HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS. 1. Introduction

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1 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS QIN LI, JIANFENG LU, AND WEIRAN SUN Abstract. We stud half-space linear kinetic equations with general boundar conditions that consist of both gien incoming data and arious tpe of reflections, etending our preious work [LLS] on halfspace equations with incoming boundar conditions. As in [LLS], the main technique is a damping adding-remoing procedure. We establish the well-posedness of linear or linearized half-space equations with general boundar conditions and quasi-optimalit of the numerical scheme. The numerical method is alidated b eamples including a two-species transport equation, a multi-frequenc transport equation, and the linearized BGK equation in D elocit space.. Introduction In this paper we propose an efficient numerical method for linear half-space kinetic equations with general boundar conditions f + Lf =, in, V, f > =. h + K f <, on =, where the densit function f, R m with m for [, and =, =,,, d V. Tpical eamples for the elocit space V include the whole space R d, as in the case of the Boltzmann equation, and V = [, ] as in the case of the transport equation. B allowing higher-dimensions in f and, we include multi-species models and models with multi-dimensional elocit ariables such as the linearized Boltzmann and linearized BGK equations. The setup also includes the multi-frequenc case where the frequenc ariable can be treated as an inde for multi-species after discretization. The operator L in. is a linear operator, eamples of which include the scattering operator in the linear transport equations, the collision operator in the linearized Boltzmann equations and the linearized BGK equation. The operator K is the boundar operator which characterizes arious tpes of reflections at the boundar. Two classical eamples for the reflections are the diffuse and specular reflections. It will be discussed in details in Section. that our method applies to a general class of boundar operators including Mawell boundar condition linear combination of the diffuse and specular reflection, bounceback reflection, and also the more general linearized Cercignani-Lampis boundar condition. It is well known that to ensure the well-posedness of equation., one needs to prescribe suitable boundar conditions at =. The precise conditions were first formulated in [CGS88] for the linearized Boltzmann equations with prescribed incoming data. This tpe of well-posedness result has been etended to general linear/linearized half-space equations and weakl nonlinear half-space equations with both incoming and Mawell boundar conditions see e.g., [Gol8, ST, UYY3] and also to discrete Boltzmann equation Date: September 9,. Mathematics Subject Classification. 3Q; N3. We would like to epress our gratitude to the support from the NSF research network grant RNMS-7 KI-Net. The research of Q.L. was supported in part b the AFOSR MURI grant FA and the National Science Foundation under award DMS The research of J.L. was supported in part b the Alfred P. Sloan Foundation and the National Science Foundation under award DMS-39 and DMS-939. The research of W.S. was supported in part b the Simon Fraser Uniersit President s Research Start-up Grant PRSG and NSERC Discoer Indiidual Grant #.

2 QIN LI, JIANFENG LU, AND WEIRAN SUN with general boundar conditions [Ber8,Ber]. This is also the setting that we use for deeloping numerical methods. Now we briefl eplain the details of the formulation of the boundar condition at infinit. Denote the null space of L as Null L which is assumed to be finite-dimensional. Let P be the L -projection operator onto Null L and P as the corresponding orthogonal projection operator such that Define the operator P : Null L Null L as P : L dσ Null L, P = I P. P f = P f for f Null L. It is clear that P is a smmetric operator on a finite-dimensional space, and hence all its eigenalues are real. Denote the eigenspaces of P associated with positie, negatie, and zero eigenalues as H +, H, H respectiel. Then Null L is decomposed as Null L = H + H H. Using these notations, we prescribe the boundar conditions at = in a similar wa as in [CGS88] such that lim f H + H. The complete form of the kinetic equation considered in this paper reads f + L f =, f > = h + K f <, =,. lim f H + H, More specific assumptions regarding L and K to guarantee the well-posedness of. will be discussed in Section. Half-space equations with general boundar conditions are frequentl encountered in electric propulsion for satellites [GK8] and photon transport in solid state deices [HM,HRB], among man other applications. The standard treatment of this tpe of equations is the Monte Carlo method [HRB]. There are also special cases where analtical solutions are possible [HM]. In [LLS] we deeloped a direct sstematic method to sole half-space equations in the case of pure incoming boundar condition when K =. There are also other direct numerical approaches for this case proposed in [Cor9, GK9]. Compared with our approach, these methods suffer from seere Gibbs phenomena and lack of error analsis or sstematic strateg to reduce numerical errors. We also note that the method for linearized discrete equations in [Ber8] can be applied to sole the continuous half-space equation b approimating it using discrete elocit models. Unlike [Ber8] which focuses on the analsis of the discrete model, our goal here is to approimate the solutions to the continuous half-space equation using a spectral tpe method with conergence analsis. The present work etends our preious method to the case when arious reflections are inoled. The main difficulties that we need to oercome are the degenerac of L, the deriation of a proper weak formulation inoling K, and the fact that the boundar conditions at = are part of the solution instead of being prescribed. To this end, we appl similar procedure proposed in [LLS], which combines and etends the ideas of een-odd decomposition [ES] and a damping adding-remoing procedure [UYY3, Gol8]. More specificall, we first modif L b adding damping terms to it see Section.. This will remoe the degenerac of L and ensure that the end-state of the damped solution at = is zero. Both analsis and numerical schemes are then performed on the weak formulation of the damped equation, which is deried b appling an een-odd decomposition with mied regularit [ES] of the damped solution f. One

3 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS 3 important adantage of the een-odd decomposition is that it leads to a natural wa of constructing a famil of basis functions that captures the possible jump discontinuit of the solution at =, b the odd etension of the basis functions constructed for positie. This discretization of the elocit space based on een-odd decomposition turns out to be equialent to the double-p n method deeloped in the literature of soling neutron transport equations, see e.g., [Tho3]. We also comment that the appearance of the boundar operator K introduces etra difficult into formulating the weak form of the half-space equation. The difficult comes from the fact that onl the een part of the solution f + has enough regularit to define a trace on the boundar. Our main idea here is to use the properties assumed for K in Section to represent the odd part f on the boundar in terms of f +. Our numerical method is spectral in nature: we appl Galerkin approimations to the weak formulation and use Babuška-Aziz lemma to show that the damped equation is well-posed and the finite-dimensional approimation is quasi-optimal. It will be clear that the damping plas a crucial role here. Finall, we make use of the linearit and use proper superposition of certain special solutions to the damped equation to recoer the original undamped solution. A b-product of the aboe procedure is that we obtain a unified proof for the well-posedness of the halfspace equations with general boundar conditions. This well-posedness theor is general enough to include multi-species and multi-dimensional in elocit half-space equations. The laout of the paper is as follows. In Section we eplain all the assumptions for the linear operator L and the boundar operator K. In Section 3 we proe the well-posedness of the half-space equation using the damping adding-remoing procedure. In Section we show three numerical eamples which coer the three cases of multi-species, multi-frequenc transport equations and a multi-dimensional in elocit linearized BGK equation.. Main Assumptions for L and K In this section we collect the conditions on the linear operator L and the boundar operator K. Notation. In this paper we denote f, g = V f g dσ, and f, g, = R V f g dσ d, where dσ is a measure in the elocit space. Throughout this paper we assume that the measure dσ is smmetric with respect to... Main Assumptions for L. In this subsection we state the general assumptions for the collision operator L. First, define the weight function attenuation coefficient a = + κ. for some κ. The first four basic assumptions for the linear operator L are as follows: PL L : DL L dσ m is self-adjoint with its domain DL gien b DL = { f L dσ m a f L dσ m } L dσ m, where a is defined in.. Such space arises naturall for linear/linearized collision operator since in man cases L has the structure as L = ai + L, where L is a bounded or een compact operator.

4 QIN LI, JIANFENG LU, AND WEIRAN SUN PL L : L a dσ m L a dσm is bounded, that is, there eists a constant C > such that Lf L C L f. a dσm a dσ m PL3 Null L is finite dimensional and Null L L p dσ m for all p [,. PL L is nonnegatie: for an f L a dσ m, f Lf dσ.. V Assumptions PL-PL are general enough to include man classical models such as the linearized Boltzmann operators around Mawellians with hard-potentials, the linearized BGK operator, and linear transport operators for single- or multi-species. In fact, these classical operators satisf an een stronger coercieness propert: f Lf P dσ c f,.3 L a dσ m V where recall that P = I P and P : L dσ m Null L is the projection onto Null L. We need one last essential assumptions on the coerciit of a damped ersion of L on the whole L dσ m but not just Null L. To properl eplain this assumption, we introduce seeral definitions related to the null space of L. Recall that P : Null L Null L is the operator gien b P f = Pf for an f Null L. Note that P is a smmetric operator on the finite dimension space Null L. Therefore, its eigenfunctions form a complete basis of Null L. Denote H +, H, H as the eigenspaces of P corresponding to positie, negatie, and zero eigenalues respectiel and denote their dimensions as ν + = dim H +, ν = dim H, ν = dim H. Let X +,i, X,j, X,k be the associated orthornormal eigenfunctions with i ν +, j ν, and k ν. Note that if an of ν ±, ν is equal to zero, we simpl do not hae an eigenfunction associated with the corresponding eigenspace. B definition, these eigenfunctions satisf X τ,γ, X τ,γ = δ ττ δ γγ, X τ,γ, X τ,γ = if τ τ or γ γ, X,j, X,k =, X +,j, X +,i >, X,j, X,j <, where τ {+,, }, γ {i, j, k}, i ν +, j ν, and k ν. Our method relies on full coerciit of the collision/scattering operator on L a dσ m instead of the partial one in.3 on Null L. Hence, instead of working directl with L, we add in the damping terms on the modes in Null L and define the damped linear operator L d as ν + L df =Lf + α X +,k X +,k, f k= ν + α X,k X,k, f k= + α k= ν k= X,k X,k, f ν + α L X,k L X,k, f, where α > is some constant damping coefficient to be determined later. The motiation of defining L d in such a form is as follows: the operator L normall will proide bounds for the orthogonal component of f in Null L. With the added damping terms to dissipate the modes in Null L, we epect that L d will satisf certain full coerciit condition on L dσ m. On the other hand, this added damping effect can be eentuall remoed using linearit of the equations. The precise assumption of L regarding its coerciit states.

5 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS PL There eist two constants α, σ > such that the damped operator L d satisfies f L df dσ σ f. L a dσ m for an f L a dσ m. V It will be shown in Lemma. that the coerciit condition in.3 combined with the form of L d in. implies PL, and hence PL is a natural assumption for man eamples... Main Assumptions for K. In this part we specif conditions for the boundar operator K. These conditions are stated in rather general forms and are satisfied b a large class of boundar operators. Recall that we hae denoted =, =,,, d. Denote the incoming and outgoing parts of the elocit space as V + = { =, > }, and V = { =, < }. We consider the general case where the boundar operator K consists of arious tpes of reflections in the sense that there eists a coefficient α r [, and a scattering kernel k r which is a positie measure such that K = α r K r, [K rf ] = k r, f dσ for V +.. The main assumption for such K is < PK The reflection operator K r satisfies that K rf dσ > < f dσ..7 There is a large famil of reflection boundar operators K r that satisf PK. In the literature, the reflection boundar operator for nonlinear kinetic equations of a single species is usuall written as [K r F ] = R, F d, V + < for some scattering kernel R. If we consider the linearization around the equilibrium state such that then the linearized ersion has the form [K r f] = M F = M + Mf, < R, f M d.8 We show in the following lemma that as long as K r satisfies the classical normalization and reciprocit conditions, then the main assumption PK holds: Lemma.. Suppose M is a scalar equilibrium state and dσ = M d where d is the Lebesgue measure. Suppose K r has the form as in.8. If R satisfies the normalization and reciprocit conditions: then PK holds. M R, = MR,, V +, V,.9 R, d =, V,. >

6 QIN LI, JIANFENG LU, AND WEIRAN SUN Proof. Note that an immediate ariation of. is R, d =, V +.. B the definition of K r, we hae K r f M d = >.9 =. = > > > < > M R, f M d < R, f M d d < < R, M M d d R, f M d R, d d < < < R, f M d d = f M < < The condition PK follows as dσ = M d in this case. Eamples that satisf.9 and. include R, d d. = f M d. < the specular reflection condition where R, = δ + δ ; the bounce-back condition where R, = δ + ; the pure diffuse condition for BGK or linearized Boltzmann equation where R, = e π d ; cone combinations of the aboe three; and more generall, the linearized Cercignani-Lampis collision operator with R gien b R, = πα n α t α t ep + α n ep α t αn J, α n α t α t α n where < α n <, < α t <, and J = π π e cos φ dφ. Hence our method applies to all of these classical cases for single species. Remark.. In all of our numerical eamples in Section, we use either the Dirichlet boundar condition with gien incoming data or the classical Mawell boundar condition where with the accommodation coefficients α d, α s satisfing K = α d K d + α s K s α d, α s, α d + α s <. The two operators K d, K s are the diffuse and specular reflection operators respectiel. notation in., we can choose in this case α r = α d + α s, K r = α d α r K d + α s α r K s. In terms of the Since K s automaticall satisfies PK with an equal sign, we onl need to check in each numerical eample that K d satisfies PK as well. Below we state two essential consequences of assumption PK, which will guarantee the well-posedness of the half-space equation and proide the foundation for the numerical scheme.

7 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS 7 Lemma.. Suppose K satisfies PK. Then the half-space equation. has at most one solution f C[, ; L dσ m. Proof. B the linearit of the equation we onl need to proe that i = in the boundar condition of., then the onl solution to. is zero. B the non-negatiit of L, we hae V f, dσ is decreasing in. Since f H + H, we hae V f dσ. Hence V f, dσ for all. In particular this shows f, dσ.. B assumption PK, at the boundar = we hae f dσ = α r K r f < dσ αr Therefore, > V > B. and that αr >, we deduce that f, dσ =, V < f dσ. f, dσ αr f, dσ. < < and hence > f, dσ =. Therefore, at = we hae f,. B the uniqueness of solutions to. with onl the incoming data [CGS88] that is, α r =, we hae that. has at most one solution. Remark.. The assumption that α r < in. is necessar for the uniqueness of the solution. For eample, if α r = or α d + α s = in the boundar operator K for the linear transport equation in.9 considered in our numerical eamples, then an multiple of X is a solution to the half-space equation with zero incoming data. The second consequence of assumption PK is Lemma.3. Suppose the measure dσ in the elocit space is smmetric with respect to. operator K : L > dσ m L > dσ m such that Define the where K r is defined as K r f, = > K = α r K r,.3 k r,,, f, dσ,, >, where k r is the reflection kernel of K r. Note that we hae reflected the component of in the kernel. Then a I + K is inertible on L > dσ m. b There eists a constant β > such that the operator I + K I K satisfies that f, I + K I K f β f, f for an f L > dσ m. > >. Proof. a Denote g, = f,. Then K rf = Kr g b the smmetr of dσ with respect to. Hence, K rf dσ = K r g dσ g dσ = f dσ. > > < >

8 8 QIN LI, JIANFENG LU, AND WEIRAN SUN Therefore, K LL > dσ m α r <.. This shows I + K is inertible on L > dσ m. Furthermore, we hae the bound I + K LL > dσ m. K LL > α dσ m r. c Denote g = I + K f L > dσ m. Then f, I + K I K f = I + K g, I K g > = g, g > K g, K g > > = g L > dσ m K g L > dσ m. α r g L > dσ m. Obsere that f I + K LL g. L > dσ m > dσ m L > dσ + α m r g L > dσ. m We conclude b combining the preious two estimates such that f, I + K I K f αr + αr f L. > dσ m > Hence. holds with β = α r + αr. 3. Well-posedness In this section we establish the well-posedness of equation. based on the assumptions for L and K in the preious section. The framework is similar to [LLS]: first we add damping terms to L and show that the damped equation has a unique solution. This will be achieed b using the Babuška-Aziz lemma. Then we show how to recoer the solution to the original kinetic equation using suitable superpositions with special solutions. The damped kinetic equation has the form f + Ld f =, f > = h + K f <, >, 3. where the damped operator L d is defined in.. f, as, 3.. Weak Formulation. In order to show the well-posedness of 3., we consider the weak formulation of the equation using the een-odd decomposition. Recall that we hae denoted =,, d. For an scalar function g,, let g +, g be its een and odd parts with respect to respectiel such that Therefore we hae g +, = For the ector-alued function f, denote g, + g,, g, = g, g, g +, + g, = g,, g +, g, = g,. f + = f +, f +,, f + d T, f = f, f,, f d T.,

9 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS 9 The solution space for. and 3. is Γ = { f L dσ d m f + L dσ d m }. for some dσ such that the term f V f + is well-defined. The norm in Γ is defined as f = f + f Γ L dσ d m L dσ d m One eample of dσ, dσ is that dσ = a d and dσ = a d where a is the attenuation coefficient defined in.. For the operator defined in., we hae dσ = dσ. This tpe of solution space Γ with mied regularit is introduced in [ES]. For a general function f Γ, the trace of f + at = is well-defined while the trace of f ma not. Due to this lack of regularit for f, when deriing the weak formulation we will represent f in terms of f + on the boundar. Recall that the boundar condition is gien b Using the een-odd decomposition, we hae f + + f > = h + K f + < + K f < = h + α r = h + α r < > = h + K f + K f, f > = h + K f <. k r,,, f + dσ + α r k r,,, f + dσ α r < > k r,,, f dσ k r,,, f dσ where K is defined in.3. Note that in order to get the third line we hae used that dσ is smmetric with respect to. Hence, the boundar condition has been reformulated as I + K f = h I K f +, >. For an operator K that satisfies assumption PK, we hae shown in Lemma.3 that I + K is inertible on L > dσ m. Thus, f is related to f + as f > = I + K h I + K I K f + >. 3.3 Hence when deriing the weak formulation of the half-space, the boundar term at = becomes f, φ = φ +, f = φ +, I + K h φ +, I + K I K f +. > > > Define the bilinear form B f, φ = f, φ + + φ, f + + φ, Ldf,,, + φ +, I + K I K f +. > = and let l be the linear functional on L > dσ m such that l φ = φ +, I + K h > The preious calculations then show that the weak formulation of equation 3. has the form B f, φ = l φ for an φ Γ. 3. The main tool that we use to show well-posedness and quasi-optimalit is the Babuška-Aziz lemma which we recall below:

10 QIN LI, JIANFENG LU, AND WEIRAN SUN Theorem 3. Babuška-Aziz. Suppose Γ is a Hilbert space and B : Γ Γ R is a bilinear operator on Γ. Let l : Γ R be a bounded linear functional on Γ. a If B satisfies the boundedness and inf-sup conditions on Γ such that there eists a constant c > such that Bf, g c f Γ g Γ for all f, g Γ; there eists a constant δ > such that for some constant δ >, sup Bf, ψ δ ψ Γ, for an ψ Γ, f Γ= sup Bf, ψ δ f Γ, ψ Γ= then there eists a unique f Γ which satisfies for an f Γ Bf, ψ = lψ, for an ψ Γ. b Suppose Γ N is a finite-dimensional subspace of Γ. If in addition B : Γ N Γ N R satisfies the inf-sup condition on Γ N, then there eists a unique solution f N such that Bf N, ψ N = lψ N, for an ψ N Γ N. Moreoer, f N gies a quasi-optimal approimation to the solution f in a, that is, there eists a constant κ such that f f N Γ κ inf f w Γ. w Γ N Now we erif that B, and l defined in 3. and 3. satisf the conditions in Theorem 3.. Proposition 3.. Suppose the measure dσ in the elocit space is smmetric with respect to. Suppose the linear operators L satisfies the assumptions PL-PL and the boundar operator K satisfies assumption PK. Then a the bilinear form B : Γ Γ R satisfies the boundedness and inf-sup conditions and the linear functional l is bounded on Γ. Therefore, equation 3. has a unique solution f Γ. Thus f is a strong solution to the damped half-space equa- b Moreoer, f L a dσ dm. tion Proof. For each f Γ the proof is done b finding an appropriate test function φ f Γ such that B φ, f satisfies the inf-sup condition: B φ f, f ĉ f, φ Γ f ĉ f Γ. Γ The particular choice of φ f is the same as in [LLS] such that φ f = δ φ + φ with δ > large enough and φ = f, φ = + κ f +. Using such φ f together with the coerciit of the damped operator L d in PL, we hae identical estimates for the interior terms in B φ f, f as in the proof of [LLS, Proposition 3.]. Moreoer, the positiit of the boundar term f +, I + K I K f + is guaranteed b Lemma.3. Hence b the same argument > as in [LLS], we hae that B satisfies the inf-sup condition. Boundedness of B and l can be shown b direct applications of the Cauch-Schwarz inequalit. Thus the weak formulation 3. has a unique solution. This also implies that the half-space equation. has a unique solution in the distributional sense. In addition, the half-space equation itself shows f = Lf L a dσ dm where a is the attenuation coefficient defined in.. Hence the full trace of f in L dσ is well-defined.

11 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS As in [LLS] we will sole the damped equation 3. b a Galerkin method. Proposition 3.3 Approimations in R d. Suppose {ψ k ψ n ψ n d d d } k,n, n d = is an orthonormal basis of L dσ such that ψ n ψ n is odd and ψ is een in for an n ; n span{ψ,, ψ n+ Define the closed subspace Γ NK as { m Γ NK = g, Γ g, = where e i =,,,,,, T Then i= } for each n. N+ k= K n,,n d = a there eists a unique f NK Γ NK such that which satisfies f NK, = m i= g i k,n,,n d ψ k } ψ n ψ n d d d e i, is the standard i th basis ector with i m and g i k,n n d H R +. N+ k= K n,,n d = a i k,n,,n d ψ k ψ n ψ n d d d e i, 3.8 B f NK, g = l g for eer g Γ NK, 3.9 where B and l are defined in 3. and 3. respectiel. The coefficients {a i k,n,,n d } satisf that a i k,n,,n d C [, H,, k N +, n,, n d K, i m. b The approimation is quasi-optimal, that is, there eists a constant κ > such that f f NK Γ κ inf f w Γ. w Γ NK Proof. Part a and b follow directl from the Babuška-Aziz lemma as long as we erif that B, satisfies the inf-sup condition oer Γ NK. The onl modification is in the choice of φ f where φ is projected onto Γ NK. The proof again follows along the same line to the proof of Proposition 3. in [LLS] using the positiit of the boundar term guaranteed b Lemma.3. The following Proposition reformulates 3.9 into an ODE with eplicit boundar conditions. Proposition 3.. Let A = Define the d+-tensors A and B as ψ k, ψ j. N+ N+ A = A I I I = A ik δ nl δ nd l d δ pq, N+ K K m ψ n ψ n d d d e p, L d ψ B il l dq kn n d p = ψ k i ψ l ψ d l d d e q 3. for i, k N +, n,, n d K, l,, l d K, and p, q m. Then the ariational form 3.9 is equialent to the following ODE for the coefficients a p kn n d : m p= N+ k= K n,,n d = A il l dq kn n d p a p kn n d = m p= N+ k= K n,,n d = B il l dq kn n d p ap kn n d, 3.

12 QIN LI, JIANFENG LU, AND WEIRAN SUN together with the boundar conditions at = : N+ k= ψ k, ψ i a q k,n,,n d + ψ q i,n,,n d, I + K I K f NK > = I + K h q ψ i,n,,n d dσ > 3. for i =,, N, n,, n d =,,, K, and q =,, d. Here f NK is defined in 3.8 at = and the basis function ψ q i,n,,n d is ψ q i,n,,n d = ψ i ψ n ψ n d d d e q. Proof. Equation 3. is obtained b choosing the test function g in 3.9 as g = g ψ k ψ n ψ n d d d e p for each basis function ψ k ψ n ψ n d d d e p and for an arbitrar g Cc,. The boundar condition 3. is deried b choosing the test functions as g = g ψ k ψ n ψ n d d d e p, for each basis function ψ k ψ n ψ n d d d e p in Γ NK and for an arbitrar g Cc [,. Since the tensors A, B are the same as in [LLS], we hae that there are mnk d positie, mnk d negatie, and mk d generalized eigenalues of A, B. Note that there are mn + K d unknowns in the ODE sstem 3. and mnk d boundar conditions. This is again the correct number of boundar conditions for 3. to hae a unique decaing solution. 3.. Recoer. In this part we show the procedures to recoer the solution to the original kinetic equation.. To this end, let f be the solution to the damped equation 3.. For all i ν and j γ +, let g,i, g,j be the solution to 3. with h = X,i K X,i < and h = X +,j K X +,j < respectiel. More eplicitl, for each i ν, and for each j ν +, g,i + L d g,i =, g >,i = X,i K X <,i + K g <,i, >, 3.3 g,i, as, g +,j + L d g +,j =, g > +,j = X+,j K X < +,j + K g < +,j, >, 3. g +,j, as, The ke idea is that the damping terms in L d anish for a proper linear combination of f, g,i s, and g,j s. The recoering procedures rel on the uniqueness of solutions to the original kinetic equation.. Proposition 3.. There eists a unique sequence of constants c,i, c +,k R for i γ and j γ + such that if we define γ g = c,i g,i + i= γ + i= c +,j g +,j, 3.

13 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS 3 then X,i, f g =, for all and all i γ, i ± γ ±. X ±,i±, f g =, L X,i, f g = 3. Proof. Since the proof of this proposition onl depends on the structure of the kinetic equation instead of the particular form of the boundar condition, the details are the same as in Proposition 3.8 in [LLS]. We eplain the main idea here. The ke structure we utilize here is that the coefficients in the added damping terms onl depends on the aerage of the damped solution against X α,i or L X. Hence to remoe the damping effect, we onl need to choose g carefull such that f g will hae zero aerages. Bearing this in mind, we denote and U + = X +,, f,, X T +,ν+, f, U = X,, f,, X T,ν, f, U = X,, f,, X T,ν, f, U L, = L X,, f,, + ul + ux, X,ν, f T, U f = U T +, U T, U T, U T L, T. 3.7 B multipling X +,j, X,i, X,k, L χ,m to 3. and integrating oer R d, we hae U + A U =, 3.8 where the coefficient matri A is A = αd + αa αd αa αb αa T αa T I + αb αd, 3.9 where D ± are positie diagonal matrices and A,ik = X +,i, L X,k, A,jk = X,j, L X,k γ + γ B ij = X,i, L X,j, D ij = L X,i, L X,j γ γ γ γ, γ γ, where B is smmetric positie definite and D is smmetric. Thus A is a matri of size γ + + γ + γ γ + + γ + γ. The proof of Proposition 3.8 in [LLS] shows that A has γ + γ negatie eigenalues { i } γ +γ i=. Moreoer, A is of rank γ + + γ since the original kinetic equation. satisfies the uniqueness propert in Lemma.. Note that b the boundedness, all the solutions to the damped equation 3. will be orthogonal to span{ i } γ +γ i=. Hence for an solution f to the damped equation, there eists a unique set of {c,i } γ i= {c +,j} γ+ i= such that for g defined in 3. with these coefficients, we hae A U g = A U f, which is equialent to 3.. Now we can construct the solution to the original kinetic equation.

14 QIN LI, JIANFENG LU, AND WEIRAN SUN Proposition 3.. Let f be the solutions to the damped equation 3. with h and g the function defined in Proposition 3.. Let c,i s and c +,i+ s be the coefficient of g gien in Proposition 3.. Let η = f ν g + c,ix,i + Then η is the unique solution to the original half-space equation.. Proof. Note that η = f g satisfies η + L η =, η > = h + K η < Hence η defined in 3. satisfies ν i= c,ix,i > + i= ν + i= ν + i= c +,j X+,j. 3. c +,j X+,j >, =, >, η, as. η + L η =, η = h + K η, =, >, ν η c,ix,i + i= ν + i= c +,j X+,j, as. where c,i, c,j are the coefficients defined in Proposition 3.. We thereb hae recoered η as the unique solution to.. Combining the error estimate in Proposition 3.3 and the damping terms, we derie the final error estimate for our method as follows: Proposition 3.7. Suppose η is constructed as in 3. in Proposition 3. with f, g +,i, g,j being numerical approimations obtained in Proposition 3.3 to the damped equation with appropriate boundar conditions. Suppose f h is the unique solution to the equation.. Then there eists a constant C such that f h η Γ C inf fh w + inf Γ f w + δ NK L f, w Γ NK Γ w Γ NK a d d m where Γ is the norm defined in 3. and δ NK := ν + i= inf w Γ NK g +,i w Γ + ν j= inf w Γ NK g,j w Γ. Proof. The proof of this proposition onl depends on the recoer procedures and the quasi-optimalit shown in Proposition 3.3. In particular, it does not depend on the specific form of the boundar conditions. Hence it is identical to the proof for Proposition 3.9 in [LLS] and we omit the details.. Numerical eamples In this section we show the numerical results of our algorithm for three models, which coer the cases for multi-species, multi-dimensional in the elocit ariable, and multi-frequenc sstems. The three eamples are: a linear transport equation with two species, linearized BGK/Boltzmann equations with elocit in R, and a linearized transport equation with multi-frequenc. We treat these three cases in order. Recall the general form of the half-space equation: f + Lf =, in, V, f > = h + K f <, at =..

15 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS As mentioned before, the boundar conditions for all these eamples are either the Dirichlet condition with gien incoming data or the classical Mawell boundar condition such that K = α r K r = α d K d + α s K s, where K d is the diffuse reflection and K s the specular reflection. For the conenience of numerical computation, we list out two properties of such K which can be erified b direct calculation: Lemma.. Let K the Mawell boundar operator and let K, K s, K d be the operators defined in Lemma.3. Then a K s = I; b I + K has the eplicit form as I + K = + α s I γd K d,. where γ d = α d + αd + α s... Linear Transport Equation with Two Species.... Formulation. The first eample that we consider is the stead radiatie transfer equation RTE with Thomson Raleigh scattering and polarization effect in planar geometr see [Pom73, Section.]. In this model, the ariables I and Q denote the total intensit and the intensit difference of light. The sstem [Pom73, Eq.., page 3] depends on the frequenc which onl seres as a parameter. Hence we simpl ignore the frequenc dependence here. In this case the scattering coefficients σ, σ s in [Pom73] are both constants. We consider a pure scattering case with no source such that σ = σ s and rescale σ to be one. The speed of light c is also normalized to be one. Then the RTE has the form I + I + p p I d p p Q d =, Q + Q p p I d +.3 p p Q d =, where p = 3 is the second-order Legendre polnomial.... Properties of L and K. Denote f = I, Q T. Then the collision operator L has the form Lf = f Σ, f, Σ, + = p p p p p p p, p. where we recall the notation g, g = g g d. In this case, we hae dσ = d. First we check that Lemma.. The scattering operator L defined in. satisfies PL-PL. Proof. The attenuation coefficient a in this case is a =. One can then directl check that L is self-adjoint and Null L = span{ X } with X =, T and L X = X.. Propert PL is also readil erified b the boundedness of Σ,. Furthermore, we can show b direct calculation again that there eists a constant β > such that f Lf P d β f,. L d m where P is the projection onto Null L. Hence PL holds. V

16 QIN LI, JIANFENG LU, AND WEIRAN SUN To show that L satisfies PL, we proe a more general statement: suppose L satisfies PL-PL3 together with., then there eist constants α, σ > such that PL holds. Indeed, write f = f + f + f + + f, where ν f = P f, f = X,i, f i= X,i, f± = ν ± i= X±,j, f X ±,j. Recall that X,i, X ±,j are defined in Section.. Then b Cauch-Schwarz there eist c, c > such that f, Lf β f, X,i, f + L X,i, f, L dσ m i i L X,i, f c f c f + f + + f, L i dσ L dσ L dσ L dσ X +,j, f + X,k, f c f+ + f c f. L j dσ L dσ L dσ k Therefore, if the coefficient α in the definition of L d in. is small enough, then f L df β dσ f + c α f+ + f L dσ m L dσ m Furthermore, f L df dσ α V V i L X,k, f c α f c α L dσ m Hence b multipling.7 b ma{ c c, cα β V f + f + L dσ m }, we hae f L d f dσ σ f L dσ m L dσ m + L dσ m f..7 L dσ m for some σ > which depends on α. Appling the aboe estimates to L gien in. note that in this case we onl hae H or X, we conclude that such L satisfies assumptions PL-PL...8 The boundar operator K is defined as K f < = α d K df + αs K s f < = α d f X d < X > X X d + α s f <,.9 where X is gien in., and α d, α s, α d + α s <. We hae Lemma.3. The boundar opeartor K defined in.9 satisfies PK. Proof. Since Lemma. onl coers the scalar case, we show the details of proof for the current ector case. As commented in Remark., we onl need to show that K d satisfies PK. To this end, we use the smmetr of X in and Cauch-Schwarz to obtain that K df d = < X f X d d > > > X X f d d, < which is the desired propert PK.

17 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS 7 The aboe two Lemmas show that the theor in Section 3 applies to equation.3. The details for terms in the weak formulation are as follows. First, L df = Lf α X X, f αl X L X, f = Lf α X X, f α X X, f. The Mawell boundar condition is recall.9 f > = h + K f < = h + α d K d + α s K s.. Thus, b Lemma., the boundar operators in the weak formulation B φ, f = l φ are I + K = α d I K d, + α s + α d + α s I + K I K = α s α d I + α s α s + α d + α s K d, where K df = Kd I, Q T = I, d, T.. > Let g be the special solution to the damped equation with the boundar condition g > = X K X < + K g <, =. Then the true solution is gien b = f c h g + c h X, where f is the solution to the damped equation with the boundar condition. and c h = X, f X, g...3. Algorithm. For this eample, we choose half-space Lagendre polnomials as the basis functions for each component of I, Q T. Namel, we first find the half-space Lagendre polnomials b: φ m φ n d = δ mn, m, n =,,. Then we use the een-odd etension to obtain the basis functions for the finite-dimensional space oer [, ]: where φ m, [, ], φ m = φ m, [, Γ N, = span {φ m } N m=, φ m, [, ], and φ m = φ m, [,. In this multi-species case, f is a two dimensional ector and the basis function for f is chosen to be: { } N Γ N = span φm with φ m = φ m and φ m+n = Using these basis functions, the ODE sstem becomes A d α = B α, d m= φ m, m =,, N.

18 8 QIN LI, JIANFENG LU, AND WEIRAN SUN. error regression I,Q. error slope = 3.87 I Q N Figure. Eample. For the two-species RTE.3, we set h =, T and α d = α s =. The left panel shows the numerical solutions at = for both species using 3 i.e. N = basis functions. The right panel shows the conergence rate of the Q component of the RTE. where A mn = φ m φ n d, B mn = φ m L d φn d for m, n =,, N. Therefore both A and B are of size N N.... Numerical Results. In this part we show the numerical results regarding this multi-species model with pure incoming data. Eample. Incoming boundar condition. Set h =, T and α d = α s =. The numerical solutions at = for both components are shown on the left in Figure. The plot on the right in Figure shows the conergence rate of the second component Q where we obsere an algebraic conergence rate. This is within epectation, as een though the een-odd decomposition captures the jump discontinuit at =, the solution still has a weak deriatie discontinuit near = [CLT, TF3]... Linearized BGK Equation.... Formulation. The second eample we consider is the time-independent linearized BGK equation for a single species with its elocit =, R. Here we normalize the wall temperature and denote the absolute Mawellian M as the wall Mawellian such that M = π e. Suppose F is the densit function for the nonlinear BGK equation. Let f be the perturbation such that Then the linearized collision operator L has the form F = M + Mf. Lf = f Pf,. where P : L M d span{,, } is the projection operator. In this case, d is the usual Lebesgue measure and dσ = M d.

19 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS 9 The boundar operator K is gien b Kf = α d K d f + α s K s f < = α d < = α d π f M d > M d + α sf < R < f M d + α s f < R, where R =,, α d, α s and α d + α s <. We hae.3 Lemma.. The linear operator L defined in. satisfies PL-PL and the boundar opeartor K defined in.3 satisfies PK. Proof. It is classical to show that L satisfies PL-PL and also the stronger coercieness.. Therefore b the proof of Lemma., condition PL also holds. The boundar operator K d in this case fits the form in Lemma., which guarantees that PK holds. Remark.. Our method can also be applied to the linearized BGK equation for multi-species. The linearization of these species is chosen slightl differentl depending on whether there is diffusion reflection or not. In the case where there is nontriial diffuse reflection from the wall, the equilibrium state for different particles will all be the same, which is the Mawellian M gien b the wall. We can then linearize the ector-alued densit function F = F,, F m as F = M,, T + Mf,, f m T. On the other hand, if there is onl the incoming data and/or the specular reflection, i.e., α d =, then we allow the equilibrium states of different species to be different. In this case, we linearize F = F,, F m as F i = M i + M i f i, i =,, m. The adantage of this linearization is that the function space for f is gien b L d m instead of the weighted-l b arious Mawellians M i for each component f i. To define the particular damped operator in the case of linearized BGK equation, we compute the eigenmodes in Null L: X, = +, X, =, The associated eigenspaces are Moreoer, L X,k is computed as X + = + +, X = +. H = span{x,, X, }, H ± = span{x ± }. L X, = X, = +, L X, = X, =. Hence, the damped operator has the form L d f =Lf + αx + X +, f + αx X, f + α X,k X,k, f + α X,k X,k, f, k= where f, f = f R f M d. The boundar condition is gien as f > = h + K f <, =, k=

20 QIN LI, JIANFENG LU, AND WEIRAN SUN where recall that K is gien in.3. B Lemma., the boundar operators in the weak formulation are I + K = α d I K d, + α s + α d + α s I + K I K = α s α d I + α s α s + α d + α s K d, where K d f = π > fm d. In order to obtain the original solution to., we construct the special solutions g,, g,, g + such that the satisf the damped equation and the boundar conditions respectiel: g >, = X, KX <, + Kg <,, g >, = X, KX <, + Kg <,, g > + = X + KX < + + Kg < +. The true solution is then gien as = f c, g, + c, g, + c + g + + c, X, + c, X, + c + X +, where the coefficients c,, c,, c + satisf that X,, g = X,, f, X,, g = X,, f, X +, g = X +, f with g = c, g, + c, g, + c + g +. The unique solabilit of g is guaranteed b Proposition Algorithm: For the D-BGK case, we build the basis functions upon Hermite polnomials. Since the solution is regular in, we use the full Hermite polnomials on R for. To take into account of the jump discontinuit in =, we appl een/odd etensions oalf-space Hermite polnomials on [,. Specificall, the half-space Hermite polnomials {B m } n= satisf B m B n e d = δmn. Performing the een and odd etensions of {B m } n= gies B B E n /, >, B = B n / and B O n /, >, =, < B n /, <. Then the set of basis functions for the finite dimensional space in is gien b The set of basis functions in is Γ,N = {φ,n } N+ n= {φ,n} N n= = { B O n Γ,N = {φ,n } N n=, where } N+ n= { B E n } N n=. φ,m φ n, e d = δ mn. The basis for the approimation solution f N is then epanded b Γ,N Γ,N such that The ODE sstem still has the form f N, = N+ m= n= N β mn φ,m φ,n. A d d α = Bα,

21 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS where A pq mn = φ,m φ,n d δ pq, R B pq mn = φ,m φ,p [ L d φ,n φ,q ], d d. R..3. Numerical Results. Eamples are shown in Figure and Figure 3 for both the pure incoming data and the Mawell cases. Eample.. Incoming boundar condition. In the first eample, we use incoming boundar condition. Figure erifies that i H H +, then the solution is simpl = h, as epected from the theor. Eample.. Mawell boundar condition. In the second eample show in Figure 3, we set the accommodation coefficients to be α, α 3 =.3,.. This time i is chosen such that h = X,k K X <,k k =, or h = X + K X < +, then the solution is = X,k or X +, which again is consistent with the theor..3. Multi-frequenc linearized transport equation..3.. Formulation. In this third eample, we consider a linearized BGK-tpe of equation that models phonons with a continuous range of frequencies [MCMY]. We consider the time-independent half-space equation. Let ω, ω m be the angular frequenc of phonons and F be the densit function such that F = F,, ω. The stationar nonlinear equation has the form ω F = F F BE τω,. where ω is the group elocit and τω is the relaation time. The are both frequenc-dependent. We assume that ω >, although it ma not hae a positie lower bound. The equilibrium state F BE is gien as the Bose-Einstein distribution function such that F BE ω, T = e ω/k BT,. where is the reduced Planck constant, k B is the Boltzmann constant, and T is the temperature. Gien a reference temperature T, we linearized F ω, T around F ω, T such that F ω, T = F ω, T + Gω, T.. The resulting equation has the form ω G = G C ω T τω,.7 where C ω = e ω/k B T ω e ω/k B T k B. B the mass conseration, we hae T which gies T = ωm Θ ωm Equation.7 then has the form G ωm d dω = τω ω G = τω C ω d dω T, τω G ωm τω d dω with Θ C ω = d dω. τω G C ωm ω Θ G d dω..8 τω

22 QIN LI, JIANFENG LU, AND WEIRAN SUN X, X, X, X, X +.. -X X X Figure. Eample.. These four rows of figures demonstrate the results computed using h = X,, h = X,, h = X + and h = X, and the three columns show h, recoered solution and the difference h respectiel. For the first three cases, the solutions satisf that = h since h H H +. Solution to the last case does not satisf = h since h H. In all eamples, we use 3 basis functions along each direction.

23 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS h 3 X X h X, X, h X, X, h X X Figure 3. Eample.. These four rows show numerical results using boundar conditions proided b h = X KX < where X = X, X,, X,, X + respectiel. The accommodation coefficients are set as α d, α s =.3,.. The four columns from left to right are: h, X, recoered result, and the difference X. In the bottom three cases, one recoers X,, X,, X + as the solutions as epected. In all eamples, we use 3 basis functions along each direction. The operator on the right-hand side of.8 is not self-adjoint in the space L d dω. Howeer, we show below that it is smmetrizable. Indeed, if we define β ω = ωτω, G = C ω Θ βω f,.9

24 QIN LI, JIANFENG LU, AND WEIRAN SUN then f satisfies f = β ω f + βω ωm f dσ = βω βω f βω f βω,ω = Lf,. where dσ = Cω Θ d dω is a probabilit measure b the definition of Θ τω. Note that the linear operator L on the right-hand side of. is now self-adjoint. In the preious two eamples, both linear operators L satisf the classical coerciit condition.. Howeer, this propert ceases to hold in the current eample. Neertheless, we show that with the help of the added damping terms, condition PL is still true. Lemma.. Suppose < ω < ω m, < β ω b, and f = f, ω. Let L be the scattering operator gien b where dσ = Lf = β ω f βω ωm Cω Θ τω d dω is a probabilit measure in, ω. Then f βω dσ,. a L is self-adjoint, nonnegatie, and Null L = span{ β ω }. Moreoer, β ω H. b Denote X = β ω. Define βω dσ V L d f = Lf + α X X, f,ω + L X L X, f..,ω where g, g,ω = ω m α such that g g dσ. Then for α > small enough, there eists σ depending on Hence L satisfies all the assumptions PL-PL. Proof. B the definition of L, we hae ωm V flf dσ = = fl d f dσ σ f L dσ..3 ωm ωm f f βω βω f βω f βω f βω,ω,ω dσ dσ.. This shows L is nonnegatie and Null L = span{ β ω }. B direct calculation we hae β ω,ω =. Hence βω H. Denoting X = β ω, one can erif b direct calculation that βω dσ V L βω 3 X =. V β ω dσ Hence, L X, f =,ω βω, V β ω dσ = βω, V β ω dσ f βω,ω f f βω βω,ω,ω + α f βω,ω,

25 where α = βω >. Thus, βω dσ V ωm HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS fl d f dσ = flf,ω + α X, f,ω + α L X, f,ω flf,ω + α L X, f,ω flf,ω + αα ωm f βω,ω f βω f βω α V β β ω dσ ω, dσ + αα,ω f f βω βω,ω,ω f, βω,ω where the last inequalit follows from choosing α small enough and then appling Cauch-Schwartz and.. Let α = min{, αα }. Then b the assumption that β ω, b ], we hae ωm fl d f dσ α f σ f L dσ, σ = α, βω τ L dσ which proes the coerciit of the damped operator L d on L dσ. Hence L satisfies all the assumptions PL-PL. The tpe of boundar conditions we use here is the case where the wall does not change the frequenc of the phonon. More precisel, for each ω >, the boundar condition for the original nonlinear equation reads K F F BE ω, T > = α d ωf d > ωf BEω, T d + α sf <, ω. = α d < < ωf d + α s F <, ω. Linearizing F as in. and.9, we obtain the linearized boundar operator K as Kf = α d K d f + α s K s f < = α d f d + α s f < R,. where α d, α s, and α d + α s <. Now we erif that Lemma.. The boundar operator K defined in. satisfies PK. < Proof. Again we onl need to show that K d satisfies PK. B the definition of K d in., we hae K d f dσ = f, ω d C ω dω d > > < Θ τω f C ω, ω dω d d > < Θ τω = f, ω dσ, which shows PK holds. < In summar, the damped equation has the form f = f f αx X, f αl X L X, f.. βω βω βω,ω

26 QIN LI, JIANFENG LU, AND WEIRAN SUN The boundar condition is gien as f = = h + Kf, >,.7 where K is gien in.. We again hae where I + K = + α s I + K I K = α s + α s The special solution g is constructed as I I K d f = f d. > α d K d, + α d + α s α d α s + α d + α s K d g + L d g =, g > = X KX < + Kg <, >.,.8 Finall, the eact solution for equation. with boundar condition. is = f c h g + c h X, where f soles the damped equation. with the boundar condition.7 and c h = X, f,ω X, g,ω..3.. Numerical Results. We discretize the ω-ariable uniforml and replace the integral in ω b the trapezoidal rule. The resulting sstem can be iewed as a multi-species sstem. Hence the construction of basis functions is the same as in Section..3. We again show eamples with both pure incoming data and Mawell boundar condition. For computational conenience, we modif the equilibrium state as F BE ω, T = e ω/k BT. Eample.3. Incoming boundar condition. In the first eample, we set C ω τω = ω ep ω/, τωω =, ω [, 8]. ω Then Null L = span{ τωω} = span{ /ω}. Figure shows that i = X, then the numerical solution is in good agreement with the analtical solution where = X. Eample.3. Mawell boundar condition. In the second eample, we take the same C ω, τω and ω as in the preious eample and set the accommodation coefficients as α d, α s =.3,.. Once again if h = X KX <, the eact solution must be = X. The numerical solution demonstrated in Figure shows a good match with the eact solution. References [Ber8] N. Bernhoff, On half-space problems for the linearized discrete Boltzmann equation, Ri. Mat. Uni. Parma 9 8, 73. [Ber] N. Bernhoff, On half-space problems for the weakl non-linear discrete Boltzmann equation, Kinet. Relat. Models 3, 9. [CGS88] F. Coron, F. Golse, and C. Sulem, A classification of well-posed kinetic laer problems, Comm. Pure Appl. Math. 988, 9 3. [CLT] I-K. Chen, T.-P. Liu, and S. Takata, Boundar singularit for thermal transpiration problem of the linearized Boltzmann equation, Arch. Rational Mech. Anal., 7 9.

27 HALF-SPACE KINETIC EQUATIONS WITH GENERAL BOUNDARY CONDITIONS ω ω ω ω 7 8 Figure. Eample.3. The two figures on the left are boundar data and the difference = h obtained for h = /ω Null L. The two figures on the right are the solution at the boundar and the difference,, ω h obtained for h = ω. In this case we epect the difference to be of order O. In both cases, we use 3 basis functions in direction and sample 8 grid points along ω ω ω ω ω 7 8 Figure. Eample.3. In the second row we use h = X KX < as the incoming Dirichlet data. The four plots show h, X, numerical result and the recoer difference X. The accommodation coefficients are set as α d, α s =.3,.. We recoer the eact solution X as epected. In this eample, we use 3 basis functions in direction and sample 8 grid points along ω. Note that the domain size for h is onl half of that for X. [Cor9] F. Coron, Computation of the asmptotic states for linear half space kinetic problems, Transport Theor Statist. Phs. 9 99, no., 89. [ES] H. Egger and M. Schlottbom, A mied ariational framework for the radiatie transfer equation, Math. Models Methods Appl. Sci.,. [GK8] D.M. Goebel and I. Katz, Fundamentals of electric propulsion: Ion and hall thrusters, JPL Space Science and Technolog Series, Wile, 8. [GK9] F. Golse and A. Klar, A numerical method for computing asmptotic states and outgoing distributions for kinetic linear half-space problems, J. Stat. Phs. 8 99, no., 33. [Gol8] F. Golse, Analsis of the boundar laer equation in the kinetic theor of gases, Bull. Inst. Math. Acad. Sin. N.S. 3 8, no.,. [HM] C. Hua and A. J. Minnich, Analtical Green s function of the multidimensional frequenc-dependent phonon Boltzmann equation, Phs. Re. B 9 December, no.. [HRB] N. G. Hadjiconstantinou, G. A. Radtke, and L. L. Baker, On Variance-Reduced Simulations of the Boltzmann Transport Equation for Small-Scale Heat Transfer Applications, J. Heat Transfer 3, no.,. [LLS] Q. Li, J. Lu, and W. Sun, A conergent method for linear half-space kinetic equations,. preprint, arxi:8.3.