Kinetic/Fluid micro-macro numerical scheme for a two component gas mixtures

Kinetic/Fluid micro-macro numerical scheme for a two component gas mixtures

A. Crestetto, C. Klingenberg, and M. Pirner
Abstract

This work is devoted to the numerical simulation of the BGK equation for two species in the fluid limit using a particle method. Thus, we are interested in a gas mixture consisting of two species without chemical reactions assuming that the number of particles of each species remains constant. We consider the kinetic two species model proposed by Klingenberg, Pirner and Puppo in [16], which separates the intra and interspecies collisions. We want to study numerically the influence of the two relaxation term, one corresponding to intra, the other to interspecies collisions. For this, we use the method of micro-macro decomposition. First, we derive an equivalent model based on the micro-macro decomposition (see Bennoune, Lemou and Mieussens [2] and Crestetto, Crouseilles and Lemou [6]). The kinetic micro part is solved by a particle method, whereas the fluid macro part is discretized by a standard finite volume scheme. Main advantages of this approach are: (i) the noise inherent to the particle method is reduced compared to a standard (without micro-macro decomposition) particle method, (ii) the computational cost of the method is reduced in the fluid limit since a small number of particles is then sufficient.

111We certify that the general content of the manuscript, in whole or in part, is not submitted, accepted or published elsewhere, including conference proceedings.

Keywords: Two species mixture, kinetic model, BGK equation, micro-macro decomposition, particles method.

AMS subject classsification: 65M75, 82C40, 82D10, 35B40.

1 Introduction

We want to model a gas mixture consisting of two species. The kinetic description of a plasma is based on the BGK equation. In [6], Crestetto, Crouseilles and Lemou developed a numerical simulation of the Vlasov-BGK equation in the fluid limit using particles. They consider a Vlasov-BGK equation for the electrons and treat the ions as a background charge. In [6] a micro-macro decomposition is used as in [2] where asymptotic preserving schemes have been derived in the fluid limit. In [6], the approach in [2] is modified by using a particle approximation for the kinetic part, the fluid part being always discretized by standard finite volume schemes. Other approaches where kinetic description of one species is written in a micro-macro decomposition can be seen in [7, 8].
In this paper, we want to model two species by a system of two BGK equations. Such a multi component kinetic description of the gas mixture has for example importance in modelling applications in air, since air is a gas mixture. We want to consider applications where the gas mixture is close to a fluid in some regions, but the kinetic description is mandatory in some other regions. For this, we want to use the approach in [6], since it has the following advantages: the presented scheme has a much less level of noise compared to the standard particle method and the computational cost of the micro-macro model is reduced in the fluid regime since a small number of particles is needed for the micro part.
From the modelling point of view, we want to describe this gas mixture using two distribution functions via the BGK equation with interaction terms on the right-hand side. BGK models give rise to efficient numerical computations, see for example [18, 12, 11, 2, 10, 3, 6]. In the literature one can find two types of models for gas mixtures. The Boltzmann equation for gas mixtures contains a sum of collision terms on the right-hand side. One type of BGK model for gas mixtures also has a sum of collision terms in the relaxation operator. One example is the model of Klingenberg, Pirner and Puppo [16] which we will consider in this paper. It contains the often used models of Gross and Krook [13] and Hamel [14] as special cases. The other type of model contains only one collision term on the right-hand side. Example of this is the well-known model of Andries, Aoki and Perthame in [1].
In this paper we are interested in the first type of models, and use the model developed in [16]. In this type of model the two different types of interactions, interactions of a species with itself and interactions of a species with the other one, are kept separated. Therefore, we can see how these different types of interactions influence the trend to equilibrium. From the physical point of view, we expect two different types of trends to equilibrium. For example, if the collision frequencies of the particles of each species with itself are larger compared to the collision frequencies related to interspecies collisions, we expect that we first observe that the relaxation of the two distribution functions to its own equilibrium distribution is faster compared to the relaxation towards a common velocity and a common temperature. This effect is clearly seen in the model presented in [16] since the two types of interactions are separated.

The outline of the paper is as follows: In section 2 we present the model for a gas mixture consisting of two species and write it in dimensionless form. In section 3 we derive the micro-macro decomposition of the model presented in section 2. In section 4 we prove some convergence rates in the space-homogeneous case of the distribution function to a Maxwellian distribution and of the two velocities and temperatures to a common value which we will verify numerically later on. In section 5, we briefly present the numerical approximation, based on a particle method for the micro equation and a finite volume scheme for the macro one. In section 6, we present some numerical examples. First, we verify numerically the convergence rates obtained in section 4. Then, in the general case, we are interested in the evolution in time of the system. We consider different possibilities for the values of the collision frequencies. When the collision frequencies are very large we observe relaxations towards Maxwellian distributions. Finally, if we vary the relationships between the different collision frequencies, we observe a corresponding variation in the speed of relaxation towards Maxwellians and the relaxation towards a common value of the mean velocities and temperatures. Finally, section 7 presents a brief conclusion.

2 The two-species model

In this section we present in 1D the BGK model for a mixture of two species developed in [16] and mention its fundamental properties like the conservation properties. Then, we present its dimensionless form.

2.1 1D BGK model for a mixture of two species

We consider a gas mixture consisting of two species denoted by the index and 2. Thus, our kinetic model has two distribution functions and where , are the phase space variables and the time.

Furthermore, for any with
, we relate the distribution functions to macroscopic quantities by mean-values of ,

 ∫fk(v)⎛⎜⎝1vmk|v−uk|2⎞⎟⎠dv=:⎛⎜⎝nknkuknkTk⎞⎟⎠,k=1,2, (1)

where is the mass, the number density, the mean velocity and the mean temperature of species , . Note that in this paper we shall write instead of , where is Boltzmann’s constant.

We want to model the time evolution of the distribution functions by BGK equations. Each distribution function is determined by one BGK equation to describe its time evolution. The two equations are coupled through a term which describes the interaction of the two species. We consider binary interactions. So the particles of one species can interact with either themselves or with particles of the other species. In the model this is accounted for introducing two interaction terms in both equations. Here, we choose the collision terms as BGK operators, so that the model writes

 ∂tf1+v∂xf1+F1m1 ∂vf1=ν11n1(M1−f1)+ν12n2(M12−f1),∂tf2+v∂xf2+F2m2 ∂vf2=ν22n2(M2−f2)+ν21n1(M21−f2), (2)

with the mean-field or external forces and and the Maxwell distributions

 Mk(x,v,t)=nk√2πTkmkexp(−|v−uk|22Tkmk),k=1,2,Mkj(x,v,t)=nkj√2πTkjmkexp(−|v−ukj|22Tkjmk),k,j=1,2,k≠j, (3)

where and are the collision frequencies of the particles of each species with itself, while and are related to interspecies collisions. To be flexible in choosing the relationship between the collision frequencies, we now assume the relationship

 ν12=εν21,ν22=β2ν21=β2εν12,0<ε≤1, β1,β2>0. (4)

The restriction on is without loss of generality. If , exchange the notation and and choose In addition, we take into account an acceleration due to interactions using a mean-field or a given external forces . In the following we will omit the forces and for simplicity, but the following work can be extended to the equations with forces in a straightforward way.

The functions are submitted to the following periodic condition

 fk(0,v,t) =fk(Lx,v,t), for everyv∈R,t≥0,

together with an initial condition

 fk(x,v,0)=f0k(x,v),for everyx∈[0,Lx],v∈R.

The Maxwell distributions and in (3) have the same moments as and respectively. With this choice, we guarantee the conservation of mass, momentum and energy in interactions of one species with itself (see section 2.2 in [16]). The remaining parameters and will be determined using conservation of total momentum and energy, together with some symmetry considerations.

If we assume that

 n12=n1andn21=n2, (5) u12=δu1+(1−δ)u2,δ∈R, (6) T12=αT1+(1−α)T2+γ|u1−u2|2,0≤α≤1,γ≥0, (7)

we have conservation of the number of particles, of total momentum and total energy provided that

 u21 =u2−m1m2ε(1−δ)(u2−u1),and (8) T21=[εm1(1−δ)(m1m2ε(δ−1)+δ+1)−εγ]|u1−u2|2+ε(1−α)T1+(1−ε(1−α))T2, (9)

see theorem 2.1, theorem 2.2 and theorem 2.3 in [16].

In order to ensure the positivity of all temperatures, we need to impose restrictions on and given by

 0≤γ≤m1(1−δ)[(1+m1m2ε)δ+1−m1m2ε],and (10) m1m2ε−11+m1m2ε≤δ≤1, (11)

see theorem 2.5 in [16].

2.2 Dimensionless form

We want to write the BGK model presented in subsection 2.1 in dimensionless form in order to do the numerical experiments with dimensionless quantities. The principle of non-dimensionalization can also be found in chapter 2.2.1 in [19] for the Boltzmann equation and in [4] for macroscopic equations. First, we define dimensionless variables of the time , the length , the velocity , the distribution functions , the number densities , the mean velocities , the temperatures and of the collision frequency per density . Then, dimensionless variables of the other collision frequencies can be derived by using the relationships (4). We start with choosing typical scales denoted by a bar.

 t′=t/¯t,   x′=x/¯x,   v′=v/¯v,

where is the typical order of magnitude of the density of species 1 and the typical order of magnitude of the density of the species 2 in the volume . Further, we choose

 n′1=n1/¯n1,   n′2=n2/¯n2,
 u′1=u1/¯u1,   u′2=u2/¯u2,   ¯u2=¯u1=¯v,
 T′1=T1/¯T1,   T′2=T2/¯T2,   ¯T2=¯T1=m1¯v2,
 ν′ie=νie/¯νie.

We want to make the following assumptions on the gas mixture regime.

Assumptions 2.1.

We assume

 ¯n1=¯n2=:¯n,¯u1=¯u2=¯v,¯T:=¯T2=¯T1=m1¯v2,

and the assumptions on the collision frequencies (4).

Now, we want to write equations (2) in dimensionless variables. We start with the Maxwellians (3) and with (6)-(9). We replace the macroscopic quantities and in by their dimensionless expressions and obtain

 M1=n′1¯n√2π¯T1T′1m1exp(−|v′¯v−u′1¯u1|2m12T′1¯T1) (12)

by using the first assumption of assumptions 2.1. By the third assumption of assumptions 2.1, we obtain

 M1=¯n¯vn′1√2πT′1exp(−|v′−u′1|22T′1)=:¯n¯vM′1. (13)

In the Maxwellian we again assume the first and third assumption in assumptions 2.1 and obtain in the same way as for

 Me=¯n¯v(memi)12n′e√2πT′eexp(−|v′−u′e|22T′ememi)=:¯n¯vM′e. (14)

Now, we consider the Maxwellian in (3), its velocity in (6) and its temperature in (7). Now, we use the first, second and third assumption of assumptions 2.1 and obtain

 u12=δu′1¯u1+(1−δ)u′2¯u2=(δu′1+(1−δ)u′2)¯v=:¯vu′12,T12=αT′1¯T1+(1−α)T′2¯T2+γ|¯v|2|u′1−u′2|2=m1|¯v|2[αT′1+(1−α)T′2+γm1|u′1−u′2|2]=:|¯v|2m1T′12,M12=n′1¯n√2π¯v2T′12exp(−|v′−u′12|22T′12)=:¯n¯vM′12. (15)

With the same assumptions we obtain for , and in a similar way the expressions

 u21 =[(1−m1m2ε(1−δ))u′2+m1m2ε(1−δ)u′1]¯v=:u′21¯v, T21 =[(1−ε(1−α))T′2+ε(1−α)T′1]¯T +(εm1(1−δ)(m1m2ε(δ−1)+δ+1)−εγ)|u′1−u′2|2|¯v|2 =[(1−ε(1−α))T′2+ε(1−α)T′1]|¯v|2m2m1m2 +(εm1(1−δ)(m1m2ε(δ−1)+δ+1)−εγ)|u′1−u′2|2|¯v|2=:|¯v|2m2m1m2T′21, M21 =¯n¯vm2m1n′2√2πT′21exp(−|v′−u′21|22T′21m2m1)=:¯n¯vM′21.

Now we replace all quantities in (2) by their non-dimensionalized expressions. For the left-hand side of the equation for the species 1 we obtain

 ∂tf1+v∂xf1=1¯t¯n¯v∂t′f′1+1¯x¯n¯v¯vv′∂x′f′1 (16)

and for the right-hand side using (4), (13) and (15), we get

 ν11n1(M1−f1)+ν12n2(M12−f1)=ν12β1n1(M1−f1)+ν12n2(M12−f1)=β1¯ν12¯n2¯vν′12n′1(M′1−f′1)+¯ν12¯n2¯vν′12n′2(M′12−f′1). (17)

Multiplying by and dropping the primes in the variables leads to

 ∂tf1+¯t¯v¯xv∂xf1=β1¯ν12¯t ¯n ν12n1(M1−f1)+¯ν12¯t ¯n ν12n2(M12−f1).

In a similar way we obtain for the second species

 ∂tf2

and the non-dimensionalized Maxwellians given by

 M1(x,v,t)=n1√2πT1exp(−|v−u1|22T1),M2(x,v,t)=n2√2πT2(m2m1)12exp(−|v−u2|22T2m2m1),M12(x,v,t)=n1√2πT12exp(−|v−u12|22T12),M21(x,v,t)=n2√2πT21(m2m1)12exp(−|v−u21|22T21m2m1), (18)

with the non-dimensionalized macroscopic quantities

 u12 =δu1+(1−δ)u2, (19) T12 =αT1+(1−α)T2+γm1|u1−u2|2, (20) u21 =(1−m1m2ε(1−δ))u2+m1m2ε(1−δ)u1, (21) T21=[(1−ε(1−α))T2+ε(1−α)T1]+(ε(1−δ)(m1m2ε(δ−1)+δ+1)−εγm1)|u1−u2|2. (22)

Defining dimensionless parameters

 A=¯t¯v¯x,1ε1=β1¯ν12¯t ¯n,1~ε1=¯ν12¯t ¯n,1ε2=β2ε¯ν12¯t ¯n,1~ε2=1ε¯ν12¯t ¯n, (23)

we get

 ∂tf1+A v∂xf1=1ε1ν12n1(M1−f1)+1~ε1ν12n2(M12−f1),∂tf2+A v∂xf2=1ε2ν12n2(M2−f2)+1~ε2ν12n1(M21−f2). (24)

In the sequel, parameters , , and are referred to as Knudsen numbers. In addition, we want to write the moments (1) in non-dimensionalized form. We can compute this in a similar way as for (2) and obtain after dropping the primes

 ∫fkdv=nk,∫vfkdv=nkuk,k=1,2,1n1∫|v−u1|2f1dv=T1,m2m11n2∫|v−u2|2f2dv=T2. (25)

3 Micro-Macro decomposition

In this section, we derive the micro-macro model equivalent to (24).

First, we take the dimensionless equations (24) and choose . The choice means .

Now, we propose to adapt the micro-macro decomposition presented in [2] and [6]. It is used for numerical methods to solve Boltzmann-like equations for mixtures to capture the right compressible Navier-Stokes dynamics at small Knudsen numbers. The idea is to write each distribution function as the sum of its own equilibrium part (verifying a fluid equation) and a rest (of kinetic-type). So, we decompose and as

 f1=M1+g11,f2=M2+g22. (26)

Let us introduce and the notation . Since and (resp. and ) have the same moments: (resp. ), then the moments of (resp. ) are zero:

 ∫m(v)g11dv=∫m(v)g22dv=0. (27)

With this decomposition we get from equation (24) of species 1 in dimensionless form

 (28)

and a similar equation for species 2.

Now we consider the Hilbert spaces such that , , with the weighted inner product . We consider the subspace span , . Let the orthogonal projection in on this subspace . This subspace has the orthonormal basis

 ~Bk={1√nkMk,(v−uk)√Tkm1/mk1√nkMk,(|v−uk|22Tkm1/mk−12)1√nkMk}=:{bk1,bk2,bk3}.

Using this orthonormal basis of , one finds for any function the following expression of

 ΠMk(ϕ)=3∑n=1(ϕ,bkn)bkn =1nk[⟨ϕ⟩+(v−uk)⋅⟨(v−uk)ϕ⟩Tkm1/mk +(|v−uk|22Tkm1/mk−12)2⟨(|v−uk|22Tkm1/mk−12)ϕ⟩]Mk. (29)

This orthogonal projection has some elementary properties.

Lemma 3.0.1 (Properties of ΠMk).

We have, for ,

 (\mathds1−ΠMk)(Mk)=(\mathds1−ΠMk)(∂tMk)=0, ΠMk(gkk)=ΠMk(∂tgkk)=0,

and

 ΠM1(M12)=(1 +(v−u1)(u12−u1)T1 +(|v−u1|22T1−12)(T12T1+|u12−u1|2T1−1))M1, (30) ΠM2(M21)=(1 +(v−u2)(u21−u2)T2m1/m2 +(|v−u2|22T2m1/m2−12)(T21T2+|u21−u2|2T2m1/m2−1))M2. (31)
Proof.

The proof of the first five equalities is analogue to the one species case and is given in [2]. Besides, using the explicit expression of , , given by (29) we obtain (30)-(31) by direct computations. ∎

Now we apply the orthogonal projection to (28), use lemma 3.0.1 and obtain

 ∂tg11 +(\mathds1−ΠM1)(v∂xM1)+(\mathds1−ΠM1)(v∂xg11) =1~εiν12n2(M12−ΠM1(M12))−(1ε1ν12n1+1~ε1ν12n2)g11.

Again with lemma 3.0.1 we replace by its explicit expression

 ∂tg11+(\mathds1−ΠM1)(v∂xM1)+(\mathds1−ΠM1)(v∂xg11)=1~ε1ν12n2(M12−(1+(v−u1)(u12−u1)T1+(|v−u1|22T1−12)(T12T1+1T1|u12−u1|2−1))M1)−(1ε1ν12n1+1~ε1ν12n2)g11. (32)

We take the moments of equation (28), use (27), and we get

 ∂t⟨m(v)M1⟩+∂x⟨m(v)vM1⟩+∂x⟨m(v)vg11⟩=1~ε1ν12n2(⟨m(v)(M12−M1)⟩). (33)

In a similar way, we get an analogous coupled system for species 2 which is coupled with the system of the ions

 ∂tg22+(\mathds1−ΠM2)(v∂xM2)+(\mathds1−ΠM2)(v∂xg22)=1~ε2ν12n1(M21−(1+(v−u2)(u21−u2)T2m2m1+(|v−u2|22T2m2m1−12)(T21T2+m2m1T2|u21−u2|2−1))M2)−(1ε2ν12n2+1~ε2ν12n1)g22, (34) ∂t⟨mM2⟩+∂x⟨m(vM2)⟩+∂x⟨m(vg22)⟩=1~ε2ν12n1(⟨m(M21−M2)⟩). (35)

Now we have obtained a system of two microscopic equations (32), (34) and two macroscopic equations (33), (35). One can show that this system is an equivalent formulation of the BGK equations for species 1 and species 2. This is analogous to what is done in [6].

4 Space-homogeneous case

In this section, we consider our model (24) in the space-homogeneous case, where we can prove an estimation of the decay rate of , and

In the space-homogeneous case, the BGK model for mixtures (2) simplifies to

 ∂tf1=1ε1ν12n1(M1−f1)+1~ε1ν12n2(M12−f1),∂tf2=1ε2ν12n2(M2−f2)+1~ε2ν12n2(M21−f2), (36)

and we let the reader adapt the micro-macro decomposition (32)-(33)-(34)-(35) to this case.

4.1 Decay rate for the BGK model for mixtures in the space-homogeneous case

We denote by the entropy of a function and by the relative entropy of and .

Theorem 4.1.1.

In the space homogeneous case we have the following decay rate of the distribution functions and

 ||fk−Mk||L1(dv)≤4e−12Ct[H(f01|M01)+H(f02|M02)]12,k=1,2,

where is a constant.

Proof.

We consider the entropy production of species defined by

 D1(f1,f2)=−∫1ε1ν12n1lnf1(M1−f1)dv−∫1~ε1ν12n2lnf1(M