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A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion

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A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion
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  Acta Mathematicae Applicatae Sinica, English SeriesVol.18, No.1 (2002) 37–62 A Relaxation Scheme for Solving the Boltzmann EquationBased on the Chapman-Enskog Expansion Shi Jin 1 , Lorenzo Pareschi 2 , Marshall Slemrod 3 1 Department of Mathematics, University of Wisconsin, Madison, Van Vleck Hall, WI 53706, USA.(E-mail: jin@math.wisc.edu) 2 Department of Mathematics, University of Ferrara, Via Machiavelli 35, I-44100, Italy & Department of Math-ematics, University of Wisconsin-Madison, Van Vleck Hall, WI 53706, USA. (E-mail: pareschi@dm.unife.it) 3 Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53715-1149, USA.(Email: slemrod@math.wisc.edu) Abstract In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jinand Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weaklyparabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. Itagrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion.We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system forthe Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme,and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and bythe extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments showthat the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equationobtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the otherhydrodynamic systems. Keywords Boltzmann equation, Chapman-Enskog expansion, Burnett equations, relaxation, central schemes 2000 MR Subject Classification 35L99, 65M06, 76P05, 82C40 1 Introduction Dynamics of a moderately rarefied gas of monatomic molecules is often represented by theBoltzmann equation. Observable quantities such as density, velocity, temperature, etc., arederived as expectations of a probability density function f  ( xxx,ξξξ,t ) satisfying the Boltzmannequation (see [7,36]) f  + ξξξ ·∇ f  =1 εQ ( f,f  ) , where xxx denotes the position of a particle at time t moving with velocity ξξξ , Q is the integral col-lision operator, and  is the Knudsen number which is proportional to the mean free path of thegas. The main numerical difficulty to solve the Boltzmann equation is its high dimensionality.There are two practical methods being used in applications. One is the DSMC (Direct Simula-tion Monte-Carlo [3 , 30] and the others are moment methods that provide continuum equationsfor the observable macroscopic equations. DSMC offers much less computational cost than a Manuscript received October 8, 2001. 1 Supported by NSF grant DMS-0196106; 3 Supported by NSF grant DMS-9803223 and DMS-00711463.  38 Shi Jin, et al. deterministic method, but on the other hand it yields low accuracy and statistically fluctuat-ing results and the convergence in general is very slow. Moment methods, among them theGrad’s thirteen moment equations [13] and the extended Thermodynamics equations [26] , definedin physical space, are generally faster than DSMC but the results deviate from that of theBoltzmann at high Mach numbers.In this paper we propose a new numerical scheme based on the Chapman-Enskog expansion(see [7,10,36]) for the Boltzmann equation. This scheme is a numerical discretization of therelaxation approximation proposed by Jin and Slemrod [16 , 17] and its conceptual basis is indeedthe Chapman-Enskog expansion.The classical Chapman-Enskog procedure for the Boltzmann equation is a well known toolfor bridging the gap between kinetic theory as described by the Boltzmann equation for theevolution of a monatomic gas and continuum mechanics. The Chapman-Enskog expansion is aformal power series ordered by the viscosity µ which is itself proportional to the non-dimensionalKnudsen number, i.e., T T T  = −  pI I I  − P P P , p = Rρθ,P P P  = µ ΠΠΠ (1) + µ 2 ΠΠΠ (2) + µ 3 ΠΠΠ (3) + ··· , qqq = µ ΞΞΞ (1) + µ 2 ΞΞΞ (2) + µ 3 ΞΞΞ (3) + ··· . The coefficients ΠΠΠ ( j ) , ΞΞΞ ( j ) , j = 1 , 2 , ··· are obtained from the Boltzmann equation and havebeen determined up to j = 2 (Burnett order) (cf. [10,36]) and in one space dimension up to  j = 3 (super-Burnett order) (cf. [12]). (We remind the readers that all physical quantities inthis paper and their mathematical definitions are given in the Nomenclature at the end of thepaper.)In practice however the Chapman-Enskog expansion as a tool for solving the Boltzmannequation has had limited practical value. Truncation at first order yields the Navier-Stokesequations which as µ ceases to be small becomes a poor approximation to solutions of theBoltzmann equation (cf. [21,26]). Truncation at order µ 2 yields the Burnett equations whichpossesses the unphysical property of yielding linearly unstable rest states (cf. [1,5,23,24,25]).Simply by expanding to the higher order will not remove this instability (cf. [31]).In addition, the Chapman-Enskog expansion destroys the material frame indifference at theBurnett order (cf. [4]).Despite the linear instability of the Burnett equations, numerical solutions on augmentedBurnett equations (cf. [1,11,36]) suggest that they provide more accurate solutions in theshock layer than those of the Navier-Stokes equations when compared with the direct simulationMonte-Carlo method of the Boltzmann equation. In [1,11,36] the augmented Burnett equationswere obtained either by removing the unstable term from or by adding linearly stabilizingterms of the super Burnett order to the stress and heat flux. Unfortunately the augmentedBurnett equations possess two drawbacks. First, numerically they require resolution of thesuper-Burnett stabilizing terms which practically means numerical resolution of derivatives upto fourth order. This is rather a cumbersome approach in several space dimensions. Secondly,the augmented Burnett equations have not been shown to have a globally defined entropypossessing the usual property of satisfying an entropy inequality.In [16], a visco-elastic relaxation approximation was introduced as an approximation to theBoltzmann equation. This relaxation system has the following properties.(i) It requires at most resolution of second derivatives in spatial variables;(ii) it possesses a globally defined “entropy” like function;(iii) when expanded via the Chapman-Enskog expansion, it matches the classical Chapman-Enskog expansion for the Boltzmann equation to the Burnett order.Specifically the pressure deviator and heat flux were relaxed by rate equations to obtain a systemof local equations that can recover the Burnett equations via the Chapman-Enskog expansion  A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion  39 with a correction at the super-Burnett order. By doing this, a system of thirteen local equationswere obtained that is linearly stable. This system is weakly parabolic with a linearly hyperbolicconvection part. Moreover, it is endowed with a generalized entropy inequality. The nonlinearentropy inequality guarantees the irreversibility of the relaxation process. The localness of thissystem is attractive for a robust numerical approximation to the gas dynamics valid to theBurnett order.In recent years, relaxation approximations have been used as an effective tool to designnumerical methods — known as the relaxation schemes. In [18] a generic way to relax a generalsystem of hyperbolic conservation laws was introduced by Jin and Xin, which induced a classof relaxation schemes free of Riemann solver and local characteristic decomposition for inviscidgas dynamics. A physically natural pressure relaxation method was developed by Coquel andPerthame for an inviscid general gas [9] .In this paper, for one-dimensional problem, we propose a class of relaxation schemes forthe Boltzmann equation based on the relaxed Burnett system by Jin and Slemrod. There aretwo main difficulties when discretizing this system. First, the equations for the stress deviatorand heat flux are not in conservative form, thus canonical shock capturing methods, developedby hyperbolic systems of conservation laws, cannot be applied directly. Secondly, the stiff relaxation terms need to be discretized properly so the scheme is efficient even for small meanfree paths.Our relaxation schemes combine a conservative solver for the conserved part of the system(balance laws for density, momentum and energy), while for equations of  P P P  and qqq we discretizethe spatial derivatives using slope limiters and central differences. These discretizations arecarried out conveniently using a staggered grid, as in a staggered non-oscillatory central scheme.We compare the numerical results obtained by this relaxation scheme with those obtained byDSMC, the Navier-Stokes equations and the extended thermodynamics (with thirteen moments)for one-dimensional stationary shocks with various Mach numbers. Our results show that therelaxed Burnett system offers more accurate shock profiles compared to the DSMC than otherhydrodynamic theories.The paper is divided into five sections after this Introduction. Section 2 reviews the relaxedBurnett system introduced by Jin and Slemrod. We also derive boundary conditions for thissystem using the moment definition from the probability density distribution. Section 3 reviewsseveral main properties of the relaxed Burnett system, and computes the linear dispersionrelation. In Section 4 we introduce the numerical discretization for the one-dimensional relaxedBurnett system. In section 5 we solve a one-dimensional stationary shock problem by thenewly introduced relaxation scheme, and compare it with DSMC, Navier-Stokes and extendedthermodynamics. We end the paper with a few concluding remarks in Section 6. 2 The Relaxed Burnett System 2.1 The Field Equations of Balance The field equations of balance for continuum fluid dynamics in the absence of heat sources areas follows˙ ρ + ρ div uuu = 0 (mass conservation) , (1) ρ ˙ uuu + grad  p + div P P P  = ρbbb (linear momentum conservation) , (2) P P P  = P P P  T  (rotational momentum conservation) , (3) ρ ˙e + p div uuu + P P P  · S S S  + div qqq = 0 (energy conservation) , (4)  40 Shi Jin, et al. wheree = ψ − θ∂ψ∂θ, η = − ∂ψ∂θ, p = ρ 2 ∂ψ∂ρ. (5)Differentiation of the expression for the Helmholtz free energy ψ = ε − θη yields ρθ ˙ η = ρ ˙ e − ρ ˙ ρ∂ψ∂ρ, which when combined with (1) and (4), yields the entropy production equation ρθ ˙ η = − P P P  · S S S  − div qqq. (6)Division by θ yields the total entropy product rate of a fluid occupying domain B⊂ IR 3 ddt   B ρηdV  = −   B P P P  · S S S θ + qqq · grad θθ 2 dV  −   ∂  B qqq · nnnθdA. (7)The Clausius-Duhem inequality is a common albeit not universally accepted form of thesecond law of thermodynamics. It asserts ddt   B ρηdV  +   ∂  B qqq · nnnθdA ≥ 0 , which in turn from (7) requires P P P  , qqq to satisfy   B P P P  · S S S θ + qqq · grad θθ 2 dV  ≤ 0for all fluid domains B . However the classical Clausius-Duhem inequality is inconsistent with P P P  , qqq delivered by the Chapman-Enskog expansion beyond Navier-Stokes order. 2.2 The Chapman-Enskog Expansion The Chapman-Enskog expansion for a monatomic gas of spherical molecules yields the consti-tutive relationse =32 Rθ, p = Rρθ, µ = µ ( θ ) , (8) ψ = Rθ log ρ − 32 Rθ log θ +32 Rθ − aθ + b, (9) η = − R log ρ +32 R log θ + a. (10)where a,b are constants of integration.In addition the expansion provides representations for the pressure deviator tensor P P P  andheat flux vector qqq in terms of a series which may be ordered via powers of the viscosity µ interms of the total number of space plus time derivatives. Following the notation of Ferzigerand Kaper [10] we record P P P  = µP P P  (1) + µ 2 P P P  (2) + ··· , (11) qqq = µqqq (1) + µ 2 qqq (2) + ··· , (12)where the expressions for P P P  (1) , P P P  (2) , qqq (1) , qqq (2) are as follows P P P  (1) = − 2 S S S , (13)  A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion  41 qqq (1) = − 32 M R grad θ, (14) P P P  (2) = ω 1 1  p (div uuu ) S S S  + ω 2 1  p  ˙ S S S  − LLLS S S  − S S S L T  +23tr( S S S L T  ) I I I   + ω 3 1 ρθ  grad 2 θ − 13∆ θI I I   + ω 4 1 ρpθ  12grad  p ⊗ grad θ +12grad θ ⊗ grad  p − 13grad  p · grad θI I I   + ω 5 1 ρθ 2  grad θ ⊗ grad θ − 13 | grad θ | 2 I I I   + ω 6 1  p  S S S  2 − 13tr( S S S  2 ) I I I   , (15) qqq (2) = θ 1 1 ρθ (div uuu )grad θ + θ 2 1 ρθ  (grad θ ) • − LLL T  grad θ  + θ 3 1  pρ ( S S S  grad  p ) + θ 4 1 ρ div S S S  + θ 5 1 ρθS S S  grad θ. (16)One drawback of the Chapman-Enskog expansion is that, if truncated at the Burnett orhigher order, it destroys the property of material frame indifference. In particular, in (15) and(16), the ω 2 term in P P P  (2) and the θ 2 term in qqq (2) are both material frame dependent. It cannotbe recovered by replacing the material derivative with the space derivative using the Euler orNavier-Stokes equations [4] .The coefficients ω 1 , ··· ,ω 6 ,θ 1 , ··· ,θ 5 are functions of  θ and are not independent. For agas of spherical molecules the following universal relations have been derived by Truesdell andMuncaster [36] generalizing more specialized relations: ω 3 = θ 4 ,θ 1 =23  72 − µ  ( θ ) µ ( θ ) θ  θ 2 − 13 θ∂θ 2 ∂θ,ω 1 =23  72 − µ  ( θ ) µ ( θ ) θ  ω 2 − 13 θ∂ω 2 ∂θ. (17)Furthermore for gases of ideal spheres in which the collisions are purely elastic or satisfy aninverse k th -power attraction between molecules, the coefficients ω 1 ,ω 2 , ··· ,θ 5 are independentof  θ . In addition the relations θ 1 θ 2 = ω 1 ω 2 =  23  3 k − 5 k − 1  for inverse k th power molecules , 2 for ideal sphereshold.Exact determination of  ω 1 ,ω 2 , ··· ,θ 5 has only been accomplished for a gas of Maxwellian( k = 5) molecules. For the more general case only approximations to ω 1 ,ω 2 , ··· ,θ 5 have beenobtained. The classical approximation result (say as found in [10, p.149]) is ω 2  2 , ω 3  3 , ω 4  0 , ω 5  µ  ( θ ) θω 3 µ ( θ ) , ω 6  8 ,θ 2  458 , θ 3 − 3 , θ 4  3 , θ 5  3  354+ θµµ  ( θ )  , M 52 . (18)For Maxwell molecules the relations (18) are exact θ 1 θ 2 = ω 1 ω 2 =53 ,θµ  ( θ ) µ ( θ )= 1 , and µ is linear in θ .
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