the mixture can be obtained as follows: n = L (cid:88) s =1 n s , ρ = L (cid:88) s =1 ρ s , ρ s = m s n s , s = 1 , · · · , Lu = 1 ρ L (cid:88) s =1 ρ s u s , nK B T = 3 L (cid:88) s =1 n s K B T s + L (cid:88) s =1 ρ s | u s − u | The equilibrium solution to (2.1) is given by the Maxwellian which shares a common velocity u and temperature T : f eqs = n s M (cid:18) v ; u, K B Tm s (cid:19) where M ( v ; a, b ) ≡ (cid:18) πb (cid:19) / exp (cid:18) − b | v − a | (cid:19) , a ∈ R , b > . (2.3)The structure of the collision operators Q s allows to guarantee fundamental properties ofthe Boltzmann equations for inert gas mixtures: conservation laws, uniqueness of equilibriumsolutions, H- theorem. 3. BGK-type models
In this section we briefly recall the three different BGK models for eqns. (2.1)-(2.2) whichhave been compared in this paper.3.1.
The BGK Model of Andries, Aoki and Perthame (AAP model).
In [2], a BGK-type model was proposed, that allows to reproduce the same exchange rate in momentumand energy of the Boltzmann equation (2.1). In this model the collision term Q s (2.2) ofthe Boltzmann equation is replaced by a relaxation operator which drives the evolutiontowards an attracting auxiliary Maxwellian M s , depending on fictitious parameters. Inspite of such structural change, this BGK model still satisfies the main properties of theBoltzmann equation such as conservation law, H -theorem, indifferentiability principle. Thescaled model equations are described by ∂f s ∂t + v · ∇ x f s = ν s ε ( n s M s − f s ) , s = 1 , · · · , L, (3.1)where ε is the Knudsen number, ν s is the collision frequency for s -species gas and M s is theattracting Maxwellian: M s = M (cid:18) v ; u s , K B T s m s (cid:19) , where M is defined in (2.3). Notice that M s is defined in terms of fictitious parameters u s , T s (different from the actual fields u s , T s ), which, under the assumption of Maxwell molecules S. BOSCARINO, S. Y. CHO, M. GROPPI, AND G. RUSSO interaction potential, are explicit functions of the actual moments of the distribution functionas follows: u s = u s + 1 m s n s ν s L (cid:88) k =1 ξ sk u k ,T s = T s − m s K B (cid:0) | u s | − | u s | (cid:1) + 23 n s K B ν s L (cid:88) k =1 γ sk T k + 23 n s K B ν s L (cid:88) k =1 ν sk m s m k n s n k ( m s + m k ) ( m s u s + m k u k ) ( u k − u s ) , (3.2)where ξ sk = ν sk m s m k n s n k m s + m k − δ sk L (cid:88) r =1 ν sr m s m r n s n r m s + m r γ sk = 3 K B ν sk m s m k n s n k ( m s + m k ) − δ sk L (cid:88) r =1 K B ν sr m s m r n s n r ( m s + m r ) . (3.3)The collision frequency ν sk(cid:96) is defined by ν sk(cid:96) = 2 π | g | (cid:90) π g ( ω )(1 − cos ω ) (cid:96) sin ωdω, and satisfies ν sk(cid:96) ≤ ν sk for (cid:96) = 1 ,
2. We remark that this model is well defined with thechoice ν s = L (cid:88) k =1 ν sk n k , which guarantees the positivity of temperature. Moreover, for consistency, we hereafterassume the following case: ν sk = ν sk =: λ sk . (3.4)3.2. The BGK model preserving global conservations (GS model).
Another BGK-type model with one attracting Maxwellian for each species has been proposed in [6, 22]. Thefictitious parameters are adjusted to impose the same conservation laws of the Boltzmannequation (2.1), namely species number densities, global momentum, total kinetic energy. In[6, 22], the model has been originally designed to describe a bimolecular reversible chemicalreaction in a four species mixture; the model has been then adapted to a general L speciesinert gas mixture in [22]. The scaled model reads ∂f s ∂t + v · ∇ x f s = ν s ε ( n s M sGS − f s ) . (3.5)where ν s is the collision frequency for s -species gas and M sGS is the attracting Maxwellian: M sGS = M (cid:18) v ; ¯ u, K B ¯ Tm s (cid:19) , (3.6) GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 5 where M is defined in (2.3). Note that it depends only on the auxiliary parameters ¯ u and¯ T , which are determined by imposing the conservation of total momentum and energy: L (cid:88) s =1 (cid:90) R m s (cid:18) v | v | / (cid:19) ( n s M sGS − f s ) d v = 0 , Consequently, we obtain the following representation of ¯ u and ¯ T in terms of the actualmacroscopic fields u s and T s : ¯ u = (cid:80) Ls =1 ν s m s n s u s (cid:80) Ls =1 ν s m s n s , ¯ T = (cid:80) Ls =1 ν s n s (cid:0) m s (cid:0) | u s | − | ¯ u | (cid:1) + 3 K B T s (cid:1) K B (cid:80) Ls =1 ν s n s . In [22], it is proved that the positivity of auxiliary temperature ¯ T is guaranteed and theH-theorem holds for the space homogeneous case.3.3. A general consistent BGK model for inert gas mixtures (BBGSP model).
In [11], authors introduce a different BGK-type model whose BGK operators Q s mimic thestructure of the Boltzmann ones, namely are sums of bi-species operators Q sk , each of themprescribing the same exchange rates of the corresponding term of the Boltzmann equations.This model also satisfies the main qualitative properties of Boltzmann equation such asconservation laws, H -theorem, indifferentiability principle. The scaled equations read ∂f s ∂t + v · ∇ x f s = 1 ε L (cid:88) k =1 ν sk ( n s M sk − f s ) , s = 1 , · · · , L, (3.7)with M sk = M (cid:18) v ; u sk , K B T sk m s (cid:19) , where M is defined in (2.3). The auxiliary parameters u sk and T sk are defined by u sk = (1 − a sk ) u s + a sk u k T sk = (1 − b sk ) T s + b sk T k + γ sk K B | u s − u k | (3.8)with a sk = λ sk n k m k ν sk ( m s + m k ) , b sk = 2 a sk m s m s + m k , γ sk = m s a sk (cid:18) m k m s + m k − a sk (cid:19) . (3.9)In [11], authors proved that the positivity of T sk is guaranteed by T s > , T k > , ν sk ≥ λ sk n k . Considering this, throughout this paper, we set ν sk = ν sk n k = λ sk n k . This implies a sk = m k m s + m k , b sk = 2 a sk m s m s + m k , γ sk = m s a sk (cid:18) m k m s + m k − a sk (cid:19) . S. BOSCARINO, S. Y. CHO, M. GROPPI, AND G. RUSSO Discrepancy between BGK-type models
In this section, our goal is to check the discrepancy between AAP model (3.1) and BBGSPmodel (3.7). For this, we start from multiplying the two models (3.1) and (3.7) by εn s , andsubtract the resulting equations. After then, we expand two Maxwellians M s , M sk around u s and T s to obtain ν s M s − L (cid:88) k =1 ν sk M sk = L (cid:88) k =1 ν sk (cid:34) (cid:32) M s + ∂M∂u (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) ( u s − u s ) + ∂M∂T (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) ( T s − T s ) + h.o.t. (cid:33) − (cid:32) M s + ∂M∂u (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) ( u sk − u s ) + ∂M∂T (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) ( T sk − T s ) + h.o.t. (cid:33) (cid:35) , where M s := M (cid:18) v ; u s , K B T s m s (cid:19) ,∂M∂u (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) := m s ( v − u s ) K B T s M s ,∂M∂T (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) := (cid:18) − T s + m s | v − u s | K B T s (cid:19) M s . Then, we have ν s M s − L (cid:88) k =1 ν sk M sk = (cid:32) ∂M∂u (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) (cid:33) E u + (cid:32) ∂M∂T (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) (cid:33) E T + h.o.t.where E u := L (cid:88) k =1 ν sk ( u s − u sk ) , E T := L (cid:88) k =1 ν sk ( T s − T sk ) . (4.1)Here E u and E T are the contributions of leading order errors with respect to the derivativeof u and T , and all the remainders are denoted by h.o.t..In the following Proposition, we provide the explicit forms of E u and E T (the proof canbe found in Appendix A.1.) Proposition 4.1.
Suppose that M s in (3.1) and M sk in (3.7) are sufficiently smooth withrespect to macroscopic variables, velocity and temperature. Assuming that ν s = (cid:80) ν sk , theleading error terms E U and E T in (4.1) are given by (1) E u = 0(2) E T = m s K B L (cid:88) k =1 ν sk ( a sk ) | u s − u k | − m s K B (cid:32) ν s L (cid:88) r =1 ν sr a sr ( u r − u s ) (cid:33) · (cid:32) L (cid:88) r =1 ν sr a sr ( u r − u s ) (cid:33) . GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 7
Remark . In a similar manner, we can compare the AAP model (3.1) and the GS model(3.5) as follows: ν s M s − ν s M sGS = (cid:32) ∂M∂u (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) (cid:33) ¯ E u + (cid:32) ∂M∂T (cid:12)(cid:12)(cid:12)(cid:12) ( u,T )=( u s ,T s ) (cid:33) ¯ E T + h.o.t.,where ¯ E u := ν s ( u s − ¯ u ) , ¯ E T := ν s (cid:0) T s − ¯ T (cid:1) . (4.2)Note that E u in this case does not vanish:¯ E u = (cid:88) r (cid:54) = s (cid:32) ν sr m r n r m s + m r − ν s ν r m r n r (cid:80) Lr =1 ν r m r n r (cid:33) ( u r − u s ) . Due to the complexity of ¯ E T , we provide its form in Appendix A.2. This analysis showsa more pronounced discrepancy between AAP and GS models; it will be confirmed andquantified in the next section at Navier-Stokes level, and discussed later in the numericaltests in section 6.1.5. Hydrodynamic limits at Navier–Stokes (NS) level
Here we describe the Navier-Stokes asymptotics that can be derived from the differentBGK models to O ( ε ).5.1. NS equations with global velocity and temperature.
In [3], the hydrodynamiclimit of the BBGSP model (3.7)-(3.9) at the Navier-Stokes level is derived using the Chapman-Enskog expansion in a collision dominated regime. The equations for macroscopic variables n , u and T , obtained as ε -order closure of the macroscopic equations (moments of the BGKones), are given by ∂n s ∂t + ∇ · ( n s u ) + ε ∇ · ( n s u (1) s ) = 0 , s = 1 , · · · , L∂∂t ( ρu ) + ∇ · ( ρu ⊗ u ) + ∇ ( nK B T ) + ε ∇ · ( P (1) ) = 0 ,∂∂t (cid:18) ρ | u | + 32 nK B T (cid:19) + ∇ · (cid:20)(cid:18) ρ | u | + 52 nK B T (cid:19) u (cid:21) + ε ∇ · ( P (1) · u ) + ε ∇ · q (1) = 0 , (5.1)where u (1) s , P (1) , q (1) are first order corrections with respect to ε . The diffusion velocity u (1) s takes the following form: u (1) s = L (cid:88) k =1 L sk ρ s ρ k ∇ ( n k K B T )(5.2)where the symmetric matrix L is computed as L = ˜ M − Ω , Ω sk = ρ s δ sk − ρ s ρ k ρ , ˜ M sk = M sk − κ, where κ = min s (cid:54) = k M sk M sk = λ sk ρ s m s + m k − δ sk L (cid:88) r =1 λ sr ρ r m s + m r . (5.3) S. BOSCARINO, S. Y. CHO, M. GROPPI, AND G. RUSSO
The first order corrections for pressure tensor P (1) is of the following form: P (1) αβ = − µ (cid:18) ∂u α ∂x β + ∂u β ∂x α − ∇ · uδ αβ (cid:19) , ≤ α, β ≤ , (5.4)where the viscosity coefficient µ is given by µ := L (cid:88) s =1 n s K B T (cid:80) Ls =1 ν (0) sk . (5.5)Here we denote by ν (0) sk the leading order in the expansion of the collision frequency ν sk .The heat flux q (1) is given by q (1) = 52 K B T L (cid:88) s =1 n s u (1) s − λ ∇ T, (5.6)where λ is the thermal conductivity coefficient: λ = 52 K B T L (cid:88) s =1 n s m s (cid:80) Lk =1 ν (0) sk . (5.7)For detailed description of this model, we refer to [3]. It is remarkable that such results arein complete agreement with those obtained from the AAP model (3.1)-(3.2) [2]. This is notsurprising since these results are indeed exact for the Boltzmann equations with Maxwellmolecules.The same structure of the Navier-Stokes equations (5.1), with first order corrections(5.2),(5.4) and (5.6), is reproduced also by the ε -order asymptotics of the GS model (3.5)-(3.6). However, the matrix M involved in the Fick’s law (5.2) for diffusion velocities isdifferent from (5.3); indeed, in case of BGK model (3.5)-(3.6), such matrix accounts forall mechanical interactions via the inverse relaxation times ν s , whereas for the other BGKmodels considered above only the bi-species collision frequencies ν sk = λ sk are involved. Itsexpression for GS model is given by [7] M GSsk = ν s ν k (cid:80) Lr =1 ρ k ν k ρ s − ν s δ sk and consequently the diffusion velocities u (1) s in the Navier-Stokes equations are quantita-tively different from the previous ones. As regards the transport coefficients, namely vis-cosity η and thermal conductivity λ , they are exactly given by (5.5) and (5.7), respectively,also for the BGK model (3.5)-(3.6). GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 9
Representation of NS equations for n s , u and T . The system (5.1) can be rewrittenin the following form, which is more convenient for its numerical treatment: ∂n s ∂t = −∇ · ( n s u ) − ε ∇ · ( n s u (1) s ) , s = 1 , · · · , L∂u∂t = uρ (cid:32) ∇ · (cid:32) L (cid:88) i =1 ερ s u (1) s (cid:33)(cid:33) − u ∇ · u − ∇ ( nK B T ) ρ − ε ∇ · ( P (1) ) ρ∂T∂t = − | u | nK B (cid:32) ∇ · (cid:32) L (cid:88) i =1 ερ s u (1) s (cid:33)(cid:33) + 2 u nK B · (cid:16) ε ∇ · ( P (1) ) (cid:17) + Tn (cid:32) ∇ · (cid:32) L (cid:88) s =1 εn s u (1) s (cid:33)(cid:33) − ( ∇ T ) · u − T ∇ · u − nK B ε ∇ · ( P (1) · u ) − nK B ε ∇ · q (1) . (5.8)5.2. NS equations with multi-velocity and temperature.
An interesting point of(3.7) is the possibility of allowing different hydrodynamic limits, thanks to the structure ofthe BBGSP collision operators as a sum of bispecies relaxation terms. In [4], the authorsconsider the case in which intra-species collisions are the dominant process in the evolutionof the mixture. This occurs for instance in the so called ε -mixtures of heavy and lightgases [18], where molecules with very disparate masses exchange energy more slowly thanmolecules of the same species, and also in some applications to plasmas and astrophysics [37].In this case it is possible to define a proper Knudsen number and obtain the adimensionalscaled equations: ∂f s ∂t + v · ∇ x f s = 1 ε ν ss ( n s M ss − f s ) + 1 κ L (cid:88) k (cid:54) = s ν sk ( n s M sk − f s ) . (5.9)This scaling leads to a Navier-Stokes model of multi-velocity and multi-temperature for s -species gases. In this case, the equations for macroscopic variables n s , u s and T s are givenby ∂n s ∂t + ∇ · ( n s u s ) = 0 ,∂∂t ( ρ s u s ) + ∇ · ( ρ s u s ⊗ u s ) + ∇ ( n s K B T s ) + ε ∇ · ( P (1) s ) = 1 κ L (cid:88) k (cid:54) = s R sk ,∂∂t (cid:18) ρ s | u s | + 32 n s K B T s (cid:19) + ∇ · (cid:20)(cid:18) ρ s | u s | + 52 n s K B T s (cid:19) u s (cid:21) + ε ∇ · ( P (1) s · u s ) + ε ∇ · q (1) s = 1 κ L (cid:88) k (cid:54) = s S sk , (5.10)for s = 1 , · · · , L , with R sk = λ sk m sk n s n k ( u k − u s ) , S sk = λ sk m sk m s + m k n s n k [( m s u s + m k u k ) · ( u k − u s ) + 3 K B ( T k − T s )] m sk = m s m k m s + m k . Here the pressure tensor is given by P (1) s = − n s K B T s ν (0) ss H s + 1 ν (0) ss L (cid:88) k (cid:54) = s ν (0) sk ( a (0) sk ) m s n s (cid:20) ( u s − u k ) ⊗ ( u s − u k ) − | u s − u k | I (cid:21) H s,αβ = ∂u s,α ∂x β + ∂u s,β ∂x α − · ∇ u s δ αβ , and the heat flux vector takes the following form: q (1) s = − n s K B T s n s ν (0) ss ∇ T s + 5 m s n s ν (0) ss L (cid:88) k (cid:54) = s ν (0) sk ( a (0) sk ) m s + m k K B ( T k − T s )( u k − u s )+ 13 m s n s ν (0) ss L (cid:88) k (cid:54) = s ν (0) sk ( a (0) sk ) (cid:18) m k m s + m k − a (0) sk (cid:19) | u k − u s | ( u k − u s ) . Note that multi-temperature Euler equations can be obtained by putting ε = 0 in (5.10).We refer to [31, 36] where multi-temperature Euler equations are described in the frameworkof Extended Thermodynamics.It is worth noticing that, according to [30], shock structure in Helium-Argon mixturescan be better reproduced by the multi-temperature description. In numerical tests, we willalso numerically deal with the Helium-Argon mixtures.5.2.1. Representation of NS equations for n s , u and T . We rewrite (5.10) as follows (fordetail see Appendix B.2.) ∂n s ∂t = −∇ · ( n s u ) , s = 1 , · · · , L∂u s ∂t = − u s ∇ · u s − ∇ ( n s K B T s ) ρ s − ε ∇ · ( P (1) s ) ρ s + 1 ρ s L (cid:88) k (cid:54) = s R sk ∂T s ∂t = 2 u s n s K B · ε ∇ · ( P (1) s ) − L (cid:88) k (cid:54) = s R sk − ( ∇ T s ) · u s − T s ∇ · u s − n s K B ε ∇ · ( P (1) s · u ) − n s K B ε ∇ · q (1) s + 23 n s K B L (cid:88) k (cid:54) = s S sk . (5.11) 6. Numerical tests
In this section, we present several numerical examples. First, we numerically check thediscrepancies between the three BGK-type models for inert gas mixtures given by (3.1),(3.5) and (3.7). Second, with reference to the scaled model (5.9) we approximate the twocorresponding systems of NS equations (5.1) and (5.10). Third, we consider a binary mixtureof noble gases with large mass ratio, in which the multi-velocity and temperature description(5.9) and (5.10) may explain better the behavior of gases. Finally, we study the structureof a stationary shock wave for a binary mixture of noble gases.To compute numerical solutions to (3.1) and (3.5), we consider a semi-Lagrangian methodintroduced in [22]. Since the method has been introduced with a non-conservative recon-struction, to make the method conservative we adopt a technique introduced in [12, 13].In particular, we make use of Q-CWENO23 and Q-CWENO35 reconstructions, which arebased on CWENO23 [29] and CWENO35 [17], respectively. For details, we refer to [12, 13].
GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 11
For (3.7), we use a conservative semi-Lagrangian method introduced in [15]. For the timediscretization, we consider an implicit Runge-Kutta method (DIRK) and a backward dif-ference formula (BDF). In particular, we here consider a second order DIRK method and athird order BDF3 method in [15].Note that we perform numerical simulations based on Chu reduction [16] as in [15, 22],that allows to reduce the problem from 3D to 1D in velocity and space, under suitablesymmetry assumptions. Below we list the name of schemes, which will be used in thissection:(1) RK2-QCWENO23: DIRK2 with Q-CWENO23.(2) BDF3-QCWENO35: BDF3 with Q-CWENO35.For discretization of the space and velocity (1D) domain, we use N x and N v + 1 gridpoints with uniform mesh sizes ∆ x and ∆ v , respectively. Based on this, we will use gridpoints x i and v j over computation domain [ x min , x max ] × [ v min , v max ]. To fix a time step∆ t , we use a CFL number defined byCFL = max {| v min | , | v max |} ∆ t ∆ x . To compute the solutions of NS equations (5.1) and (5.10), we instead solve (5.8) and(5.11) with spectral methods, which make the treatment of the many derivatives appearingthere easier. Let us consider a general representation of the two NS equations: U t = F ( U, U x , U xx ) , (6.1)where U ≡ U ( x, t ) is assumed to be smooth and periodic on the spatial domain. Givenvalues of { U ni } , we first compute Fourier coefficients ˆ U k , k = − N x / , · · · , N x / − n th spatial derivatives of functionsinvolved in equations of interest using the inverse fast Fourier transform of { ( jk ) n ˆ U k } where j denotes the imaginary number. For the time integration in (6.1), we adopt an explicitRK4 scheme.6.1. Comparison among the BGK models.
Here we numerically investigate the dis-crepancy between the three BGK-type models for gas mixture. For this, as in [1, 22], weconsider a mixture of four monoatomic gases whose molecular masses are given by m = 58 . , m = 18 , m = 40 , m = 36 . . (6.2)We use the following mechanical collision frequencies: ν = 5 , ν = 6 , ν = 2 , ν = 7 ν = 4 , ν = 5 , ν = 8 ν = 4 , ν = 3 ν = 6(6.3)with ν sk = ν ks for s, k = 1 , . . . ,
4. Since we are comparing the three BGK models, the onlycommon NS limit is the one with global velocity and temperature.
Smooth initial data with large variance in velocity.
In section 4, Proposition 4.1 showsthat the discrepancy between the AAP model (3.1) and the BBGSP model (3.7) becomesapparent when there is a large variance in the macroscopic velocities of gases. To show theseaspects, we set as initial data Maxwellians whose macroscopic fields are given by n s ( x ) = 1 m s , T s ( x ) = 4 (cid:80) s =1 n s ,u s ( x ) = η s σ s (cid:20) exp (cid:18) − (cid:16) σ s x − s (cid:17) (cid:19) + exp (cid:18) − (cid:16) σ s x + 3 − s (cid:17) (cid:19)(cid:21) (6.4)where σ s = (10 , , ,
9) and η s = ( − , − , , s = 1 , · · · ,
4. We impose periodicboundary conditions on the space domain [ − ,
1] and truncate velocity domain by [ − , N x = 200 and N v = 60. Wefirst check the discrepancy of the three models at final time t f = 0 .
04. We use CFL = 0 . t = 0 .
004 and CFL= 2 for t ∈ [0 . , .
04] in order to be able to resolve the relaxationtowards local equilibrium.We first measure the discrepancy of three BGK-type models for different values of ε =10 − q , 2 ≤ q ≤
8, and plot the differences of two solutions to the AAP model (3.1) and theBBGSP model (3.7) using relative L -norm in Figure 1a. Here we compare the quantity g ( x, v, t ) ≡ (cid:82) R f ( x, v , t ) dv dv , which is based on the Chu reduction (for details, we referto [16]). Similarly, we report numerical results relevant to the comparison of the two modelsAAP (3.1) and GS (3.5) in Figure 1b. Figure 1a shows that the differences between the AAPand BBGSP models are of order O ( ε ) for relatively large values of ε ∈ [10 − , − ], whilethey become of order O (cid:0) ε (cid:1) for small values of ε ∈ [10 − , − ]. On the contrary, we canonly observe differences of order O ( ε ) between AAP and GS models in Figure 1b. Thesenumerical evidences support the analytical result that the three models share the same Eulerlimit, and in addition AAP (3.1) and BBGSP (3.7) models have the same hydrodynamiclimits at the NS level. On the contrary, the GS model (3.5) has a quantitatively differenthydrodynamic limit, as shown in section 5.1. Although we obtain similar results for globalvelocity u and temperature T , we omit them here for brevity. -12 -10 -8 -6 -4 -2 Kn=10 -2 Kn=10 -3 Kn=10 -4 Kn=10 -5 Kn=10 -6 Kn=10 -7 Kn=10 -8 (a) AAP and BBGSP models -12 -10 -8 -6 -4 -2 Kn=10 -2 Kn=10 -3 Kn=10 -4 Kn=10 -5 Kn=10 -6 Kn=10 -7 Kn=10 -8 (b) AAP and GS models Figure 1.
Time evolution of relative L -norm of the differences in thedistribution functions g between BGK models for various values of ε . GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 13 -1 -0.5 0 0.5 10.60.811.21.41.61.82 (a) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 10.60.811.21.41.61.82 (b) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 1-0.8-0.6-0.4-0.200.20.40.60.81 (c) Velocity u s , s = 1 , , , -1 -0.5 0 0.5 1-0.8-0.6-0.4-0.200.20.40.60.81 (d) Velocity u s , s = 1 , , , -1 -0.5 0 0.5 120304050607080 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4AAP s=1AAP s=2AAP s=3AAP s=4GS s=1GS s=2GS s=3GS s=4 (e) Temperature T s , s = 1 , , , -1 -0.5 0 0.5 120304050607080 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4AAP s=1AAP s=2AAP s=3AAP s=4GS s=1GS s=2GS s=3GS s=4 (f) Temperature T s , s = 1 , , , Figure 2.
Comparison of the three BGK models for ε = 10 − (Left) and ε = 10 − (Right) with initial data in (6.4).In Figs 2-3, we plotted the numerical solutions of the four gases obtained with the threeBGK methods at t f = 0 .
2, using CFL= 0 . t = 0 .
02 and CFL= 2 for t ∈ [0 . , . ε = 10 − , especially between GS modelsand the other ones, which instead remain closer to each other, but all species velocities andtemperatures are close to global velocity and temperature already from ε = 10 − , whiledensities equalize slower. Although we took smooth initial data, it is noticeable that shocksappear as time flows around x = 0 .
27. Figure 3a shows even that densities are almost -1 -0.5 0 0.5 10.60.811.21.41.61.82 (a) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 1-0.8-0.6-0.4-0.200.20.40.60.81 (b) Velocity u s , s = 1 , , , Figure 3.
Comparison of the three BGK models for ε = 10 − with initialdata in (6.4).overlapped for ε = 10 − . Here we omit the profiles of temperature since they show similartrends of Figure 2f.6.2. Comparison with NS equations.
Here we focus on the different hydrodynamiclimits that can be obtained from the BGK model (3.7) in order to highlight their peculiarbehaviors.6.2.1.
Case 1: global velocity and temperature.
In this test, we check the capability to cap-ture hydrodynamic limit (5.1) by solving model (3.7). For this, we consider the same testproposed in [1, 22]. Here we consider the mixture of four monoatomic gases with molecularmasses in (6.2) and collision frequencies in (6.3).We set initial data as Maxwellians whose macroscopic fields are n s ( x ) = 1 m s , T s ( x ) = 4 (cid:80) s =1 n s ,u s ( x ) = sσ s (cid:20) exp (cid:18) − (cid:16) σ s x − s (cid:17) (cid:19) + exp (cid:18) − (cid:16) σ s x + 3 − s (cid:17) (cid:19)(cid:21) (6.5)for s = 1 , · · · ,
4, where σ s = (10 , , , ε = 10 − k , k = 2 , ,
4. Herewe compute numerical solutions to (3.7) using N x = 500 and N v = 60. We take CFL= 0 . t = 0 .
02, and CFL= 2 in t ∈ [0 . , . ε = 10 − . However, both solutions give similarvalues of macroscopic quantities as we take smaller Knudsen numbers. For ε = 10 − ,global velocity u and temperature T are almost overlapped, contrary to species densities.In Figure 5, we note that even species densities become identical for a sufficiently smallKnudsen number ε = 10 − .6.2.2. Case 2: multi velocity and temperature.
Here we consider the case in which intra-species collisions are dominant, and hence we impose κ (cid:54) = ε in (5.9) so that the scaled versionof the BBGSP model (3.7) leads to the multi-velocity and multi-temperature NS equations GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 15 -1 -0.5 0 0.5 10.0150.020.0250.030.0350.040.0450.050.0550.060.065 t=0.2000
BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (a) Number density n s , s = 1 , , , -1 -0.5 0 0.5 10.0150.020.0250.030.0350.040.0450.050.0550.060.065 t=0.2000 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (b) Number density n s , s = 1 , , , -1 -0.5 0 0.5 10.90.920.940.960.9811.021.041.061.081.1 t=0.2000 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (c) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 10.90.920.940.960.9811.021.041.061.081.1 t=0.2000 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (d) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 1-0.12-0.1-0.08-0.06-0.04-0.0200.020.040.060.08 t=0.2000 BBGSPNSE (e) Velocity u -1 -0.5 0 0.5 1-0.12-0.1-0.08-0.06-0.04-0.0200.020.040.060.08 t=0.2000 BBGSPNSE (f) Velocity u -1 -0.5 0 0.5 13030.53131.53232.53333.534 t=0.2000 BBGSPNSE (g) Temperature T -1 -0.5 0 0.5 13030.53131.53232.53333.534 t=0.2000 BBGSPNSE (h) Temperature T Figure 4.
Comparison of BGK model (3.7) and NS equations (5.1) for ε = 10 − (Left) and ε = 10 − (Right) with initial data in (6.5). -1 -0.5 0 0.5 10.0150.020.0250.030.0350.040.0450.050.0550.060.065 t=0.2000 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (a) Number density n s , s = 1 , , , -1 -0.5 0 0.5 10.90.920.940.960.9811.021.041.061.081.1 t=0.2000 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (b) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 1-0.12-0.1-0.08-0.06-0.04-0.0200.020.040.060.08 t=0.2000 BBGSPNSE (c) Velocity u -1 -0.5 0 0.5 13030.53131.53232.53333.534 t=0.2000 BBGSPNSE (d) Temperature T Figure 5.
Comparison of BGK model (3.7) and NS equations (5.1) for ε = 10 − with initial data in (6.5).(5.10) for small values of ε . Here, we take Knudsen number ε = 10 − k , k = 2 , , κ = 1. We compute numerical solutions with the same numerical setting of sect. 6.2.1.In Figures 6-7, we can observe, contrary to the previous case (that can be reproducedby taking κ = ε ), a clear separation between macroscopic quantities of the different species;moreover, the behavior of the BGK solutions of (5.9) are similar to those of multi-velocityand multi-temperature NS description (5.10). We note that in Figure 7 for ε = 10 − thekinetic and macroscopic solutions show very good agreement in each species velocity andtemperature.6.3. Shock problem for binary gas mixture.
In this test, we consider a shock problemfor binary mixture of two noble gases: Helium (He) and Argon (Ar), for which mass ratiobecomes relatively large. We aim at checking in this real case which Navier-Stokes descrip-tion (for global or species macroscopic fields) is better capable to capture the behavior ofthe mixture.To perform the realistic simulation, we consider molecular mass of He and Ar as follows: m m ≈ × − kg/mol ( He ) , m m ≈ × − kg/mol ( Ar ) . GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 17 -1 -0.5 0 0.5 10.010.020.030.040.050.060.07 t=0.2000 (a) Number density n s , s = 1 , , , -1 -0.5 0 0.5 10.010.020.030.040.050.060.07 t=0.2000 (b) Number density n s , s = 1 , , , -1 -0.5 0 0.5 10.850.90.9511.051.11.151.2 t=0.2000 (c) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 10.850.90.9511.051.11.151.2 t=0.2000 (d) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 1-0.2-0.100.1 t=0.2000 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (e) Velocity u s , s = 1 , , , -1 -0.5 0 0.5 1-0.2-0.100.1 t=0.2000 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (f) Velocity u s , s = 1 , , , -1 -0.5 0 0.5 1282930313233343536 t=0.2000 (g) Temperature T s , s = 1 , , , -1 -0.5 0 0.5 1282930313233343536 t=0.2000 (h) Temperature T s , s = 1 , , , Figure 6.
Comparison of the scaled BBGSP model (5.9) and Navier-Stokes equations (5.10) with κ = 1 for ε = 10 − (Left) and ε = 10 − (Right) with initial data in (6.5). -1 -0.5 0 0.5 10.010.020.030.040.050.060.07 t=0.2000 (a) Number density n s , s = 1 , , , -1 -0.5 0 0.5 10.850.90.9511.051.11.151.2 t=0.2000 (b) Density ρ s , s = 1 , , , -1 -0.5 0 0.5 1-0.2-0.100.1 t=0.2000 BBGSP s=1BBGSP s=2BBGSP s=3BBGSP s=4NSE s=1NSE s=2NSE s=3NSE s=4 (c) Velocity u s , s = 1 , , , -1 -0.5 0 0.5 1282930313233343536 t=0.2000 (d) Temperature T s , s = 1 , , , Figure 7.
Comparison of the scaled BBGSP model (5.9) and Navier-Stokes equations (5.10) with κ = 1 for ε = 10 − with initial data in (6.5).In view of this, we rewrite the BBGSP model (3.7) in terms of mole. We divide f s by theAvogadro’s number N a ≈ . × . Now, we denote the molecular density by f ms ( x , v , t ) := f ( x , v , t ) N a for s = 1 ,
2. Then, we can rewrite (3.7) as ∂f ms ∂t + v · ∇ x f ms = 1 ε L (cid:88) k =1 ν sk ( n s M sk − f ms ) , M sk = M ( v ; u msk , RT msk m ms ) , (6.6)with macroscopic quantities at the level of mole: n ms = (cid:104) f ms , (cid:105) , n ms u ms = (cid:104) f ms , v (cid:105) , n ms RT ms = m ms (cid:104) f ms , | v − u s | (cid:105) . Here we define the universal gas constant R as R := K B N a ≈ . J/mol and the molec-ular mass m ms as m ms = m s N a . Note that the values of u msk and T msk are samely defined asin (3.8) by using m ms , n ms , u ms , T ms . For global macroscopic variables we use the following GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 19 expressions: n m = L (cid:88) s =1 n ms , ρ m = L (cid:88) s =1 ρ ms , ρ ms = m ms n ms , s = 1 , · · · , Lu m = 1 ρ m L (cid:88) s =1 ρ ms u ms , n m RT m = 3 L (cid:88) s =1 n ms RT ms + L (cid:88) s =1 ρ ms | u ms − u m | Let us consider room temperature T ms = 300 K . As in [5], here we use the collision frequen-cies corresponding to T ms = 300 K based on the following formula: ν ss = 43 Tµ s ( T ) , s = 1 , ν = 2 √
23 ( m + m ) ( m m ) T ( µ ( T ) µ ( T )) , (6.7)where viscosity coefficients µ s for noble gases are provided in [25]. In case of Helium andArgon, we have ν = 19 . , ν = 1 . ν = 1 . , ν = 17 . . Note that we set ν sk = ν sk n k . Now, we consider a shock problem by taking the Maxwellianas initial data which reproduces the following macroscopic variables:( ρ m , u m , T m ) = (cid:40) (1 . , , , x < . . , , , x > . ρ m , u m , T m are kg/m , m/s , K , respectively. Now, we set( ρ m , ρ m ) = (cid:40) (0 . , . , x < . . , . , x > . . Note that the density of Helium and Argon gases for T m = 300 K for p m = 1 bar are givenby ρ m = 0 . kg/m , ρ m = 1 . kg/m . For numerical simulation, we assume the free-flow boundary condition on the spatial domain x ∈ [ − ,
6] with velocity domain v ∈ [ − , N x = 600 and N v = 320 up to t f = 0 .
06. Here we use CFL=1 . ε = 10 − q , q = 3 , , κ = 10 − , the panels on the left column showthat the global velocity and temperature is closely related to the dynamics of Argon gas.This is because its density is ten times bigger than that of Helium gas. Thus, it is difficultto describe the dynamics of mixtures involving Helium gas with global velocity and temper-ature description. On the other hand, the panels on the right column enables us to capturethe behaviors of Helium gas and this is the case where multi-velocity and multi-temperaturedescription can be a suitable model for a better description of this dynamics. In the fol-lowing Figures 9-10, both species velocities and temperatures are close to global velocityand temperature. Here the global velocity and temperature Euler system can be already asuitable choice for describing the dynamics of binary mixtures. In the right panels of Figures 8-10, we observe that the species velocity and temperaturefor light gas show very different behaviors as both ε and κ becomes smaller. For a betterunderstanding of these observations with the scaled BBGSP model (5.9), let us considerbinary gas mixtures with m = rm where r < r →
0, we obtain M → M (cid:18) v ; u , K B T m (cid:19) , M → M (cid:18) v ; u , K B T m + 13 | u − u | (cid:19) , while M ss = M (cid:16) v ; u s , K B T s m s (cid:17) for s = 1 ,
2. The form of M and M implies that as theintra-species collisions becomes dominant, light gas tends to follow the behavior of heavygas: ∂f ∂t + v · ∇ x f = ν κ (cid:18) n M (cid:18) v ; u , K B T m + 13 | u − u | (cid:19) − f (cid:19) + ν ε ( n M − f ) , while heavy gas tends to behave like a single gas: ∂f ∂t + v · ∇ x f = ν ε ( n M − f ) + ν κ (cid:18) n M (cid:18) v ; u , K B T m (cid:19) − f (cid:19) . Furthermore, the formula (6.7) for noble gases gives ν = 2 √ (cid:18) m m (cid:19) (cid:16) m m (cid:17) ( m ) T ( µ ( T ) µ ( T )) = 2 √ r (1 + r ) ( m ) T ( µ ( T ) µ ( T )) , in which, for a fixed value of m > r → ν →
0. If ν << κ , thescaled BBGSP model (5.9) formally reduces to two independent equations: ∂f ∂t + v · ∇ x f = ν ε (cid:18) n M (cid:18) v ; u , K B T m (cid:19) − f (cid:19) ∂f ∂t + v · ∇ x f s = ν ε (cid:18) n M (cid:18) v ; u , K B T m (cid:19) − f (cid:19) . where ν ss = ν ss n s .6.4. Stationary shock.
In this test, we consider a classical problem of gas dynamics,namely the shock wave structure obtained by the Navier-Stokes description in a binarymixture of two noble gases: Neon (Ne) and Argon (Ar). This problem has been faced in[5] by using the qualitative theory of dynamical systems applied to the time-independentversion of the NS equations, which can be rewritten as a suitable system of first orderODEs. Here we alternatively obtain the stationary shock solution as asymptotic solution ofthe time-dependent Navier-Stokes equations (5.1) and also by solving the BBGSP model.For this test, we consider molecular masses: m = 20(Ne) , m = 40(Ar) . Based on (6.7), we set collision frequencies for T = 300K by ν = 12 . , ν = 15 . ν = 15 . , ν = 17 . . GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 21 -6 -4 -2 0 2 4 60.0150.020.0250.030.0350.040.045 t=0.0600 (a) Number density n s , s = 1 , -6 -4 -2 0 2 4 60.0150.020.0250.030.0350.040.045 t=0.0600 (b) Number density n s , s = 1 , -6 -4 -2 0 2 4 6-0.500.511.522.533.544.5 t=0.0600 (c) Velocity u -6 -4 -2 0 2 4 6-0.500.511.522.533.544.5 t=0.0600 (d) Velocity u -6 -4 -2 0 2 4 6250260270280290300310320330340350 t=0.0600 (e) Temperature T -6 -4 -2 0 2 4 6250260270280290300310320330340350 t=0.0600 (f) Temperature T Figure 8.
Comparison of the scaled BBGSP model (5.9) for ε = κ = 10 − with: (left) global velocity and temperature Euler system (5.1) for ε = 0and (right) multi-velocity and multi-temperature Euler system (5.10) for ε = 0, κ = 10 − . We use the initial data in (6.3).We take initial data by the Maxwellian whose macroscopic fields reproduce n s ( x, u s ( x, T s ( x, = E sL + E sR − E sL (cid:0) tanh( ax ) + 1 (cid:1) , x ∈ [ − , -6 -4 -2 0 2 4 60.0150.020.0250.030.0350.040.045 t=0.0600 (a) Number density n s , s = 1 , -6 -4 -2 0 2 4 60.0150.020.0250.030.0350.040.045 t=0.0600 (b) Number density n s , s = 1 , -6 -4 -2 0 2 4 6-0.500.511.522.533.544.5 t=0.0600 (c) Velocity u -6 -4 -2 0 2 4 6-0.500.511.522.533.544.5 t=0.0600 (d) Velocity u -6 -4 -2 0 2 4 6250260270280290300310320330340350 t=0.0600 (e) Temperature T -6 -4 -2 0 2 4 6250260270280290300310320330340350 t=0.0600 (f) Temperature T Figure 9.
Comparison of the scaled BBGSP model (5.9) for ε = κ = 10 − with: (left) global velocity and temperature Euler system (5.1) for ε = 0and (right) multi-velocity and multi-temperature Euler system (5.10) for ε = 0, κ = 10 − . We use the initial data in (6.3).where a is a parameter which adjusts the slope of the smooth jump associated to the initialdata (we take a = 2). The two states E sL and E sR , s = 1 ,
2, are chosen according to the
GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 23 -6 -4 -2 0 2 4 60.0150.020.0250.030.0350.040.045 t=0.0600 (a) Number density n s , s = 1 , -6 -4 -2 0 2 4 6-0.500.511.522.533.544.5 t=0.0600 (b) Velocity u -6 -4 -2 0 2 4 6260270280290300310320330340350 t=0.0600 (c) Temperature T Figure 10.
Comparison of the scaled BBGSP model (5.9) for ε = κ =10 − with multi-velocity and multi-temperature Euler system (5.10) for ε = 0, κ = 10 − . We use the initial data in (6.3).Rankine-Hugoniot conditions as follows [5]: E sR = ( n ∞ s , u ∞ , T ∞ ) ,E sL = (cid:18) Ma Ma + 3 n ∞ s , Ma + 34 Ma u ∞ , (5 Ma − Ma + 3)16 Ma T ∞ (cid:19) c ∞ = (cid:115) n ∞ T ∞ ρ ∞ = (cid:115) n ∞ + n ∞ ) T ∞ ρ ∞ + ρ ∞ ) Ma = (cid:114) u ∞ c ∞ , where Ma is the Mach number. We consider concentrations χ = n n = 0 . χ = n n =0 .
9, and set n = χ n, n = χ n, Ma = √ . , T ∞ = 300 . In this problem, we impose the inflow and outflow boundary conditions. We consider velocitydomain [ − ,
32] and compute numerical solutions with N v = 80, N x = 200 and CFL= 0 . ε = 1 for this problem. -5 0 50.60.70.80.91 n1 -5 0 50.60.70.80.91 n2 -5 0 511.21.4 u -5 0 50.70.80.91 T Figure 11.
BDF3-QCWENO35 for ε = κ = 10 − . Neon and Argon with n = 0 . m , n = 0 . m . Black dashed lines are reference NS solutionsand solid lines are BGK solutions.In Figure 11, we plot normalized macroscopic fields: n s /n ∞ s , u/u ∞ , T /T ∞ . Our solution shows very good agreement between the long-time solution of the BBGSPmodel and the reference solution asymptotically obtained from the NS equations (5.1) dis-cretized by a MacCormack scheme ( N x = 800). Moreover, the results are in accordancewith the steady shock wave solution that can be obtained by solving the stationary (ODEs)system of Navier-Stokes equations [5]. For other relevant tests, we refer to [5]. Acknowledgement
S. Y. Cho has been supported by ITN-ETN Horizon 2020 Project ModCompShock, Mod-eling and Computation on Shocks and Interfaces, Project Reference 642768. S. Y. Cho, S.Boscarino and G. Russo would like to thank the Italian Ministry of Instruction, Universityand Research (MIUR) to support this research with funds coming from PRIN Project 2017(No. 2017KKJP4X entitled “Innovative numerical methods for evolutionary partial differ-ential equations and applications”). S. Boscarino has been supported by the University ofCatania (“Piano della Ricerca 2016/2018, Linea di intervento 2”). S. Boscarino and G.Russo are members of the INdAM Research group GNCS. M. Groppi thanks the support by
GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 25 the University of Parma, by the Italian National Group of Mathematical Physics (GNFM-INdAM), and by the Italian National Research Project “Multiscale phenomena in Contin-uum Mechanics: singular limits, off-equilibrium and transitions” (PRIN 2017YBKNCE).
Appendix A. Leading error terms in Proposition 4.1 and Remark 4.1
A.1.
Proof of Proposition 4.1.
Proof. • Proof of (1) : To show this, we first express u s and u sk in terms of u s and u k in E u : L (cid:88) k =1 ν sk ( u s − u sk ) = L (cid:88) k =1 ν sk (cid:20) (cid:32) u s + 1 m s n s ν s L (cid:88) r =1 ξ sr u r (cid:33) − ((1 − a sk ) u s + a sk u k ) (cid:21) = L (cid:88) k =1 ν sk (cid:20) (cid:32) m s n s ν s L (cid:88) r =1 ξ sr u r (cid:33) − a sk ( u k − u s ) (cid:21) . By the definition of ξ sr in (3.3) and a sk in (3.9), we obtain L (cid:88) k =1 ν sk ( u s − u sk ) = L (cid:88) k =1 ν sk (cid:20) (cid:32) m s n s ν s L (cid:88) r =1 ξ sr u r (cid:33) − a sk ( u k − u s ) (cid:21) = L (cid:88) r =1 (cid:32) ν sr m r n r m s + m r − δ sr L (cid:88) (cid:96) =1 ν s(cid:96) m (cid:96) n (cid:96) m s + m (cid:96) (cid:33) u r − L (cid:88) k =1 (cid:20) λ sk m k n k m s + m k ( u k − u s ) (cid:21) = 0 . In the last line, we use the relation ν s(cid:96) = λ s(cid:96) in (3.4). Proof (2) : To prove this, we begin by splitting in E T (4.1) into two parts: L (cid:88) k =1 ν sk ( T s − T sk ) = I + II + III, where I = L (cid:88) k =1 ν sk (cid:20) n s K B ν s L (cid:88) r =1 γ sr T r − b sk ( T k − T s ) (cid:21) II = 2 m s K B L (cid:88) r =1 ν sr (cid:20) ν sr m r n r ν sr ( m s + m r ) ( m s u s + m r u r ) ( u r − u s ) (cid:21) − L (cid:88) k =1 ν sk γ sk K B | u s − u k | III = − m s K B L (cid:88) k =1 ν sk (cid:20) | u s | − | u s | (cid:21) Inserting γ sr in (3.3) into I , we have I = 23 n s K B L (cid:88) r =1 γ sr T r − L (cid:88) k =1 ν sk b sk ( T k − T s )= 2 L (cid:88) r =1 (cid:32) ν sr m s m r n r ( m s + m r ) − δ sr L (cid:88) (cid:96) =1 ν s(cid:96) m s m (cid:96) n (cid:96) ( m s + m (cid:96) ) (cid:33) T r − L (cid:88) k =1 ν sk b sk ( T k − T s )= 0 . In the last line, we use b sk = 2 a sk m s m s + m k = 2 λ sk m s m k n k ν sk ( m s + m r ) and ν sk = λ sk . Next, we simplify II as II = 2 m s K B L (cid:88) r =1 ν sr (cid:20) a sr (cid:18) u s + m r m s + m r ( u r − u s ) (cid:19) · ( u r − u s ) (cid:21) − L (cid:88) k =1 ν sk γ sk K B | u s − u k | = 2 m s K B L (cid:88) r =1 ν sr (cid:20) a sr u s · ( u r − u s ) + a sr m r m s + m r | u r − u s | (cid:21) − L (cid:88) k =1 ν sk γ sk K B | u s − u k | This combined with γ sk in (3.9) gives II = 2 m s K B L (cid:88) r =1 ν sk a sk (cid:20) u s · ( u k − u s ) (cid:21) + m s K B L (cid:88) k =1 ν sk ( a sk ) | u s − u k | . Now, the following relation | u sk | = | a sk ( u s − u k ) − u s | = ( a sk ) | u s − u k | − a sk ( u s − u k ) · u s + | u s | implies II + III = − m s K B L (cid:88) k =1 ν sk (cid:20) | u s | − | u sk | (cid:21) . To simplify further this, recall the relation u s = u s + 1 ν s L (cid:88) r =1 ν sr a sr ( u r − u s ) , (A.1)and use this to get II + III = − m s K B L (cid:88) k =1 ν sk (cid:34) u s + 1 ν s L (cid:88) r =1 ν sr a sr ( u r − u s ) − a sk ( u s − u k ) (cid:35) · (cid:34) ν s L (cid:88) r =1 ν sr a sr ( u r − u s ) + a sk ( u s − u k ) (cid:35) = A + B where A = − m s K B L (cid:88) k =1 ν sk (cid:34) u s + 1 ν s L (cid:88) r =1 ν sr a sr ( u r − u s ) (cid:35) (cid:34) ν s L (cid:88) r =1 ν sr a sr ( u r − u s ) (cid:35) = − m s ν s K B (cid:34) u s ν s + L (cid:88) r =1 ν sr a sr ( u r − u s ) (cid:35) · (cid:34) L (cid:88) r =1 ν sr a sr ( u r − u s ) (cid:35) B = − (cid:32) m s u s K B · L (cid:88) k =1 ν sk a sk ( u s − u k ) − m s K B L (cid:88) k =1 ν sk ( a sk ) | u s − u k | (cid:33) . To be more concise, we can rewrite A and B as A = − m s u s K B · X − m s ν s K B | X | B = 2 m s u s K B · X + m s K B Y GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 27 where X = L (cid:88) r =1 ν sr a sr ( u r − u s ) , Y = L (cid:88) r =1 ν sr ( a sr ) | u r − u s ) | . Then, we have II + III = m s K B L (cid:88) k =1 ν sk ( a sk ) | u s − u k | − m s ν s K B (cid:32) L (cid:88) r =1 ν sr a sr ( u r − u s ) (cid:33) · (cid:32) L (cid:88) r =1 ν sr a sr ( u r − u s ) (cid:33) , which gives the desired result. (cid:3) A.2.
Calculation of ¯ E u and ¯ E T terms in (4.2) . Proof. • Proof of (1) : We first rewrite ¯ u as¯ u = u s + 1 (cid:80) Lr =1 ν r m r n r (cid:88) r (cid:54) = s ν r m r n r ( u r − u s ) . This, combined with the expression of u s in (A.1), gives¯ E u = ν s ( u s − ¯ u )= ν s ν s (cid:88) r (cid:54) = s (cid:18) ν sr m r n r m s + m r (cid:19) ( u r − u s ) − (cid:80) Lr =1 ν r m r n r (cid:88) r (cid:54) = s ν r m r n r ( u r − u s ) = (cid:88) r (cid:54) = s (cid:32) ν sr m r n r m s + m r − ν s ν r m r n r (cid:80) Lr =1 ν r m r n r (cid:33) ( u r − u s ) . Proof (2) :From the definition of T s and ¯ T , we have¯ E T = ν s ( T s − ¯ T )= ν s (cid:20) T s − m s K B (cid:0) | u s | − | u s | (cid:1) + 23 n s K B ν s L (cid:88) r =1 γ sr T r + 23 n s K B ν s L (cid:88) r =1 ν sr m s m r n s n r ( m s + m r ) ( m s u s + m r u r ) ( u r − u s ) − (cid:32) T s − (cid:80) Lr =1 ν r n r m r (cid:0) | ¯ u | − | u r | (cid:1) K B (cid:80) Lr =1 ν r n r + (cid:80) r (cid:54) = s ν r n r ( T r − T s ) (cid:80) Lr =1 ν r n r (cid:33) (cid:21) = J + J + J , where J = ν s (cid:34) T s + 23 n s K B ν s L (cid:88) r =1 γ sr T r − (cid:32) T s − (cid:80) Lr =1 ν r n r m r (cid:0) | ¯ u | − | u r | (cid:1) K B (cid:80) Lr =1 ν r n r + (cid:80) r (cid:54) = s ν r n r ( T r − T s ) (cid:80) Lr =1 ν r n r (cid:33)(cid:35) J = ν s (cid:34) (cid:80) Lr =1 ν r n r m r (cid:0) | ¯ u | − | u r | (cid:1) K B (cid:80) Lr =1 ν r n r (cid:35) J = L (cid:88) k =1 ν sk (cid:20) − m s K B (cid:0) | u s | − | u s | (cid:1) + 23 K B ν s L (cid:88) r =1 ν sr m s m r n r ( m s + m r ) ( m s u s + m r u r ) ( u r − u s ) (cid:21) . For J , we use γ sr = 3 K B ν sr m s m r n s n r ( m s + m r ) − δ sr K B L (cid:88) (cid:96) =1 ν s(cid:96) m s m (cid:96) n s n (cid:96) ( m s + m (cid:96) ) , to obtain J = ν s (cid:20) ν s (cid:32) ν ss m s m s n s ( m s + m s ) − L (cid:88) (cid:96) =1 ν s(cid:96) m s m (cid:96) n (cid:96) ( m s + m (cid:96) ) (cid:33) T s + 2 ν s (cid:88) r (cid:54) = s ν sr m s m r n r ( m s + m r ) T r − (cid:32) (cid:80) r (cid:54) = s ν r n r ( T r − T s ) (cid:80) Lr =1 ν r n r (cid:33) (cid:21) = (cid:88) r (cid:54) = s (cid:32) ν sr m s m r n r ( m s + m r ) − ν s ν r n r (cid:80) Lr =1 ν r n r (cid:33) ( T r − T s ) . Next, for J , we use | ¯ u | − | u r | = (cid:32) (cid:80) L(cid:96) =1 ν (cid:96) m (cid:96) n (cid:96) ( u (cid:96) − u r ) (cid:80) L(cid:96) =1 ν (cid:96) m (cid:96) n (cid:96) (cid:33) · (cid:32) (cid:80) L(cid:96) =1 ν (cid:96) m (cid:96) n (cid:96) ( u (cid:96) + u r ) (cid:80) L(cid:96) =1 ν (cid:96) m (cid:96) n (cid:96) (cid:33) . Then, we have J + J = (cid:88) r (cid:54) = s (cid:32) ν sr m s m r n r ( m s + m r ) − ν s ν r n r (cid:80) Lr =1 ν r n r (cid:33) ( T r − T s )+ ν s (cid:80) Lr =1 ν r n r m r (cid:104)(cid:16)(cid:80) L(cid:96) =1 ν (cid:96) m (cid:96) n (cid:96) ( u (cid:96) − u r ) (cid:17) · (cid:16)(cid:80) L(cid:96) =1 ν (cid:96) m (cid:96) n (cid:96) ( u (cid:96) + u r ) (cid:17)(cid:105) K B (cid:80) Lr =1 ν r n r (cid:12)(cid:12)(cid:12)(cid:80) L(cid:96) =1 ν (cid:96) m (cid:96) n (cid:96) ) (cid:12)(cid:12)(cid:12) For J , we use | u s | − | u s | = ν s (cid:88) r (cid:54) = s (cid:18) ν sr m r n r m s + m r (cid:19) ( u r − u s ) · u s + 1 ν s (cid:88) r (cid:54) = s (cid:18) ν sr m r n r m s + m r (cid:19) ( u r − u s ) to get J = ν s (cid:20) − m s K B (cid:0) | u s | − | u s | (cid:1) + 23 K B ν s L (cid:88) r =1 ν sr m s m r n r ( m s + m r ) ( m s u s + m r u r ) ( u r − u s ) (cid:21) = − m s K B (cid:88) r (cid:54) = s (cid:18) ν sr m r n r m s + m r (cid:19) ( u r − u s ) · u s + 1 ν s (cid:88) r (cid:54) = s (cid:18) ν sr m r n r m s + m r (cid:19) ( u r − u s ) + 2 m s K B (cid:88) r (cid:54) = s ν sr m r n r m s + m r (cid:18) u s + m r m s + m r ( u r − u s ) (cid:19) ( u r − u s )= − m s K B (cid:88) r (cid:54) = s (cid:18) ν sr m r n r m s + m r (cid:19) ( u r − u s ) · ν s (cid:88) r (cid:54) = s (cid:18) ν sr m r n r m s + m r (cid:19) ( u r − u s ) + 2 m s K B (cid:88) r (cid:54) = s ν sr m r n r ( m s + m r ) | u r − u s | . To sum up, we rewrite ¯ E u in terms of u s and T s with ¯ E u = J + J + J . GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 29 (cid:3)
Appendix B. Representation of NS equations for n s , u and T B.1.
Derivation of (5.8) . We begin with the first equation in (5.1): ∂n s ∂t = −∇ · ( n s u ) − ε ∇ · ( n s u (1) s ) , s = 1 , · · · , L. The sum of these L equations leads to ∂n∂t = −∇ · ( nu ) − ∇ · (cid:32) L (cid:88) s =1 εn s u (1) s (cid:33) ,∂ρ∂t = −∇ · ( ρu ) − ∇ · (cid:32) L (cid:88) i =1 ερ s u (1) s (cid:33) . (B.1)Next, we rewrite the second equation in (5.1) as u ∂ρ∂t + ρ ∂u∂t + u ∇ · ( ρu ) + ρu ∇ · u + ∇ ( nK B T ) + ε ∇ · ( P (1) ) = 0 . This together with (B.1) gives ∂u∂t = uρ (cid:32) ∇ · (cid:32) L (cid:88) i =1 ερ s u (1) s (cid:33)(cid:33) − u ∇ · u − ∇ ( nK B T ) ρ − ε ∇ · ( P (1) ) ρ . (B.2)For the third equation in (5.1), we transform it into the following form: ρu · ∂u∂t + u · ∂∂t ( ρu ) + 3 K B T ∂n∂t + 3 nK B ∂T∂t + ∇ (cid:18) ρ | u | (cid:19) · u + (cid:18) ρ | u | (cid:19) ∇ · u + ∇ (cid:18) nK B T (cid:19) · u + (cid:18) nK B T (cid:19) ∇ · u + ε ∇ · ( P (1) · u ) + ε ∇ · q (1) = 0 . (B.3)To simplify this, we use ∇ ( a · b ) = a × ( ∇ × b ) + b × ( ∇ × a ) + ( a · ∇ ) b + ( b · ∇ ) a , (B.4)to have ∇ (cid:18) ρ | u | (cid:19) = ∇ (cid:16) ρu · u (cid:17) = (cid:16) ρu × ( ∇ × u ) + u × (cid:16) ∇ × ρu (cid:17) + (cid:16) ρu · ∇ (cid:17) u + ( u · ∇ ) ρu (cid:17) . Also, we use the following decomposition: ∇ (cid:18) nK B T (cid:19) · u = 1 ρ ∇ ( nK B T ) · ρu ∇ (cid:18) nK B T (cid:19) · u + ∇ (cid:18) nK B T (cid:19) · u. Using these relations, we write (B.3) in the following form: ρu · (cid:18) ∂u∂t + u · ∇ u + 1 ρ ∇ ( nK B T ) (cid:19) + u · (cid:18) ∂∂t ( ρu ) + ∇ · ( ρu ⊗ u ) + ∇ ( nK B T ) (cid:19) + 3 K B T (cid:18) ∂n∂t + ∇ n · u + n ∇ · u (cid:19) + 3 nK B ∂T∂t + (cid:16) ρu × ( ∇ × u ) + u × (cid:16) ∇ × ρu (cid:17)(cid:17) · u + (cid:16) ( u · ∇ ) ρu (cid:17) · u + (cid:18) ρ | u | (cid:19) ∇ · u − ∇ · ( ρu ⊗ u ) · u (cid:18) nK B ∇ T (cid:19) · u + nK B T ∇ · u + ε ∇ · ( P (1) · u ) + ε ∇ · q (1) = 0 . Recalling (B.2), the second equation in (5.1), (B.1) and (cid:16) ρu × ( ∇ × u ) + u × (cid:16) ∇ × ρu (cid:17)(cid:17) · u = 0 (cid:16) ( u · ∇ ) ρu (cid:17) · u + (cid:18) ρ | u | (cid:19) ∇ · u = ∇ · ( ρu ⊗ u ) · u u · (cid:32) u (cid:32) ∇ · (cid:32) L (cid:88) i =1 ερ s u (1) s (cid:33)(cid:33)(cid:33) − u · (cid:16) ε ∇ · ( P (1) ) (cid:17) + 3 K B T (cid:32) −∇ · (cid:32) L (cid:88) s =1 εn s u (1) s (cid:33)(cid:33) + 3 nK B ∂T∂t + (cid:18) nK B ∇ T (cid:19) · u + nK B T ∇ · u + ε ∇ · ( P (1) · u ) + ε ∇ · q (1) = 0 . This gives the expression in (5.8).B.2.
Derivation of (5.11) . We begin with the first equation in (5.10): ∂n s ∂t = −∇ · ( n s u s ) , s = 1 , · · · , L. (B.5)which implies ∂ρ s ∂t = −∇ · ( ρ s u s ) . Next, we use this to simplify the second equation in (5.10) as − ρ s ∂u s ∂t = u s (cid:18) ∂ρ s ∂t + ∇ · ( ρ s u s ) (cid:19) + ρ s u s ∇ · u s + ∇ ( n s K B T s ) + ε ∇ · ( P (1) s ) − L (cid:88) k (cid:54) = s R sk = ρ s u s ∇ · u s + ∇ ( n s K B T s ) + ε ∇ · ( P (1) s ) − L (cid:88) k (cid:54) = s R sk . This reduces to ∂u s ∂t = − u s ∇ · u s − ∇ ( n s K B T s ) ρ s − ε ∇ · ( P (1) s ) ρ s + 1 ρ s L (cid:88) k (cid:54) = s R sk . (B.6) GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 31
For the third equation in (5.10), we rewrite it as ρ s u s · ∂u s ∂t + u s · ∂∂t ( ρ s u s ) + 3 K B T s ∂n s ∂t + 3 n s K B ∂T s ∂t + ∇ (cid:18) ρ s | u s | (cid:19) · u s + (cid:18) ρ s | u s | (cid:19) ∇ · u s + ∇ (cid:18) n s K B T s (cid:19) · u s + (cid:18) n s K B T s (cid:19) ∇ · u s + ε ∇ · ( P (1) s · u s ) + ε ∇ · q (1) s = L (cid:88) k (cid:54) = s S sk . (B.7)To simplify this, we use (B.4) to obtain ∇ (cid:18) ρ s | u s | (cid:19) · u s = (cid:16) ρ s u s × ( ∇ × u s ) + u s × (cid:16) ∇ × ρ s u s (cid:17) + (cid:16) ρ s u s · ∇ (cid:17) u s + ( u s · ∇ ) ρ s u s (cid:17) · u s . Using this and the following decomposition: ∇ (cid:18) n s K B T s (cid:19) · u s = 1 ρ s ∇ ( n s K B T s ) · ρ s u s ∇ (cid:18) n s K B T s (cid:19) · u s + ∇ (cid:18) n s K B T s (cid:19) · u s . we can rewrite (B.7) as ρ s u s · (cid:18) ∂u s ∂t + u s · ∇ u s + 1 ρ s ∇ ( n s K B T s ) (cid:19) + u s · (cid:18) ∂∂t ( ρ s u s ) + ∇ · ( ρ s u s ⊗ u s ) + ∇ ( n s K B T s ) (cid:19) + 3 K B T s (cid:18) ∂n s ∂t + ∇ n s · u s + n s ∇ · u s (cid:19) + 3 n s K B ∂T s ∂t + (cid:16) ρ s u s × ( ∇ × u s ) + u s × (cid:16) ∇ × ρ s u s (cid:17)(cid:17) · u s + (cid:16) ( u s · ∇ ) ρ s u s (cid:17) · u s + (cid:18) ρ s | u s | (cid:19) ∇ · u s − ∇ · ( ρ s u s ⊗ u s ) · u s (cid:18) n s K B ∇ T s (cid:19) · u s + n s K B T s ∇ · u s + ε ∇ · ( P (1) s · u s ) + ε ∇ · q (1) s = L (cid:88) k (cid:54) = s S sk . Finally, we use (B.6), the second equation in (5.10), (B.5) and the following relations: (cid:16) ρ s u s × ( ∇ × u s ) + u s × (cid:16) ∇ × ρ s u s (cid:17)(cid:17) · u s = 0 (cid:16) ( u s · ∇ ) ρ s u s (cid:17) · u s + (cid:18) ρ s | u s | (cid:19) ∇ · u s = ∇ · ( ρ s u s ⊗ u s ) · u s to derive u s · − ε ∇ · ( P (1) s ) + L (cid:88) k (cid:54) = s R sk + 3 n s K B ∂T s ∂t + (cid:18) n s K B ∇ T s (cid:19) · u s + n s K B T s ∇ · u s + ε ∇ · ( P (1) s · u s ) + ε ∇ · q (1) s = L (cid:88) k (cid:54) = s S sk . This gives the expression in (5.11).
References
1. A. Aimi, M. Diligenti, M. Groppi, C. Guardasoni, On the numerical solution of a BGK-type model forchemical reactions, Eur. J. Mech. B Fluids, (2007), 455–472.2. P. Andries, K. Aoki, B. Perthame, A consistent BGK-type model for gas mixtures, J. Stat. Phys., (2002), 993–1018.3. M. Bisi, A. V. Bobylev, M. Groppi, G. Spiga, Hydrodynamic equations from a BGK model for inert gasmixtures, In: AIP Conference Proceedings, AIP Publishing LLC, (2019), 130010.4. M. Bisi, M. Groppi, G. Martal`o, Macroscopic equations for inert gas mixtures in different hydrodynamicregimes, J. Phys. A: Math. and Theor., (2021), 085201.5. M. Bisi, M. Groppi, G. Martal`o, The evaporation–condensation problem for a binary mixture of rarefiedgases. Contin. Mech. Thermodyn., (2020), 1109–1126.6. M. Bisi, M. Groppi, G. Spiga, Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and itshydrodynamic limit, Phys. Rev. E, (2010), 036327.7. M. Bisi, G. Spiga, Navier–Stokes hydrodynamic limit of BGK kinetic equations for an inert mixture ofpolyatomic gases, In: “From Particle Systems to Partial Differential Equations V” (eds. P. Goncalvesand A. J. Soares), Springer Proceedings in Mathematics and Statistics, (2018), 13–31.8. S. Boscarino, S. Y. Cho, G. Russo and S.-B. Yun, High order conservative Semi-Lagrangian scheme forthe BGK model of the Boltzmann equation, Commun. Comput. Phys., (2021), 1–56.9. S. Boscarino, S. Y. Cho, G. Russo, S.-B. Yun, Convergence estimates of a semi-Lagrangian scheme forthe ellipsoidal BGK model for polyatomic molecules. arXiv preprint arXiv:2003.00215. (2020)10. P. L. Bhatnagar, E. P. Gross and K. Krook, A model for collision processes in gases, Phys. Rev., (1954), 511–524.11. A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga, I. F. Potapenko, A general consistent BGK model for gasmixtures, Kinet. Relat. Models, (2018), 1377.12. S. Y. Cho, S. Boscarino, G. Russo, S.-B. Yun, Conservative semi-Lagrangian schemes for kinetic equa-tions Part I: Reconstruction, J. Comput. Phys, (2021), 110159.13. S. Y. Cho, S. Boscarino, G. Russo, S.-B. Yun, Conservative semi-Lagrangian schemes for kinetic equa-tions Part II: Applications, arXiv preprint, arXiv:2007.13166, (2020).14. C. Cercignani, The Boltzmann Equation and its Applications , Springer, New York, 1988.15. S. Y. Cho, S. Boscarino, M. Groppi. G. Russo, Conservative semi-Lagrangian schemes for a generalconsistent BGK model for inert gas mixtures, arXiv preprint, arXiv:2012.02497, (2020).16. C. K. Chu, Kinetic-theoretic description of the formation of a shock wave, Phys. Fluids, (1965), 12–22.17. I. Cravero, G. Puppo, M. Semplice and G.Visconti, CWENO: uniformly accurate reconstructions forbalance laws. Math. Comp., (2018), 1689–1719.18. V. S. Galkin, N. K. Makashev, Kinetic derivation of the gas-dynamic equation for multicomponentmixtures of light and heavy particles, Fluid Dyn., (1994), 140–155.19. M. Groppi, S. Rjasanow, G. Spiga, A kinetic relaxation approach to fast reactive mixtures: shock wavestructure, J. Stat. Mech. Theory Exp., (2009), P10010.20. M. Groppi, G. Russo and G. Stracquadanio, High order semi-Lagrangian methods for the BGK equation,Commun. Math. Sci., (2016), 389–414.21. M. Groppi, G. Russo, G. Stracquadanio, Boundary conditions for semi-Lagrangian methods for theBGK model, Commun. Appl. Ind. Math., (2016), 138–164.22. M. Groppi, G. Russo and G. Stracquadanio, Semi-Lagrangian Approximation of BGK Models for Inertand Reactive Gas Mixtures, In: “From Particle Systems to Partial Differential Equations V” ((Eds.) P.Gon¸calves and A. Soares), Springer Proceedings in Mathematics and Statistics (2018), 53–80. GK MODELS FOR INERT MIXTURES: COMPARISON AND APPLICATIONS 33
23. M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures,Phys. Fluids, (2004), 4273–4284.24. J.R. Haack, C.D Hauck, M.S. Murillo, A conservative, entropic multispecies BGK model, J. Stat. Phys., (2017), 826–856.25. J. Kestin, K. Knierim, E. A. Mason, B. Najafi, S. T. Ro, M. Waldman, Equilibrium and transportproperties of the noble gases and their mixtures at low density, J. Phys. Chem. Ref. Data, (1984),229–303.26. C. Klingenberg, M. Pirner, Existence, uniqueness and positivity of solutions for BGK models for mix-tures, J. Differential Equations, (2018), 702–727.27. C. Klingenberg, M. Pirner, G. Puppo, A consistent kinetic model for a two-component mixture with anapplication to plasma, Kinet. Relat. Models, (2017), 445–465.28. M.N. Kogan, Rarefied Gas Dynamics , Plenum Press, New York, 1969.29. D. Levy, G.Puppo, G.Russo, Central WENO schemes for hyperbolic systems of conservation laws,ESAIM: Math. Model. Numer. Anal., (1999), 547–571.30. D. Madjarevi´c and S. Simi´c, Shock structure in helium-argon mixture-a comparison of hyperbolic multi-temperature model with experiment, EPL, (2013), 44002.31. T. Ruggeri and S. Simi´c, On the hyperbolic system of a mixture of Eulerian fluids: a comparison betweensingle- and multi-temperature models, Math. Methods Appl. Sci., (2007), 827–849.32. G. Russo, F. Filbet, Semilagrangian schemes applied to moving boundary problems for the BGK modelof rarefied gas dynamics, Kinet. Relat. Models,, (2009), 231–250.33. G. Russo and P. Santagati, A new class of large time step methods for the BGK models of the Boltzmannequation, arXiv preprint, arXiv:1103.5247v1, 2011.34. G. Russo, P. Santagati, S.-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of theBoltzmann equation, SIAM J. Numer. Anal., (2012), 1111–1135, .35. G. Russo and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model ofthe Boltzmann equation, SIAM J. Numer. Anal., (2018), 3580–3610, .36. S. Simi´c, M. Pavic-Colic and D. Madjarevi´c, Non-equilibrium mixtures of gases: modelling and compu-tation, Riv. di Mat. della Univ. di Parma, (2015), 135–214.37. J. Vranjes and P.S. Krstic, Collisions, magnetization, and transport coefficients in the lower solar atmo-sphere, Astron. Astrophys., (2013), A22. Sebastiano Boscarino, Department of Mathematics and Computer Science, University of Cata-nia, 95125 Catania, Italy
Email address : [email protected] Seung Yeon Cho, Department of Mathematics and Computer Science, University of Catania,95125 Catania, Italy
Email address : [email protected] Maria Groppi, Department of Mathematical, Physical and Computer Sciences, University ofParma, Parco Area delle Scienze 53/A, I–43124 Parma, Italy
Email address : [email protected] Giovanni Russo, Department of Mathematics and Computer Science, University of Catania,95125 Catania, Italy
Email address ::