The kinetic scheme for the full-burnett equations

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1 Journal of Computational Physics 2 (24) The kinetic scheme for the full-burnett equations Taku Ohwada a, *, Kun Xu b a Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyo University, Kyo 66-85, Japan b Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 4 May 23; received in revised form 25 May 24; accepted 25 May 24 Available online 6 July 24 Abstract The present paper concerns two aspects for the Burnett equations. First, we are going to theoretically show the consistency between the traditional Chapman Enskog expansion and the successive approximation for the BGK equation up to the super-burnett order. Second, we will design a numerical scheme to efficiently solve the Burnett equations. The current approach is an improvement of the BGK-Burnett scheme [K. Xu, Phys. Fluids 5 (23) 277], where the locally constant collision frequency is considered. Based on the Burnett distribution function, a high order time accurate numerical flux is derived by different approaches. The resulting scheme is tested in the problem of normal shock wave and that of force-driven Poiseuille flow. Ó 24 Elsevier Inc. All rights reserved. PACS: 65M6; 65M25; 65M5; 76P5; 76N Keywords: Kinetic scheme; Boltzmann equation; BGK equation; Chapman Enskog expansion; Burnett; Super-Burnett. Introduction The distribution functions of gas molecules derived from the Chapman Enskog expansion for the Boltzmann equation (or the BGK model equation) are often employed in the construction of kinetic schemes for gasdynamic equations. For example, the first order approximation in the expansion, the local Maxwellian, is employed for the compressible Euler equations (e.g. [5]) and the second order approximation is used for the compressible Navier Stokes equations (e.g., [3,7,,2,7]). When we try to incorporate higher order rarefaction effects by extending the distribution function to the Burnett or super- Burnett order (the third or fourth order approximation), we are faced with the theoretical problems, such as the ill-posedness of the Burnett and super-burnett equations [5] and the boundary conditions [6]. * Corresponding author. Tel.: ; fax: address: ohwada@kuaero.kyo-u.ac.jp (T. Ohwada) /$ - see front matter Ó 24 Elsevier Inc. All rights reserved. doi:.6/j.jcp

2 36 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) Despite the above difficulties, this approach is far from being useless. In fact, the Burnett and super- Burnett equations are employed in some numerical experiments of rarefied gas flows, e.g., [,4,8,9,2], and the improvements over the NS results are reported. In connection with the hybrid approach, which employs the gasdynamic equations in slowly varying regions, such as the far field, and the kinetic equations in rapidly varying regions, such as the shock layer and the Knudsen layer, the numerical study of the Burnett and super-burnett equation has been attracting attentions recently. However, it is n too much to say that the progress still remains in each specialist s skill and the clear understanding and common basis are n prevailing. For example, it is n clear whether the simple extension of the existing kinetic NS solvers is valid or n. Due to the complicated nature of the Burnett and super-burnett terms, which usually give small effects, some approaches may be based on the equations which are n consistent with the Chapman Enskog expansion. The objective of the present paper is to display the common basis and the clear understanding of the construction of kinetic schemes for the gasdynamic equations beyond the NS equations. The organization of the paper is as follows. In Section 2, we briefly review the Chapman Enskog expansion for the BGK equation. The equivalence between the standard procedure and the successive approximation has been recognized among some of the specialists in kinetic theory but its clear understanding has n yet been prevailing. It is clearly explained and demonstrated up to the super-burnett level there. The result of the successive approximation is applied to an algebraic construction of the Burnett and super-burnett distribution function in Section 3. The algebraic method for the compressible Navier Stokes equations is developed in [7] and is extended to the case of the Burnett and super-burnett equations in [9]. However, this extension is based on the assumption of locally constant collision frequency and the error caused by this simplification appears from the Burnett order. We derive the full-burnett distribution function without using this simplification. In Section 4, a kinetic scheme for the full-burnett equations is derived together with the information of the time step truncation error. The different derivations of the scheme, which are helpful for the clear understanding, are explained. In Section 5, we carry out the numerical test of the newly developed Burnett scheme in the problem of normal shock wave and that of force-driven Poiseuille flow. The first numerical test exhibits the robustness of the resulting scheme around the border of the application range of the Chapman Enskog expansion. The second numerical test exhibits the non-navier Stokes behavior [2,9,6,9,2]. 2. Chapman Enskog expansion 2.. Basic equation and nation We consider the case of a monatomic gas without external force. The dimensional BGK model equation is written as o^f o^f þ ^u o^t o^x ¼ ^mð^g ^f Þ; ^q ^g ¼ ð2pr^t Þ 3=2 exp " # 2 ð^u ^UÞ ; ð2þ 2R^T ðþ ^q ^U A ¼ 3^qR^T ^u ð^u ^UÞ 2 A^f d^u; ð3þ

3 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) where ^t is the time; ^x ¼ð^x ; ^x 2 ; ^x 3 Þ is the space rectangular coordinate system; ^u ¼ð^u ; ^u 2 ; ^u 3 Þ is the molecular velocity; ^f ð^t; ^x; ^uþ is the distribution function of gas molecules; ^q, ^U, and ^T are the density, flow velocity, and temperature of the gas, respectively; R is the specific gas constant; and ^m is the collision frequency. The collision frequency ^m is proportional to the density and is written as ^m ¼ ^A c^q: ð4þ The coefficient ^A c is usually a constant but it may depend on the temperature of the gas. In the present paper, the explanation is for the latter general case, i.e. ^A c ¼ ^A c ð^t Þ. In addition, vectors are sometimes expressed by using subscripts, e.g. ^U i is the component of ^U in the direction of ^x i. In order to carry out the Chapman Enskog expansion, we rewrite the BGK equation in a nondimensional form. For this purpose, we introduce the following nondimensional variables: x ¼ L ^x, t ¼ð2R^T Þ =2 L ^t, u ¼ð2R^T Þ =2^u, f ¼ð2R^T Þ 3=2^q ^f, q ¼ ^q ^q, U ¼ð2R^T Þ =2 ^U, T ¼ ^T ^T, where L, ^q, and ^T are the reference length, density, and temperature of the system under consideration, respectively. Then, the nondimensional BGK equation is written as ðg f Þ Df ¼ ; ð5þ s! g ¼ q 2 ðu UÞ exp ; ð6þ 3=2 ðpt Þ T where D ¼ o t þ u i o xi, s ¼ q SðT Þ p, SðT Þ¼^A c ð^t T Þ=^A c ð^t Þ [SðÞ ¼], ¼ ffiffiffi p l =ð2lþ, andl is the mean free path of gas molecules in the equilibrium state at rest with the density ^q and the temperature ^T, i.e. l ¼ 2p =2^A c ð^t Þ ^q ð2r^t Þ =2 : The nondimensional macroscopic variables h ¼ðq; qu ; qu 2 ; qu 3 ; 3qT =2 þ q½u 2 þ U 2 2 þ U 3 2 ŠÞ are given by the moments of f Z h ¼ wf du; ð7þ where w ¼ t ðw ; w ; w 2 ; w 3 ; w 4 Þ¼ t ð; u ; u 2 ; u 3 ; u 2 þ u2 2 þ u2 3Þ and the domain of the integration with respect to the molecular velocity is the whole velocity space (this is applied to all the following integrals with respect to the velocity variables) Standard procedure of Chapman Enskog expansion In this subsection, we briefly review the Chapman Enskog expansion for the BGK equation according to [8]. In the Chapman Enskog expansion, the situation where is considered. The following functional form of the solution is the postulation of this expansion f ¼ f ðh; Dh; u;þ; ð8þ where D represents the differential operators with respect to the space coordinates. Eq. (8) means that the distribution function f depends on t and x only through the macroscopic variables h and their space derivatives Dh. Multiplying bh hand sides of the BGK equation (5) by w, carrying out the integration over the whole velocity space, and substituting Eq. (8) into the result, we have the conservation equations in the form:

4 38 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) Z oh ¼ w u of du ¼ Uðh; Dh;Þ: ox ð9þ The f and U are formally expanded into the power series of : f ðh; Dh; u;þ¼f ðh; Dh; uþþf ðh; Dh; uþþ 2 f 2 ðh; Dh; uþþ; Uðh; Dh;Þ¼U ðh; DhÞþU ðh; DhÞþ 2 U 2 ðh; DhÞþ: ðþ ðþ Substituting Eqs. () and () into the BGK Eq. (5) and arranging the results in the order of power of formally, we have f ¼ g; SðT Þqf ¼ og oh U u og ox ; SðT Þqf 2 ¼ og oh U of oh U of orh ru u of ox ; ð2þ ð3þ ð4þ SðT Þqf 3 ¼ og oh U 2 of oh U of orh ru of 2 oh U of 2 orh ru of 2 or 2 h r2 U u of 2 ox ; ð5þ where r n is the abbreviation of the nth order differential operators with respect to space. Since f is the local Maxwellian, f k ðk ¼ ; 2; 3;...; Þ should satisfy the orthogonality condition Z wf k du ¼ ðk ¼ ; 2; 3;...Þ; ð6þ which is reduced to Z U k ¼ w u of k ox du ðk ¼ ; ; 2;...Þ: ð7þ Once U k is determined, we immediately obtain f kþ. The truncated conservation equations oh ¼ Xm k¼ k U k ð8þ become the compressible Euler equations for m ¼, the compressible Navier Stokes equations for m ¼, the Burnett equations for m ¼ 2, and the super-burnett equations for m ¼ 3. Incidentally, Eqs. (8) are rewritten as the evolutionary equations for the primitive variables h ¼ t ðq; U i ; T Þ, i.e. o h ¼ Xm k¼ k Uk : ð9þ We can replace [h, U k, ru k ]by[ h, U k, r U k ] in Eqs. (3) (5); the replacement makes the explicit computation of f k easier. Finally, we show the explicit form of f and the compressible NS equations o t h ¼ U þ U for reference. The f is given by

5 f ¼ 2 n i n j T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) n2 3 d oui ij þ n i n 2 5 ot q SðT Þ f ox j T =2 ; 2 ox i ð2þ where n i ¼ðu i U i Þ=T =2 and n 2 ¼ n 2 þ n2 2 þ n2 3. The corresponding NS equations are q o B qu i A þ o qu j B qu i U j þ 2 q½ 3 T þ U 2 2 k Š ox P ij A ¼ ; j q½ 3 T þ U 2 2 k ŠU j þ P kj U k þ Q j ð2þ where P ij ¼ qt d ij ½T =SðT ÞŠ ou i ox j þ ou j 2 ou k d ij ; ð22þ ox i 3 ox k Q i ¼ 5 ot ½T =SðT ÞŠ : 4 ox i ð23þ Then, we nice that the viscosity and thermal conductivity are proportional to T =SðT Þ. So, the temperature dependence of these coefficients can be controlled by SðT Þ. However, the Prandtl number is equal to unity irrespective of SðT Þ, while it is around 2/3 for real monatomic gases. This is a well-known drawback of the BGK model Successive approximation of the BGK solution The BGK equation (5) is rewritten in the form: f ¼ g sdf : ð24þ Then, we nice f ¼ g sdðg sdf Þ¼g sdg þ sd½sdðg sdf ÞŠ: So, we have the solution of the BGK equation in the form of the successive approximation f ¼ g sdg þ sdðsdgþ sd½sdðsdgþšþ: ð25þ The successive approximation (25) implies f ¼ g þ sg ðh; Dh; uþþs 2 g 2 ðh; Dh; uþþs 3 g 3 ðh; Dh; uþþ¼f ðh; Dh; u; sþ; ð26þ oh ¼ U ~ ðh; DhÞþs U ~ ðh; DhÞþs 2 U2 ~ ðh; DhÞþs 3 U3 ~ ðh; DhÞþ¼ Uðh; ~ Dh; sþ: ð27þ Thus, the successive approximation (25) is consistent with the postulation of the Chapman Enskog expansion. Therefore, we should, in principle, obtain the same result as that of the Chapman Enskog expansion, i.e. s n Un ~ ¼ n U n and s n g n ¼ n f n for n ¼ ; ; 2; 3;... We will explicitly demonstrate this up to the super-burnett level. Since h is computed from g, g i (i ¼ ; 2; 3; ) should satisfy the orthogonality condition Z wg i du ¼ ði ¼ ; 2; 3...; Þ: ð28þ

6 32 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) The term Dg is expressed as Dg ¼ og U oh ~ þ s ~U þ s 2 ~U 2 þ Then, sg in Eq. (26) is given by og sg ¼ s U oh ~ þ u og ox þ u og ox : ð29þ : ð3þ The orthogonality condition for g is reduced to ~U ¼ R w½u i ðog=ox i ÞŠdu since R w i ðog=oh j Þdu ¼ d ij. Then, we have ~U ¼ U ; ð3þ i.e. we have the compressible Euler equation system. At the same time, we have sg ¼ q SðT Þ og oh U þ u og ¼ f ; ox ð32þ i.e. we have the same NS distribution function. The term sdg is expressed as sdg ¼ f þ og oh s2 U ~ þ s 3 U2 ~ þ : ð33þ So, the term sdðsdgþ is expressed as sdðsdgþ ¼ s of oh ðu þ s ~U þþ s of orh ½rU þrðs ~U þ ÞŠ þ sd og oh ðs2 ~U þþ su of ox : By extracting the terms of the order 2 from Eq. (25), we have s 2 g 2 ¼ s 2 og U oh ~ of s oh U þ of orh ru su of ox : From the orthogonality condition for g 2, we have s ~U ¼ U ; i.e. we have the same compressible NS system. At the same time, we have s 2 g 2 ¼ 2 f 2 ; i.e. we have the same Burnett distribution function. The term sdðsdgþ is rewritten in the form sdðsdgþ ¼s 2 g 2 þ s 2 og U oh ~ þ sd og oh ðs2 U ~ þþ s of oh ðu þþ s of orh ðru þþ: So, we have sd½sdðsdgþš ¼ s 2 of 2 oh U þ of 2 orh ru þ of 2 or 2 h r2 U þ s 2 u of 2 ox og þ sd s2 U oh ~ þ: ð34þ ð35þ ð36þ ð37þ ð38þ ð39þ

7 Thus, s 3 g 3 is given by s 3 g 3 ¼ s 3 og U oh ~ 2 s 2 of oh U T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) þ of orh ru s 2 of 2 oh U þ of 2 orh ru þ of 2 or 2 h r2 U s 2 u of 2 ox : Bh of Eqs. (38) and (39) have the term sd½s 2 ðog=ohþ ~U Š. This term does n appear in Eq. (4) because of the cancellation. From the orthogonality condition for g 3,wehave ð4þ s 2 ~U 2 ¼ 2 U 2 ; ð4þ i.e. we have the same Burnett equation system. At the same time, we have s 3 g 3 ¼ 3 f 3 ; ð42þ i.e. we have the same super-burnett distribution function. 3. Algebraic construction of distribution functions In the gas-kinetic BGK scheme for the compressible NS equations [7], the distribution function f is computed without using the formulas of ~U and ~U, which are derived from the orthogonality condition. Instead, the orthogonality condition is employed in the actual numerical computation and is reduced to the computation of the linear algebra. In this section, we explain the extension of this approach to the case of Burnett and super-burnett distribution functions. 3.. NS distribution function The distribution function g is given by g ¼ og u og ox ; where ðo=þ is the time derivative evaluated by using ð43þ oh ¼ ~U : ð44þ We express ðog=þ and og=ox k as Ag and a k g, respectively, where A and a k are the linear combinations of w i (i ¼ ; ; 2; 3; 4). The a k is computed from the finite difference approximation of rh in the following way. The a k is expressed as a k ¼ X4 i¼ a i k w i: ð45þ Then, the vectors a ¼ t ða k ; a k ; a2 k ; a3 k ; a4 k Þ and d ¼ oh=ox k satisfy Ma ¼ d, where M is the matrix the element of which is defined by M ij ¼ R w i w j g du. We can easily obtained the inversion of M (see Appendix of [7]), and therefore, we have a from the finite difference approximation of d. Similarly, A is obtained from ha þ u k a k i¼; ð46þ

8 322 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) where hi ¼ R w g du. Since Eq. (46) is nhing more than the orthogonality condition for g, A so obtained satisfies Ag ¼ðog=ohÞ ~ U. The explicit computation of A is reduced to the inversion of M as in the case of a k. Thus, we have g ¼ ða þ u k a k Þg: ð47þ In the actual computation, the data of R u k w i w j g du are computed for each cell interface at the beginning of each time step. These data are employed n only in the construction of the distribution function but also in the computation of the numerical flux. Then, the computation of the numerical flux is reduced to the manipulation of the coefficients of the polynomials Burnett distribution function The sum of two terms in the bracket on the right-hand side of Eq. (35) is equal to of = with Eq. (44). By using Eq. (32), Eq. (35) is rewritten as s 2 g 2 ¼ s 2 os g 2 s þ u os g ; ð48þ ox g 2 ¼ og U oh ~ og u og ox ; where og ¼ ½B þ A 2 þ u k ðaa k þ C k ÞŠg; ðog=þ ¼ Ag, B ¼ðoA=Þ, and C k ¼ðoa k =Þ. Since o 2 g=ox k ¼ o 2 g=ox k, C k is given by C k ¼ oa=ox k. We can obtain C i as the finite difference approximation of oa=ox i or by making use of the orthogonality of og =ox i, which is reduced to hc i þ Aa i þ u j ða i a j þ b ij Þi ¼ ; where b ij ¼ oa i =ox j ¼ oa j =ox i (the symmetry of b ij follows from o 2 g=ox i ox j ¼ o 2 g=ox j ox i ) and b ij is computed from r 2 h in the same way as the computation of a k. The term ðog=ohþ ~U in Eq. (49) is written as B g, where B is a linear combination of w i (i ¼ ; ; 2; 3; 4). Then, g 2 is expressed as ð49þ ð5þ ð5þ g 2 ¼½BþA 2 þ 2u k ðaa k þ C k Þþu i u j ða i a j þ b ij ÞŠg; where B ¼ B þ B. The B is determined by the orthogonality condition for g 2, which is reduced to s 2 hb þ A 2 os þ 2u k ðaa k þ C k Þþu i u j ða i a j þ b ij Þi þ shu k ða þ u k a k Þi ¼ : ox k Thus, we have the Burnett distribution function: s 2 g 2 ¼ s 2 ½B þ A 2 os os þ 2u k ðaa k þ C k Þþu i u j ða i a j þ b ij ÞŠg þ s þ u j ða þ u k a k Þg: ox j ð52þ ð53þ ð54þ We have obtained B without computing B. The B will be employed in the construction of kinetic scheme for the Burnett equation and B ¼ B B will be employed in the computation of super-burnett distribution function. The B is obtained from the orthogonality of ðog =Þ, which is reduced to

9 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) hb þ A 2 þ u i ðaa i þ C i Þi ¼ : ð55þ Incidentally, in the case where o t s and o xk s are much smaller than s itself, i.e. os oa s ða ¼ t; x kþ; ð56þ which is equivalent to o a ½qSðT ÞŠ qsðt Þ, we can handle s as a local constant. Then, we can put g 2 ¼ g 2 and compute B from hb þ A 2 þ 2u k ðaa k þ C k Þþu i u j ða i a j þ b ij Þi ¼ : ð57þ 3.3. Super-Burnett distribution function For the super-burnett distribution function, we consider the simple case where Eq. (56) holds. In this case, we can handle s as a local constant. From Eqs. (32), (36), and (37), the super-burnett distribution function, Eq. (4), is rewritten as s 3 g 3 ¼ s 3 og U oh ~ 2 s osg oh s ~U þ osg orh rðs ~U Þ s os2 g 2 U oh ~ þ os2 g 2 orh r ~U þ os2 g 2 or 2 h r2 ~U su os2 g 2 ox : ð58þ The sum of the two terms in the first bracket on the right-hand side is equal to osg = with oh ¼ s U ~ ; ð59þ and the sum of three terms in the second bracket is equal to os 2 g 2 = with Eq. (44). Then, g 3 is expressed as s 3 g 3 ¼ s 3 og U oh ~ 2 s sð og Þ þð os Þ g s s 2 ð og 2 Þ þ 2sð os Þ g 2 su s 2 og 2 þ os2 ox ox g 2 ; ð6þ where ðo=þ is the time derivatives evaluated by using Eq. (59). Up to now, the computation is exact. From now on, we will consider the case where Eq. (56) holds, i.e. s is treated as a local constant. Under this simplification, we have s 3 g 3 s 3 og U oh ~ 2 s 2 og s 3 og 2 s 3 u og 2 ox ; where g 2 ¼ g 2 with B determined by Eq. (57). We evaluate each term in Eq. (6) below. First, we consider the time derivative ðog 2 =Þ. It is expressed as og 2 ¼ ob þ 2AB þ 2u k B a k þ AC k þ oc k þ AðB þ A 2 Þþ2u k ðaa k þ C k Þþu i u j ða i a j þ b ij Þ þ u i u j C i a j þ a i C j þ oc i ox j ð6þ g: ð62þ \The ðoc i =ox j Þ is computed as a finite difference approximation of C i or from the orthogonality condition for ðo 2 g =ox i ox j Þ, which is reduced to

10 324 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) oc i þ a i C j þ a j C i þ Aðb ij þ a i a j Þþu k d ijk ¼ ; ð63þ ox j where d ijk ¼ a i a j a k þ a i b jk þ a j b ki þ a k b ij þ o2 a i ox j ox k : ð64þ The o 2 a i =ox j ox k is computed from the finite difference approximation of r 3 h. The ðoc i =Þ is computed from the orthogonality condition for ðoðog =ox i Þ=Þ, which is reduced to oc i þða 2 þ B oc i Þa i þ 2AC i þ u j þ a i C j þ a j C i þ Aða i a j þ b ij Þ ¼ : ð65þ ox j Thus, only ðob=þ is unknown in ðog 2 =Þ. We express this unknown quantity as E, i.e. E ¼ðoB=Þ, which is a linear combination of w k (k ¼ ; ; 2; 3; 4). Next, we consider the ðog =Þ. It is expressed as og ¼ oa þ u k oa k sb ða þ u k a k Þ g: ð66þ From o 2 =ox k ¼ o 2 =ox k, we have ðoa k =Þ ¼ sðob =ox k Þ. Similar to the derivation of ðoa k =Þ, we have ob =ox k ¼ðoC k =Þ. The ob=ox k is computed from the orthogonality condition for og 2 =ox k, which is reduced to og 2 ox k ¼ ob þ 2AC k þ a k ða 2 þ BÞþ2u i ox k a i C k þ a k C i þ Aðb ik þ a i a k Þþ oc i þ u i u j d ijk ¼ : ox k Since B ¼ B þ B, we have ob =ox k ¼ ob=ox k ob =ox k. Then, only ðoa=þ is unknown in ðog =Þ.We express this unknown quantity as sf, i.e. sf ¼ ðoa=þ, where F is a linear combination of w k (k ¼ ; ; 2; 3; 4). Incidentally, the space derivatives of B and those of B can also be computed as their finite difference approximation. The term ðog=ohþ ~U 2 is expressed as Gg, where G is a linear combination of w k (k ¼ ; ; 2; 3; 4). The explicit form of the term uðog 2 =oxþ is readily obtained from Eq. (67). Summarizing the above results, the unknown quantity left in g 3 is the sum J ¼ E þ F þ G, which is a linear combination of w k (k ¼ ; ; 2; 3; 4) and is determined by the orthogonality condition for g 3. Thus, the recipe of the construction of the super-burnett distribution function is completed for the case where the variation of s with respect to time and space variables is much smaller than s itself. Although s is treated as a local constant in the construction of the distribution functions, it is varied from cell to cell and from time step to time step in the actual computation. ð67þ 4. Kinetic scheme for the Burnett equations In this section, we derive the formula of the numerical flux, which will be employed in the kinetic scheme for the Burnett equations. For simplicity, we consider the spatially one dimensional case, where the physical quantities of the gas are independent of x 2 and x 3 ; the extension to the multidimensional case can be done straightforwardly. For the simple expression, we will omit the subscripts in x and u, i.e. x ¼ x and u ¼ u [the bold symbol u still means the vector ðu ; u 2 ; u 3 Þ].

11 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) The kinetic scheme for the spatially one-dimensional case is written in the form: h j ðdtþ ¼h j ðþ Dx ½F jþ=2 F j =2 Š; ð68þ where h j ðtþ is the average of hðx; tþ over the cell ðs j =2 < x < s jþ=2 Þ, Dx ¼ s jþ=2 s j =2, and F jþ=2 is the numerical flux at x ¼ s jþ=2, which is defined by Z Dt Z F jþ=2 ¼ wuf ðs jþ=2 ; t; uþdudt: ð69þ Once the formula of f ðs jþ=2 ; t; uþ is specified, we can readily construct the scheme. We will derive the formula of f that yields a high order accurate numerical flux for the Burnett equations by different methods. 4.. Derivation by the railroad method In the conventional approach, the collisionless Boltzmann equation is employed to derive the formula of the numerical flux. For example, in Pullin s scheme for the compressible Euler equation, the distribution function employed in the computation of the numerical flux is the solution of the Cauchy problem for the collisionless Boltzmann equation from the initial data in the form of the local Maxwellian. The theory of kinetic equation for the numerical flux is studied in [,2,4] and the role of the collision effect is clarified there. In this subsection, we apply the theory, which we call the railroad method, to the case of the Burnett equations and derive the formula of the numerical flux. The information of the time step truncation error of the numerical flux is also presented. Consider the Burnett distribution function f ¼ g þ f þ 2 f 2 ð7þ the macroscopic variables of which satisfy the Burnett equations oh ¼ U þ U þ 2 U 2 : ð7þ The desired kinetic equation is derived by substituting Eq. (7) into ðo t þ uo x Þf, converting the time derivative o t f by Eq. (7), and equating the result to ðo t þ uo x Þf. Making use of Eqs. (3) and (4), we can rewrite the resulting kinetic equation in the form: of þ f of i ¼ SðTÞq½f þ f 2 ŠþRes; ð72þ ox i where Res is Oð 2 Þ and is orthogonal to w. The solution of the Burnett equations can be obtained as the moments of solution of the Cauchy problem for the kinetic equation (72) from the initial data in the form of Eq. (7). If the simplified equation of þ f of i ¼ SðTÞq½f þ f 2 Š; ox i ð73þ is employed instead of the exact kinetic equation, the error at t ¼ Dt is at most Oð 2 Dt 2 Þ because of the orthogonality of Res. The approximate solution of the Cauchy problem for the simplified kinetic equation (73) from the initial data in the form of Eq. (7) is employed in the derivation of the numerical flux. For simplicity, we consider

12 326 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) the case where the cell interface is located at x ¼, i.e. s jþ=2 ¼. As mentioned before, the difference between the solution of the exact kinetic equation (72) and that of the simplified kinetic equation (73) is Oð 2 Dt 2 Þ. So, it is sufficient to compute the numerical flux for the simplified equation (73) within the error of Oð 2 Dt 2 Þ. This gives the guideline of the approximation of the solution of the Cauchy problem, i.e. the error of the approximation for f ð; t; uþ should be at most Oð 2 tþ. The solution of Eq. (73) is expressed as the sum of two terms; one is the contribution from the initial data and the her is the integral of the right-hand side of Eq. (73) along its characteristics. For the case where Dt, the accuracy of the integration should be higher order. Otherwise, the error of the integration exceeds the intrinsic error, i.e. 2 t t 2. So, we employ the formula of higher order time integration of the kinetic equation [3]: f ¼ f ð ut; uþ tqð ut; ; uþ t2 2 D Qð; ; uþþoðt 3 Þ; ð74þ where Qðx; t; uþ ¼½SðT Þqðf þ f 2 ÞŠðx; t; uþ; ð75þ and D ¼ o t þ uo x. Under the guideline for the accuracy, we can approximate each term as follows: f ð ut; uþ ¼g utg x þ ðutþ2 2 g xx þ f uto x f þ 2 f 2 þ Oðt 2 ; 2 t; t 3 Þ; ð76þ Qð ut; ; uþ ¼SðT Þqf uto x ½SðT Þqf ŠþSðT Þqf 2 þ Oðt 2 ;tþ; D Q ¼ðo t þ uo x Þ½SðT Þqf ŠþOðÞ; ð77þ ð78þ where each term on the right-hand sides of Eqs. (76) (78) is evaluated at ðx; tþ ¼ð; Þ. Then, we have f ð; t; uþ ¼g þ f þ 2 f 2 t½ug x þ SðT Þqf Š t½uðf Þ x þ SðT Þqf 2 Š þ t2 2 ½g xx ðsðt Þqf Þ t þ uðsðt Þqf Þ x ŠþOðt 2 ; 2 t; t 3 Þ: ð79þ The time derivative in Eq. (79) is determined from the moment equation of Eq. (73) and the initial data; the time derivatives of the macroscopic variables are given by the Burnett equations. However, the guideline assures that we can employ the time derivatives evaluated by the compressible Euler equations, oh ¼ U : ð8þ Next, we express Eq. (79) using the nation employed in the algebraic construction of distribution function (Section 3). In the present one-dimensional case, we omit the subscripts in a i, b ij, and C i. Recalling ¼ SðT Þqs; n f n ¼ s n g n ; Q ¼ g þ sg 2 : ðg Þ x ¼ ½C þ aa þða 2 þ bþušg; ðg Þ t ¼ ½B þ A 2 þðc þ AaÞuŠg; B ¼ B þ B, and the formula of g 2, we find that Eq. (79) is rewritten as f ð; t; uþ ¼g½ sau þð sþtþaþðs 2 stþða 2 þ BÞþðt 2 =2ÞðA 2 þ B Þþð2s 2 stþðaa þ CÞu þ s 2 ða 2 þ bþu 2 þðau þ AÞ½ðs tþs t þ sus x ŠŠ þ Oðt 2 ; 2 t; t 3 Þ; ð8þ

13 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) where s t is evaluated by using the compressible Euler equations (8). The formula of f ð; t; uþ for the Burnett equations in [9], which is derived without taking account of the local variation of s, is now modified Derivation by Cauchy Kowalevskaya technique In this subsection, we derive the result of the previous subsection by a different method from the same starting point, i.e. the Burnett distribution function the macroscopic variables of which satisfy the Burnett equations. We consider the time evolution of such f at the cell interface x ¼. Making use of the Taylor expansion, we have the formula: f ð; t; uþ ¼g þ tg t þ t2 2 g tt þ ½f þ tðf Þ t Šþ 2 f 2 þ Oðt 3 ;t 2 ; 2 tþ: Then, we employ g t ¼ og oh ðu þ U ÞþOð 2 Þ; ð82þ ð83þ ðf Þ t ¼ of oh U þ of orh ru þ OðÞ; ð84þ and g tt is evaluated by using the compressible Euler equation (8). The above computation is nhing more than the Cauchy Kowalevskaya procedure at the level of the distribution function. From the derivation of Chapman Enskog expansion, we recall og oh U ¼ ug x SðT Þqf ; og oh U þ of oh U þ of orh ru ¼ uðf Þ x SðT Þqf 2 : The g tt for Eq. (8) is given by g tt ¼ ðsðt Þqf Þ t þ uðsðt Þqf Þ x þ u 2 g xx ; ð85þ where the time derivatives are computed by Eq. (8). After a simple computation, we will nice that Eq. (82) with the above formulas is identical to Eq. (79) Application of Cauchy Kowalevskaya technique to the macroscopic level Substituting Eq. (79) into Eq. (69), we have the formula of the numerical flux for the Burnett equations. For the smoh reconstruction of the initial data, we can directly apply the Cauchy Kowalevskaya technique to the numerical flux. The Burnett equations are written in the form oh ¼ ow ox ; where W ¼ W þ W þ 2 W 2 ; o x W i ¼ U i : ð87þ ð86þ

14 328 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) The time integration of W gives the formula of the numerical flux. Applying the Cauchy Kowalevskaya technique to W, we have Wð; tþ ¼W þ to t W þðt 2 =2Þo tt W þ ½W þ to t W Šþ 2 W 2 þ Oðt 3 ;t 2 ; 2 tþ; ð88þ where each term on the right-hand side is evaluated at ðx; tþ ¼ð; Þ, o t W is evaluated by the compressible NS equations, and o t W and o tt W are done by the compressible Euler equations. The resulting numerical flux is the same as that obtained in the previous subsections. This is similar to the case of the compressible Euler equations; the well-known Lax-Wendroff scheme, which is nhing more than the product of the Cauchy Kowalevskaya technique at the macroscopic level, is recovered in the kinetic derivation when the continuous piecewise linear reconstruction with smohness at cell interfaces is employed (see, e.g. [2]). In the case of the discontinuous reconstruction, which is employed in various shock capturing schemes, the Cauchy Kowalevskaya technique cann be employed directly. In this case, we can compute the numerical flux as a certain special average of the fluxes computed at bh sides of the cell interface. In the kinetic scheme, the averaging can be done naturally at the level of the distribution function. For the management of discontinuous reconstruction in kinetic schemes, we refer the reader to [4]. 5. Numerical test In this section, we carry out the numerical test of the kinetic scheme for the Burnett equations. The first problem is the structure of the stationary normal shock wave for the upstream Mach number equal to 2 (M ¼ 2). Fig. shows the distributions of the nondimensional density q=q and nondimensional temperature T =T (q, T, and k are, respectively, the density, temperature, and mean free path at upstream condition). Fig. 2 shows the distributions of stress and heat flux (s xx ¼ P p, q x ¼ Q, where the flow is in the x direction). The results for the Burnett equations are shown together with those for the Navier Stokes equations. The present kinetic schemes are modified by introducing the variable collision frequency and the 2.6 M=2 shock structure (hard sphere) T/T and ρ/ρ ρ/ρ : Boltzmann (Ohwada) T/T : Boltzmann (Ohwada) ρ/ρ : Navier-Stokes T/T : Navier-Stokes ρ/ρ : BGK-Burnett T/T : BGK-Burnett x/λ Fig.. Shock structures for M ¼ 2. Density and temperature distributions.

15 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) M=2 shock structure (hard sphere).5 stress and heat flux q x : Boltzmann (Ohwada) τ xx : Boltzmann (Ohwada) τ xx : Navier-Stokes q x : Navier-Stokes τ xx : BGK-Burnett q x : BGK-Burnett x/λ Fig. 2. Shock structure for M ¼ 2. Stress and heat flow distributions. Prandtl number fix technique of [7] so that the stress and the heat flux at the NS level are converted to those for hard-sphere molecules; the NS results correspond to hard-sphere molecules exactly. The results of the Boltzmann equations for hard-sphere molecules in [] are also presented as the standard. These two figures show the improvement of the Burnett solution over the NS. Without the conversion of the stress and heat flux, the agreement with the standard solution becomes poor. On the her hand, as seen in Fig. 2 of [], the deviation of the Boltzmann solution from the local Maxwellian is appreciable even for M ¼ 2; the Boltzmann solution exhibits the bimodal distribution of gas molecules for 3 K M. Small deviation from the local Maxwellian is one of the assumptions of the Chapman Enskog expansion and the present results demonstrate the robustness of the Burnett scheme around the border of the application range of the Chapman Enskog expansion. The present Burnett scheme is n stable for M 4, where the assumption is no longer valid. The second problem is the force-driven Poiseuille flow between two parallel plates. Since the appropriate boundary condition for the kinetic Burnett solver has n yet been established, we adopt the diffuse reflection as a plausible one. In this problem, the nonuniform pressure profile and the bimodal temperature distribution with the central minimum are exhibited as higher order rarefaction effects, which are n predicted by the Navier Stokes system (see, e.g. [2,9,6,9,2]). According to [2], where the systematic asymptic analysis of this problem for small Knudsen numbers is carried out on the basis of the BGK equation, the former characteristic is attributed to the Burnett approximation and the latter is done to the non-navier Stokes stress and heat flow beyond the Burnett approximation. These non-navier Stokes terms beyond the Burnett approximation are treated together with the second order slip boundary conditions. Therefore, the latter characteristic is n expected in the present framework of the Burnett equations and the diffuse reflection. The purpose of the present computation is the confirmation whether the BGK-Burnett scheme really works and captures the non-navier Stokes behavior of slightly rarefied gas flow to some extent. The set up of the problem is given in [2]; bh of the plates are at rest and have the same uniform temperature, which is taken as the reference temperature ^T ; the average density is taken as the reference density ^q ; the external force acting on the unit mass is :28 ð2rt =LÞ, where L is the

16 33 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) distance between the plates; the Knudsen number Kn HS ¼ l HS =L, where l HS is the mean free path of hardsphere molecular gas in the equilibrium state at rest with ^q ¼ ^q, is equal to.. The Burnett equations employed in the numerical test correspond to SðT Þ¼ and Kn HS ¼ : corresponds to ¼ :98 (the mean free path is related to the viscosity and the conversion of l HS to that for the BGK equation is done through the viscosity at ^T ¼ ^T ; no conversion of the heat flux is introduced). The external force is treated as the simple additional term. The results of the present computation are shown in Fig. 3 together with the DSMC results for hard-sphere molecules [2] [pð¼ qt Þ is the nondimensional pressure]. Despite the difference of the molecular model, i.e. SðT Þ¼ and Pr ¼ for the present BGK-Burnett and SðT ÞT =2 and Pr 2=3 for hard-sphere molecular gas, where Pr is the Prandtl number, the agreement with the DSMC result is fair. The curved pressure distribution is well captured around the center but the temperature minimum is n observed in the present computation as expected. The discrepancy is remarkable near the plates, which is considered to be due to the lack of the Knudsen-layer corrections and the employment of the diffuse reflection boundary condition in the present computation. In [2], the NS solution under the first order slip boundary condition (the velocity slip due to shear stress and the temperature jump due to the heat flow normal to the boundary) without the corresponding Knudsen-layer corrections is compared with the DSMC result. For the density q and flow speed U, the agreement with the DSMC is as good as that of the Burnett result. For the pressure p, however, the error is larger than that of the Burnett result; the pressure p is constant (p ¼ :34). As for the temperature T, the shape of the profile is similar to that of the Burnett solution; it takes the maximum value.38 at x ¼. Incidentally, the previous BGK-Burnett scheme [9], which does n take account of the local variation of s, yields almost the identical result to that of the full- Burnett scheme ρ x.5 U x.5 T p x x.5 Fig. 3. Flow distribution in the cross-stream direction. Circles: DSMC results [2], solid lines: BGK-Burnett solution.

17 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) Finally, we mention the computational performance. We have prepared two codes; one is based on Eq. (8) and the her is based on Eq. (88) [Eq. (79)]. The explicit formula of the numerical flux for Eq. (88) [Eq. (79)] is easily obtained if the computer algebra, such as MATHEMATICA, is employed. These two codes are theoretically identical and the numerical agreement is confirmed within the range of the round-off error. The difference is observed only in the computational cost; the code for Eq. (88) [Eq. (79)] is 2.5 times faster than that for Eq. (8) [these codes are n optimized]. This is inferred from the case of the compressible Euler solver that employs smoh reconstruction of initial data; the Lax-Wendroff scheme is faster than the kinetic scheme with the algebraic computation of the numerical flux. The algebraic method becomes useful when the discontinuous reconstruction of the initial data is considered. As mentioned before, this reconstruction is employed in shock capturing schemes. The extension of the present scheme to the case of discontinuous reconstruction is in preparation. 6. Conclusion In this paper, we clearly show the consistency between the Chapman Enskog expansion and the successive approximation for the BGK solution, where the gas distribution function up to the super-burnett order is obtained. Then, a high order numerical scheme for the Burnett equations is constructed by using different methods and the resulting scheme is validated in the shock wave problem and the force-driven Poiseuille flow problem. This paper provides a useful numerical tool for the Burnett equations, which can be used in the study of abnormal phenomena for the Navier Stokes equations, even in the continuum flow regime. Acknowledgements The present study is supported by Grant-in-aid for Scientific Research No from Japan Society for the Promion of Science, Kyo University 2st Century COE Program Center of Excellence for Research and Education on Complex Functional Mechanical Systems, and Hong Kong Research Grant Council. References [] R.K. Agarwal, K.Y. Yun, R. Balakrishnan, Beyond the Navier Stokes: Burnett equations for flows in the continuum-transition regime, Phys. Fluids 3 (2) 36. [2] K. Aoki, S. Takata, T. Nakanishi, Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force, Phys. Rev. E 65 (22) [3] V.V. Aristov, F.G. Tcheremissine, The kinetic numerical method for rarefied and continuum gas flows, in: O.M. Belserkovskii, M.N. Kogan, S.S. Kutateladze, A.K. Rebrov (Eds.), Rarefied Gas Dynamics, Plenum Press, New York, 985, pp [4] R. Balakrishnan, R.K. Agarwal, Higher-order distribution functions, BGK-Burnett equations and Boltzmann H-theorem, in: M. Hafez, K. Oshima (Eds.), CFD Review, vol. 2, 997, pp [5] A.V. Bobylev, The Chapman Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl. 27 (982) [6] C. Cercignani, The Boltzmann Equation and Its Applications, Springer, Berlin, 988. [7] S.Y. Chou, D. Baganoff, Kinetic flux-vector splitting for the Navier Stokes equations, J. Comput. Phys. 3 (997) [8] H. Grad, Principles of the kinetic theory of gases, in: S. Fl ugge (Ed.), Handbuch der physik, vol. XII, Springer, Berlin, 958, pp [9] M.M. Malek, F. Baras, A.L. Garcia, On the validity of hydrodynamics in plane Poiseuille flows, Physica A 24 (997) [] T. Ohwada, Structure of normal shock waves: direct numerical analysis of the Boltzmann equation for hard-sphere molecules, Phys. Fluids A 5 (993)

18 332 T. Ohwada, K. Xu / Journal of Computational Physics 2 (24) [] T. Ohwada, Boltzmann schemes for the compressible Navier Stokes equations, in: T.J. Bartel and M. Gallis (Eds.), Rarefied Gas Dynamics, Proceedings of the International Symposium on Rarefied Gas Dynamics, 2, Sydney; AIP Conference Proceedings 585, American Institute of Physics, 2, pp [2] T. Ohwada, On the construction of kinetic schemes, J. Comput. Phys. 77 (22) [3] T. Ohwada, Higher order approximation methods for the Boltzmann equation, J. Comput. Phys. 39 (998) 4. [4] T. Ohwada, S. Kobayashi, Management of the discontinuous reconstruction in kinetic schemes, J. Comput. Phys. 97 (24) [5] D.I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow, J. Comput. Phys. 34 (98) [6] M. Tij, A. Santos, Perturbation analysis of a stationary nonequilibrium flow generated by an external force, J. Stat. Phys. 76 (994) [7] K. Xu, A gas-kinetic BGK scheme for the Navier Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 7 (2) [8] K. Xu, Regularization of the Chapman Enskog expansion and its description of shock structure, Phys. Fluids 4 (22) 7 2. [9] K. Xu, Super-Burnett solutions for Poiseuille flow, Phys. Fluids 5 (23) [2] Y. Zheng, A.L. Garcia, B.J. Alder, Comparison of kinetic theory and hydrodynamics for Poiseuille flow, in: A.D. Ketsdever, E.P. Muntz (Eds.), Rarefied Gas Dynamics, Proceedings of the International Symposium on Rarefied Gas Dynamics, 22, Whistler; AIP Conference Proceedings 663, American Institute of Physics, 23, pp [2] X. Zhong, R.W. MacCormack, D.R. Chapman, Stabilization of the Burnett equations and application to hypersonic flows, AIAA J. 3 (993) 36.