UNIFORMLY ACCURATE SCHEMES FORHYPERBOLIC SYSTEMS WITH RELAXATIONRUSSEL E. CAFLISCHy , SHI JINz , AND GIOVANNI RUSSOxAbstract. We develop high resolution shock capturing numerical schemes for hyperbolic systemswith relaxation. In such systems the relaxation time may vary from order one to much less thanunity. When the relaxation time is small, the relaxation term becomes very strong and highly sti�,and underresolved numerical schemes may produce spurious results. Usually one can not decouplethe problem into separate regimes and handle di�erent regimes with di�erent methods. Thus it isimportant to have a scheme that works uniformly with respect to the relaxation time. Using theBroadwell model of the nonlinear Boltzmann equation we develop a second order scheme that workse�ectively, with a �xed spatial and temporal discretization, for all range of mean free path. Formaluniform consistency proof for a �rst order scheme, and numerical convergence proof for the secondorder scheme are also presented. We also make numerical comparisons of the new scheme with someother schemes. This study is motivated by the reentry problem in hypersonic computations.Key words. hyperbolic systems with relaxation, Broadwell model, sti� source, high resolutionshock capturing methodsAMS subject classi�cations. 35L65, 49M25, 34A65, 82B401. Introduction.Hyperbolic systems with relaxation are used to describe many physical problemsthat involve both convection and nonlinear interaction. In the Boltzmann equationfrom the kinetic theory of rare�ed gas dynamics, the collision (relaxation) terms de-scribe the interaction of particles. In viscoelasticity, memory e�ects are modeled asrelaxation. Relaxation occurs in water waves when the gravitational force balancesthe frictional force of the riverbed. For gas in thermo-nonequilibrium the internalstate variable satis�es a rate equation that measures a departure of the system fromthe local equilibrium. Relaxations also occur in other problems ranging from magne-tohydrodynamics to tra�c ow.In such systems, when the nonlinear interactions are strong, the relaxation rateis large. In kinetic theory, for example, this occurs when the mean free path betweencollisions is small (i.e. the Knudsen number is small). Within this regime, which isreferred to as the uid dynamic limit, the gas ow is well-described by the Euleror Navier-Stokes equations of uid mechanics, except in shock layers and boundarylayers. The characteristic length scale of the kinetic description of the gas is the micro-scopic, collision distance; in the uid dynamic limit it is replaced by the macroscopiclength scale of uid dynamics. By analogy with the kinetic theory, we shall refer tothe limit of large relaxation rate (or small relaxation time) for a general hyperbolicsystem with relaxation as the uid dynamic limit.The uid dynamic limit is challenging for numerical methods, because in thisregime the relaxation terms become sti�. In particular, a standard numerical schememight fail to give physically correct solutions once the (microscopic) relaxation dis-tance is much smaller than the spatial discretization. Although a full simulation ofy Department of Mathematics, University of California, Los Angeles, CA 90024, USA. Researchsupported in part by the Army Research O�ce under grant number DAAL03-91-G-0162.z School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. Research wassupported in part by NSF grant DMS-9404157 and by AFOSR grant F49620-92-J0098 as a visitingmember at Courant Institute.x Dipartimento di Matematica, Universit�a dell'Aquila, 67010 L'Aquila, Italy.1

2 R. E. CAFLISCH, S. JIN and G. RUSSOthe relaxation process would require a very �ne (and expensive) discretization, it maybe possible to accurately compute the solution on a coarser uid dynamic length scale.The goal of this paper is to present a class of numerical methods that work with uni-form accuracy from the rare�ed regime to the uid dynamic limit for the Broadwellmodel of kinetic theory.Numerical methods for hyperbolic systems with relaxation terms have attracted alot of attention in recent years [7], [23], [24], [16], [17]. Studying the numerical behaviorfor these problems is important not only for the physical applications, but also for thedevelopment of new numerical methods for conservation laws, such as kinetic schemes([15], [8], [25]) and relaxation schemes [18]. Most kinetic or relaxation schemes can bedescribed as fractional step methods, in which the collision step is just a projection ofthe system into a sort of discrete \local Maxwellian" or local equilibrium. Althoughthe goal of a kinetic scheme or relaxation scheme is di�erent from ours, neverthelesswe use them as guidelines for the study of the properties of a numerical scheme nearthe uid dynamic regime.In earlier works on system of hyperbolic equations with relaxation, the goal was todevelop robust numerical schemes that handle e�ectively the sti�ness of the problemand avoid spurious numerical solutions when the grid spacing underresolves the smallrelaxation time. In regions where the relaxation time is no longer small and theproblem becomes nonsti�, however, these schemes usually may not have high orderaccuracy uniformly with respect to the wide range of the relaxation time.Our motivation di�ers from these earlier approach in that we seek to developrobust numerical schemes that work uniformly for a wide range of relaxation rate.We consider a simpler model of the Boltzmann equation, and we derive a numericalscheme which is of second order uniformly in the mean free path. This is motivatedby hypersonic computations of reentry problems. The new scheme we introduce here,called NSPIF, is able to handle all di�erent regimes, from the rare�ed gas to the uidlimit (the sti� regime), with a �xed spatial and temporal grids that are independentof the mean free path. Although we develop our method based on the Broadwellmodel, this scheme can be applied to a much wider class of hyperbolic systems withrelaxation terms and to other discrete velocity kinetic models. In particular, it appliesto a class of hyperbolic systems with relaxation characterized by Liu [22] and Chen,Levermore and Liu [5].Probably the paper that is closest in spirit to our work is the one by Coron andPerthame [7]. In that paper the authors derive a numerical scheme for solving theBGK model of the Boltzmann equation under a wide range of mean free path. Theydiscretize velocity space and use a splitting scheme. The collision step is treatedby a semi-implicit method that guarantees positivity and entropy condition for thetime-discrete model. The scheme is �rst order accurate in space and time.Development of numerical methods for the problems considered here is consid-erably aided by knowledge of the equations for the uid dynamic limit. In othersti� source problems, such as those arising in reacting ow computations, the corre-sponding limit may be less well understood, and extra e�orts are necessary to developunderresolved numerical methods ([6], [21], [9], [23]).The outline of the paper is the following. In the next section we present theBroadwell model and its uid dynamic limit. Section 3 deals with the treatment ofthe convective step. Upwind methods and ux limiters are brie y recalled. Sections4-7 are devoted to the development and analysis of a second order scheme (in spaceand time) called NSP. In Section 4, the scheme is derived making use of truncation

HYPERBOLIC SYSTEMS WITH RELAXATION 3analysis, and in the following section the linear stability of this scheme is studied. InSection 6 Richardson extrapolation is used for the �rst time step, in order to guaranteesecond order accuracy even in presence of an initial layer. The NSP combined withthis initial step is called the NSPIF. In Section 7 the uid dynamic limit of theNSPIF is derived. In Section 8 we study a �rst order splitting scheme (called theSP1). Important properties, such as positivity and entropy inequality, are proved forthe scheme. The main result in this section is a uniformly (in the relaxation time)�rst order consistency error estimate. Our analysis is based on the assumption ofsmoothness of the solution. To the best of our knowledge this is the �rst estimateon numerical methods for hyperbolic systems with sti� relaxation terms, even forsmooth solutions. Also the uniformly �rst order accuracy sharpens earlier work inthis direction [14]. Estimates on non-smooth solutions or on higher order methodsfor such a inhomogeneous hyperbolic systems would require more advanced analytictechniques and are beyond the scope of this paper. In the last section we present somenumerical results, which show that NSPIF overperform several other approaches andalways give second order, physical solutions over a wide range of the mean free path.2. The Broadwell Model.A simple discrete velocity kinetic model for a gas was introduced by Broadwell[2]. It describes a 2-D (3-D) gas as composed of particles of only four (six) velocitieswith a binary collision law and spatial variation in only one direction. When lookingfor one dimensional solutions of the 2-D gas, the evolution equations of the model aregiven by @tf + @xf = 1" (h2 � fg) ;@th = �1" (h2 � fg) ;(2.1) @tg � @xg = 1" (h2 � fg) ;where " is the mean free path, f , h and g denote the mass densities of gas particleswith speed 1; 0 and �1 respectively in space x and time t. Similar equations areobtained for 1-D solutions of a 3-D gas [3].The uid dynamic moment variables are density �, momentumm and velocity ude�ned by � = f + 2h+ g ; m = f � g ; u = m� :(2.2)In addition, de�ne z = f + g :(2.3)Then the Broadwell equations can be rewritten as@t� + @xm = 0 ;(2.4) @tm+ @xz = 0 ;(2.5) @tz + @xm = 12" (�2 +m2 � 2�z) :(2.6)Note that if the uid variables �;m and z are known then f; g and h can be recovedfrom (2.2) and (2.3) asf = 12(z +m) ; g = 12(z �m) ; h = 12(� � z) :

4 R. E. CAFLISCH, S. JIN and G. RUSSOA local Maxwellian is a density function that satis�esQ(f; h; g) = h2 � fg = �2 +m2 � 2�z = 0 ;(2.7)i.e. z = zE(�;m) � 12� (�2 +m2) = 12(� + �u2)(2.8)As " ! 0 Eq. (2.1) or (2.6) gives the local Maxwellian distribution (2.8). Applying(2.8) in (2.5), one gets the uid dynamic limit described by the following model Eulerequations @t� + @x(�u) = 0 ;(2.9) @t(�u) + @x�12(�+ �u2)� = 0 :To the next order, a model Navier-Stokes equation can be derived via the Chapman-Enskog expansion [4]. For a description of the Broadwell model and its uid dynamiclimit see, for example, references [3] and [13].Previously the numerical solution of the Broadwell equations has been consideredby several authors [10, 12, 1]. In these earlier works the attention was focused ondeveloping methods that work in the rare�ed regime � = O(1). Many of these methodswill have problems when � ! 0. Among the problems arose in the uid regimesare numerical instability, poor shock and rarefaction resolutions, and even spuriousnumerical solutions. See the numerical examples in Section 9.The Broadwell equations are a prototypical example for more general hyperbolicsystems with relaxations in the sense of Whitham [29] and Liu [22]. These problemscan be described mathematically by the system of evolutional equations@tU + @xF (U ) = �1" R(U ) ; U 2 RN :(2.10)We will call this system the relaxation system. Here we use the term relaxation in thesense of Whitham [29] and Liu [22]. The relaxation term is endowed with an n � Nconstant matrix Q with rank n < N such thatQR(U ) = 0 ; for all U :(2.11)This yields n independent conserved quantities v = QU . In addition we assumethat each such v uniquely determines a local equilibrium value U = E(v) satisfyingR(E(v)) = 0 and such that QE(v) = v ; for all v :(2.12)The image of E then constitutes the manifold of local equilibria of R.Associated with Q are n local conservation laws satis�ed by every solution of(2.10) and that take the form@t(QU ) + @x(QF (U )) = 0 :(2.13)These can be closed as a reduced system for v = QU if we take the local relaxationapproximation U = E(v) ;(2.14)

HYPERBOLIC SYSTEMS WITH RELAXATION 5@tv + @xe(v) = 0 ;(2.15)where the reduced ux e is de�ned bye(v) � QF (E(v)) :(2.16)A system of conservation laws with relaxation is sti� when " is small compared to thetime scale determined by the characteristic speeds of the system and some appropriatelength scales. While we mainly concentrate on the Broadwell equation, the analysisas well as the numerical schemes can certainly be applied to this class of relaxationproblems. In fact from time to time we will use the general equation (2.10) to simplifythe notation.3. The Discretizations of the Convection Terms.We introduce the spatial grid points xj+ 12 ; j = � � � ;�1; 0; 1; � � � with uniformmesh spacing �x = xj+ 12 � xj�12 for all j. The time level t0; t1; � � � are also spaceduniformly with space step �t = tn+1 � tn for n = 0; 1; 2; � � �. Here the assumption ofa uniform grid is only for simplicity. We use Unj to denote the cell average of U in thecell [xj�12 ; xj+12 ] at time tn,Unj = 1�x Z xj+ 12xj� 12 U (tn; x) dx :(3.1)We use the method of lines { in which the time discretization and spatial dis-cretization are taken separately { for the Broadwell equations. In this section weshall discuss the spatial discretization, which concerns the linear convection terms.Note that the linear convection in the Broadwell equation is of hyperbolic type. Thusit is natural to use upwind schemes.A conservative spatial discretization to the Broadwell equations (2.4)-(2.6) is@t�j + mj+ 12 �mj� 12�x = 0 ;(3.2) @tmj + zj+ 12 � zj�12�x = 0 ;(3.3) @tzj + mj+ 12 �mj� 12�x = 12" (�2j +m2j � 2�jzj) :(3.4)Eqs. (3.2)-(3.4) can be diagonalized into@tfj + fj+ 12 � fj� 12�x = 1" (h2j � fjgj) ;@thj = �1" (h2j � fjgj) ;(3.5) @tgj � gj+ 12 � gj�12�x = 1" (h2j � fjgj) ;where f , h and g are exactly the original density functions for the Broadwell equation.The connection between (3.4) and (3.5) are established through the de�nition of the uid moment variables (2.2), (2.3). Since f and g travel along the constant charac-teristics with speeds 1 and �1 respectively, upwind schemes can be easily applied tothem.

6 R. E. CAFLISCH, S. JIN and G. RUSSO3.1. The Upwind Scheme. The upwind scheme applied to f and g givesfj+ 12 = fj ; gj+ 12 = gj+1 ;(3.6)while hj is constant. This implies(z +m)j+ 12 = (z +m)j ; (z �m)j+ 12 = (z �m)j+1 ;(3.7)or equivalently in uid moment variablesmj+ 12 = mj+1 +mj2 � zj+1 � zj2(3.8) zj+ 12 = zj+1 + zj2 � mj+1 �mj2 :Applying (3.8) in (3.3) gives the semi-discrete upwind scheme for the convectionstep: @t�j + mj+1 �mj�12�x � zj+1 � 2zj + zj�12�x = 0 ;@tmj + zj+1 � zj�12�x � mj+1 � 2mj +mj�12�x = 0 ;(3.9) @tzj + mj+1 �mj�12�x � zj+1 � 2zj + zj�12�x = 12" (�2j +m2j � 2�jzj) :3.2. van Leer's MUSCL Scheme. A second order extension of the upwindscheme is van Leer's MUSCL scheme [28]. While the upwind scheme uses piecewiseconstant interpolation (3.6), the MUSCL uses piecewise linear interpolation, alongwith slope limiters to eliminate numerical oscillations at discontinuities. ApplyingMUSCL to the Riemann Invariants f and g, respectively, we obtainfj+ 12 = fj + 12�x�+j ; gj+12 = gj+1 � 12�x��j+1:(3.10)Here �+j and ��j are the slope of f and g on the jth cell respectively. For w+ =f; w� = g, �� are de�ned by [20]��j = 1�x(w�j+1 �w�j�1)�(��) ; �� = w�j �w�j�1w�j+1 � w�j(3.11)and the slope limit function �(�) by van Leer is�(�) = j�j+ �1 + j�j :(3.12)With this limiter the MUSCL scheme is total-variation-diminishing (TVD). RewritingEq. (3.10) in the uid moment variables givesmj+ 12 = mj+1 +mj2 � zj+1 � zj2 + �x4 (�+j + ��j+1) ;(3.13) zj+ 12 = zj+1 + zj2 � mj+1 �mj2 + �x4 (�+j � ��j+1) :(3.14)

HYPERBOLIC SYSTEMS WITH RELAXATION 7Applying (3.13), (3.14) in (3.3) �nally gives@t�j + mj+1 �mj�12�x � zj+1 � 2zj + zj�12�x+ �x4 (�+j + ��j+1 � �+j�1 � ��j ) = 0 ;@tmj + zj+1 � zj�12�x � mj+1 � 2mj +mj�12�x+ �x4 (�+j � ��j+1 � �+j�1 + ��j ) = 0 ;(3.15) @tzj + mj+1 �mj�12�x � zj+1 � 2zj + zj�12�x(3.16) + �x4 (�+j + ��j+1 � �+j�1 � ��j ) = 12" (�2j +m2j � 2�jzj) :4. A Uniformly Second Order Time Discretization. In this and in the nextthree sections we develop a second order scheme, uniformly accurate in the relaxationparameter, and study its properties. Since our goal is to seek a robust scheme thatworks uniformly for all range of ", it is essential that the time discretization is stablefor every ". This is especially important near the uid regime where the mean freepath is small and the problem becomes sti�.Uniformly numerical stability can be achieved with implicit temporal integrators.Since sti�ness appears only through the relaxation term, it is natural to keep theconvection terms explicit, while the collision step has to be treated implicitly.Numerical experience for such relaxation problems shows that this problem isnot merely a numerical stability problem. Stable numerical discretization may stillproduce spurious solutions [23], [24], [16]. For example, the Crank-Nicholson schemezn+1j = znj + �t2" [(�nj )2 + (mnj )2 � �nj (znj + zn+1j )] ;(4.1)coupled with the free streaming convection with �x = �t gives a scheme that isstable independent of ", but does not have the correct uid limit.Previous results [16] demonstrate that an e�ective condition for the correct nu-merical behavior near the uid regime is that the numerical scheme should possessthe correct uid limit, in the sense that a discrete analogue of the Chapman-Enskogexpansion for the continuous equations remains valid for the numerical discretization,and the resulting numerical uid limit should be a good discretization for the modelEuler limit. A su�cient condition to achieve this is that the collision step alwaysprojects the nonequilibrium data into a local Maxwellian at every time step.We now show how to construct a uniformly second order scheme. The basic ideais to combine a high order convection step with an implicit collision step according tothe following guidelines:(i) Truncation error analysis is used to obtain second order accuracy in therare�ed regime (" = O(1)).(ii) The collision step is well-posed 8" > 0, and it relaxes the system to a localMaxwellian as "! 0.(iii) The scheme should be unconditionally stable in the collision step.(iv) Check that the limit scheme obtained as "! 0 is a second order scheme forthe model Euler equations when " = 0.

8 R. E. CAFLISCH, S. JIN and G. RUSSOIt is possible to show that if condition ii) is not satis�ed, then the scheme maygive the wrong behavior in the uid dynamic limit (this is the case, for example, ofthe scheme in [10]).Such splitting scheme, applied to system (2.10), can be written in the form:U1 = Un � ��t" R(U1) ;(4.2) U2 = U1 � ~��tDF (U1) ;(4.3) U3 = U2 � ��t" R(U3)� �t" R(U1) ;(4.4) U4 = U3 � ~��tDF (U3) ;(4.5) U5 = �Un + �U4 ;(4.6) Un+1 = U5 � ��t" R(Un+1) :(4.7)where D denotes a discrete approximation of the convection operator, such as thesimple upwind or a second (or higher order) MUSCL scheme. We assume that D is atleast second order accurate, so that the splitting scheme will be second order accurateboth in space and time.The scheme is a fractional step combination of convection and collision steps. Theparameters �; �; ; �; ~�; ~�; �; � will be determined by conditions i)-iii).Roughly speaking, this splitting scheme mimics the asymptotic process that leadsfrom the Broadwell equations to the model Euler equations. At t = t(1), the sti�source solver gives R(U1) � 0, which is the local Maxwellian. Applying it to the nextconvection step t = t(2) should give a numerical ux that approximates the ux forthe model Euler equation. Similar behavior occurs at t = t(3) and t(4). The last stepgives R(Un+1) � 0, guaranteeing the correct local Maxwellian at t = tn+1. The abovesteps are combined in a second order way, aiming at a scheme with uniform secondorder accuracy.In order to satisfy condition i) we apply the scheme to the linear system@tU + AU +BU = 0 ;(4.8)where A and B are constant matrices. The exact solution of (4.8) at time t = �t isgiven by U (�t) = e�(A+B)�tU (0):Applying scheme (4.2{4.7) to (4.8), and write the di�erence equation in the compactform Un+1 = CUn :(4.9)To achieve a second order accuracy we impose that(C(�t)� e�(A+B)�t)U (0) = O(�t3) ;then the following restrictions on the parameters should be satis�ed:� + � = 1 ;�(~�+ ~�) = 1 ;

HYPERBOLIC SYSTEMS WITH RELAXATION 9�(�+ � + ) + �(� + �) = 1 ;2�~�~� = 1 ;(4.10) 2�(�~�+ �~� + ~� + ~��) = 1 ;2�(~�� + �~�+ �~�) = 1 ;2�(�2 + � + �� + � + �2) + 2��(�+ � + ) + 2(� + �)�2 = 1 :This is a system of seven algebraic equations with eight unknowns. The system canbe explicitly solved by expressing all the parameters as function of �. The solution isgiven by: � = 0; � = 2�� 12(� � 1) ; = �2�2 � 2�+ 12�(�� 1) ;~� = 12�; ~� = � 12(� � 1) ; � = 2�2 � 2�+ 1; � = �2�(� � 1):We shall use this freedom to satisfy the second condition ii). When applying thescheme to the Broadwell model, the collision step (4.2) becomesz1 = zn + �(�t=2")(�21 +m21)1 + �(�t=")�1 :Therefore for positive data, both numerator and denominator are positive provided� � 0. The same property holds for step (4.7), provided � � 0.Step (4.4) becomes:z3 = z2 + �(�t=2")(�22 +m22) + (�t=2")(�21 +m21 � 2�1z1)1 + �(�t=")�2 :The denominator is positive for positive data, provided � � 0. We therefore look fora solution of equations (4.10) that satis�es also the restrictions:� � 0; � > 0; � > 0 :(4.11)A set of parameters satisfying equations (4.10) and the conditions (4.11) is givenby: � = 5=9; � = 4=9; � = 0; � = 1=3; � = 1=4; = 5=4; ~� = 3=2; ~� = 3=4:(4.12)Linear stability analysis of the scheme is performed in the next section. The uiddynamical limit of the scheme, which is required to meet the condition iv), will bestudied in Section 7.The construction of the time discretization that combines the convection and therelaxation term is similar in spirit to the time discretizations used in [9] and [16].Both time discretizations of [9] and [16] have negative parameters in the implicitterms, which may cause numerical breakdown in the intermediate regime �t = O(").Finally our new second order scheme (abbreviated as NSP) for the Broadwellequation (2.4)-(2.6) is (3.3)-(3.7), where R stands for the source term, DF is theMUSCL ux (3.13) and (3.14), and the coe�cients given in (4.12).

10 R. E. CAFLISCH, S. JIN and G. RUSSO5. Stability Analysis.In this section we study the stability of scheme (4.2{4.7) applied to Eqn. (4.8).The main result of this section is summarized in Proposition 2, which states that,under a certain condition on the explicit step, the scheme is unconditionally stable inthe implicit step.We interpret U as a complex function and A, B as complex numbers with positivereal part. Eqn. (4.9) becomes Un+1 = P (z1; z2)Un;(5.1)where z1 = A�t, z2 = B�t, and P is a rational function of z1 and z2:P = P0(z1) + a(1� z1)z2(1 + �z2)(1 + bz2)with P0(z1) = 1� z1 + z21=2, a = �� 1=(2� 2�), b = (1� 2�)=(2� 2�). The stabilityregion associated to scheme (5.1) is the set S � C2 : S = f(z1; z2) : jP (z1; z2)j � 1g.For B = 0 the scheme is a second order explicit scheme for the equationUt +AU = 0;(5.2)and for A = 0 the scheme is a second order implicit scheme for the equationUt + BU = 0:(5.3)We now show that for A = 0 the scheme is a A-stable scheme for the equation (5.3),provided 0 < � < 1=2.It is PB(z2) � P (0; z2) = 1 + az2(1 + �z2)(1 + bz2) :The scheme is A-stable if the stability regionSB = fz2 2 C : jP (0; z2)j < 1gcontains the complex half plane C+ � fz 2 C : <(z) > 0g. We express the stabilityresult relative to Eqn. (5.3) in the following Proposition.Proposition 5.1. If 0 < � < 1=2 then scheme (4.2{4.7) is a L-stable schemefor Eqn (5.3).Proof. First observe that if 0 < � < 1=2, then PB(z2) has two real negative poles,therefore it is analytic in the whole half plane <(z2) � 0. By the maximum principleof analytic functions, it follows that the maximum of jPB(z2)j on any region � Clies on the boundary of . Because limz2!1 PB(z2) = 0, it follows that the maximumof jPB(z2)j, z2 2 C+, lies on the imaginary axis. It is therefore enough to prove thatjPB(iy)j � 1; 8y 2 R. It isjPB(iy)j2 � 1 = � �2b2y4(1 + �2y2)(1 + b2y2) ;and therefore jPB(z2)j � 1 8z2 2 C+. This proves that the scheme is A-stable.Moreover, it is limz2!1 PB(z2) = 0, and therefore the scheme is L-stable.

HYPERBOLIC SYSTEMS WITH RELAXATION 11Now we set B = 0 and study the stability properties of the explicit scheme forthe equation Ut + AU = 0:It is P (z1; 0) = P0(z1) = 1� z1 + 12z21;therefore there is no dependence on �. The stability region for z1 is de�ned asSA = fz1 2 C : jP (z1; 0)j � 1gIt is convenient to write z1 = 1 + �ei�. Then it is P (z1; 0) = (1 + �2e2i�)=2. Theboundary of the stability region is obtained by the equation jP (z1; 0)j2 = 1, whichgives: �4 + 2�2 cos 2� � 3 = 0and therefore the two branches of the boundary are given by:�� = �q� cos 2� +pcos2 2� + 3; � 2 [0; �)Now we consider the general problem with z1 6= 0; z2 6= 0. We prove the followingProposition.Proposition 6.2. If jz1 � 1j < 1 and <(z2) > 0 then it is jP (z1; z2)j < 1.Proof. It is P (z1; z2) = P0(z1) + a(1� z1)z2(1 + �z2)(1 + bz2)If 0 < � < 1=2, it is a < 0; b > 0. For any �xed z1, P (z1; z2) is analytic in the halfplane <(z2) � 0, and it vanishes at in�nity. Therefore the function has its maximumon the imaginary axis. The square of the modulus of P (z1; z2) on the imaginary axiscan be expressed in the form: jP (z1; iy)j2 = NDwith N = 1 + 4a2�2y2 + �4 + 4�ay(1 � �2) sin � + 2�2 cos 2�;(5.4) D = 4(1 + �2y2)(1 + b2y2)(5.5)and z1 has been written as z1 = 1+ �ei�. We shall prove that if � < 1 then D�N >0; 8y 2 R.It is � � D � N = �+ � sin � � 2�2 cos 2�where � = 3� �4 + 4a2(1� �2)y2 + 4�2b2y4, � = 4�ay(�2 � 1) (we used the identity�2 + b2 = a2).The extrema of � as function of � are obtained by @�=@� = 0, and are given by

12 R. E. CAFLISCH, S. JIN and G. RUSSOi) cos � = 0ii) 8�2 sin � +� = 0We shall consider the two cases separately.i) With cos � = 0 it is � = �� � � + 2�2 � �. The minimum is obtained by� = �� if y > 0 and by � = �+ if y < 0. Because � is an even function ofy and � is an odd function of y, it is enough to consider the case � = ��,y > 0. It is��(�; y) = 3 + 2�2 � �4 � 4�(�2 � 1)ay + 4a2(1� �2)y2 + 4�2b2y4> G(�2; y) + 4�2b2y4where G(r; y) = 3 + 2r� r2 � 4(r � 1)ay + 4a2(1 � r)y2:If 0 � r � 1, then G(r; y) > 0; 8y 2 R. In fact G(r; y) = c0(r) + 4c1(r)y +4c2(r)y2 and �216 � c21 � c0c2 = a2(1� r)(r2 � 2r� 2) < 0therefore �� > 0, and D � N > 0; 8� � 1; y 2 R; cos � = 0.ii) For sin � = ��=(8�2) it is� = �� �216�2 � 2�2= 3� 2�2 � �4 + a2(1� �2)(3 + �2)y2 + 4�2b2y4;which is positive for � < 1. Therefore D � N > 0 ; 8� < 1; y 2 R, sin � =��=(8�2).Because � is positive at the extrema, it is always positive for any � 2 [0; 1); � 2[0; �); y 2 R, and therefore if jz1 � 1j < 1 and <(z2) > 0 it is jP (z1; z2)j < 1.6. An Initial Layer Fix.A good feature of the time discretization in [16] is that it has the correct initiallayer behavior, since in the �rst step it projects into the local Maxwellian, and inthe uid limit it becomes the second order TVD Runge-Kutta method for the uidequation. Our scheme here does not have such a mechanism, because as a result oftruncation analysis, � = 0 in Eq. (4.2), thus the de facto �rst step (4.4) is not aprojection into the local Maxwellian. This will introduce an initial disturbance if theinitial layer is not well resolved. Although at later time the scheme has the mechanismto project the data into the local Maxwellian, this initial error will remain at all latertimes, causing nonconservative numerical solutions, thus degrading the quality of thenumerical results. This has been demonstrated in [16]. Hence an extra care must betaken to properly handle the initial layer.One possibility to overcome this burden is simply to resolve the initial layer, byusing a time step �t� " in the �rst few steps, and then switch to a larger time step.This procedure is very expensive if the problem also contains nonsti� regions where" is not small and one does not need to resolve the initial layer.The initial layer analysis performed in [3] indicates that the initial layer projectsthe initial data into the local Maxwellian. In order to have the correct behavior, the

HYPERBOLIC SYSTEMS WITH RELAXATION 13numerical scheme should have the same projection in the �rst time step. Note that inthe �rst order splitting scheme (8.2), the �rst step, due to its fully implicit collisionterm, always projects the initial data into the local Maxwellian, thus can be used forthe �rst time step along with our new splitting scheme. The �rst order accuracy ofthis initial step will be overcome by a Richardson type extrapolation on (8.2).Let us write the exact solution to (2.10) asU (t) = S(t)U (0) ;(6.1)where S(t) is the evolutional operator and U (0) is the initial datum. Suppose thedi�erence operator is denoted by T (t). Then the numerical solution u(t) is given byu(t) = T (t)U (0) :(6.2)A �rst iteration of the splitting scheme (8.2) can be written asu1(�t) = T (�t)U (0) ;(6.3)while a two half step iterations giveu2(�t) = T (�t2 )2U (0) :(6.4)We de�ne our �rst time step asu(�t) = 2u2(�t)� u1(�t) :(6.5)This Richardson extrapolation will give a second order approximation, as will bedemonstrated now.Since the splitting scheme (8.2) is globally �rst order (locally second order), onehas u1(�t)� U (�t) = (T (�t)� S(�t))U (0) = C�t2 +O(�t3) ;(6.6)where C depends on T ;S and U (0). Of course this impliesu1(�t2 )� U (�t2 ) = (T (�t2 )� S(�t2 ))U (0) = 14C�t2 +O(�t3) :(6.7)Also, T (�t) = I + O(�t)(6.8)where I is the identity operator. Now,u(�t) = 2u2(�t)� u1(�t)= 2T (�t2 )2U (0)� T (�t)U (0)= 2T (�t2 )[S(�t2 ) + T (�t2 )� S(�t2 )]U (0)�[S(�t) + T (�t)� S(�t)]U (0)

14 R. E. CAFLISCH, S. JIN and G. RUSSOMaking use of (6.6) one has:u(�t) = 2T (�t2 )U (�t2 ) + 2T (�t2 )[14C�t2 + O(�t3)]�U (�t)� C�t2 +O(�t3)= 2[S(�t2 ) + T (�t2 )� S(�t2 )]U (�t2 )� U (�t)� 12C�t2 +O(�t3)= 2U (�t) + 2[14C�t2 + O(�t)3]� U (�t)� 12C�t2 + O(�t3)= U (�t) +O(�t3) :Thus we have a locally third, or globally second order approximation in this �rst step.In fact this second order scheme alone can be applied to system (2.10). However suchscheme would be slightly less e�cient that the one we use (4.2{4.7), since one has tointegrate the convection part three times in each time step, while (4.2{4.7) only needstwo. This Richardson extrapolation (6.5) may also lose positivity of the numericalsolution.The NSP, with the �rst step given by the initial layer �x step described in thissection, will be called NSPIF.7. The Numerical Fluid Dynamic Limits.To show that the scheme has the correct uid dynamic limit as � ! 0 (thecondition iv in Section 4), we assume that the solution is smooth in the sense thatall of the spatial derivatives are of O(1). Such an assumption is necessary since thecontinuous Chapman-Enskog expansion, which leads from the Broadwell equation tothe Euler equation, is only justi�ed for smooth solutions [3] or weak shocks [30]. Thisformal analysis, although not rigorous, does provide a e�ective tool to understandunderresolved (grid spacing does not resolve �) numerical methods for hyperbolicsystems with sti� relaxations [19], [17],[16], [18].Following [19], we distinct the grid spacing into three regimes. In the thin regime,max(�x;�t) = o(�). In the intermediate regime, max(�x;�t) = O(�). In the coarseregime, �=max(�x;�t) ! 0. In the thin regime, the grid space resolves �, thus thetruncation error analysis in Section 4 is already su�cient to show the second orderaccuracy of the scheme (if the MUSCL is applied for the convection term). It is in theintermediate and coarse regimes that the uid dynamic limit analysis will be applied.7.1. The Coarse Regime. Assume that "=�t� 1; "=�x� 1. The asymptoticanalysis will be performed with �xed �t and �x, while letting "! 0.We �rst analyze the limiting behavior of the two spatial discretizations discussedin Section 3 and defer the time discretization to later paragraphs. Here we are con-cerned with space discretization, and therefore we do not perform any discretizationin time.First, for the upwind discretization (3.9) of the Broadwell equation, as " ! 0,then the third equation of (3.9) gives the leading term approximationz = 12� (�2 +m2) +O(") :(7.1)Applying this to the �rst two equations in (3.9) implies (after ignoring the O(") term)@t�j + mj+1 �mj�12�x � 12 (� + �u2)j+1 � (� + �u2)j + 12 (�+ �u2)j�12�x = 0 ;

HYPERBOLIC SYSTEMS WITH RELAXATION 15@tmj + 12(� + �u2)j+1 � 12(� + �u2)j�12�x � mj+1 � 2mj +mj�12�x = 0 :(7.2)This gives the limiting scheme of the upwind scheme. It can be easily seen that thisis a �rst order centered conservative discretization of the model Euler equation (2.9).Thus the upwind scheme has the correct asymptotic limit which is only �rst order.For the MUSCL discretization (3.15), as " ! 0, the last equation again gives(7.1), which can be applied to the �rst two equations in (3.15) to yield the limitingscheme of the MUSCL:@t�j + mj+1 �mj�12�x � 12 (�+ �u2)j+1 � (� + �u2)j + 12(� + �u2)j�12�x+ �x4 (�E+j + �E�j+1 � �E+j�1 � �E�j ) = 0 ;(7.3) @tmj + 12 (� + �u2)j+1 � 12 (� + �u2)j�12�x � mj+1 � 2mj +mj�12�x+ �x4 (�E+j � �E�j+1 � �E+j�1 + �E�j ) = 0 ;(7.4)where �E�j = ��j jz= 12� (�2+m2) :(7.5)Note that ��j is de�ned by �;m and z, and the right hand side of (7.5) simply meansapplying the local Maxwellian z = (�2 +m2)=(2�) into ��j . De�ningwE+ = f jz= 12� (�2+m2) = 12(z +m)���z= 12� (�2+m2) = 12 � 12� (�2 +m2) +m� ;(7.6) wE� = gjz= 12� (�2+m2) = 12(z �m)���z= 12� (�2+m2) = 12 � 12� (�2 +m2)�m� :(7.7)Then from (3.11),�E�j = 1�x(wE�j+1 �wE�j�1)�(�E�) ; �E� = wE�j �wE�j�1wE�j+1 � wE�j(7.8)One can see that this is a second order centered conservative discretization of themodel equations (2.9). Thus the MUSCL also has the correct uid limit which issecond order to (2.9).We now demonstrate that the new temporal splitting scheme (4.2)-(4.7) has thecorrect uid limit in the sense to be speci�ed below. We always assume that R(U ) isLipschitz continuous in U , and I+��t" R0 is invertible for all " and �t where R0 is theJacobian matrix of R. This invertibility is true for general hyperbolic systems withrelaxations classi�ed by Liu [22]. In particular, it is easy to check that the collisionterm of the Broadwell equations satis�es this condition.By the initial layer �x, we can assume thatR(Un) = O(") :(7.9)Now, we want to show that this condition will imply thatR(U1) � 0 ; R(U3) � 0 ; R(Un+1) � 0 ;(7.10)

16 R. E. CAFLISCH, S. JIN and G. RUSSOup to some error terms depending on " and �t. First, since � = 0, thus U1 = Un,and R(U1) = R(Un) = O(")(7.11)Using (4.3), U2 � U1 = O(�t)(7.12)By (4.4),U3 � U2 = ���t" (R(U3) �R(U2)) � ��t" R(U2) +O(�t)= ���t" R0(U�)(U3 � U2)� ��t" (R(U1) + O(�t)) +O(�t)= ���t" R0(U�)(U3 � U2) +O(�t+ �t2" )(7.13)where U� lies between U2 and U3. This impliesU3 � U2 = (I + ��t" R0(U�))�1O(�t+ �t2" )(7.14) = O( "�t )O(�t+ �t2" ) = O("+�t) = O(�t) :(7.15)Applying this back to (4.4) then impliesR(U3) = O(") :(7.16)Similar arguments also imply R(Un+1) = O(") :(7.17)By this argument, for any initial data, with the initial layer �xing, one always hasR(U3) = O(") ; R(Un) = O(") ; for all n � 1 :(7.18)Thus this scheme projects the numerical data into the local Maxwellian at every timestep. This completes the study of the uid dynamic limit of the new splitting scheme.Finally we combine the spatial and temporal discussions above to get the limitingscheme for the fully discrete method that combines the MUSCL and the new splittingscheme. Applying (7.18) into (4.3), (4.5) respectively, after reordering the indices, weget U1 = Un � ~��tDF (Un)jR(Un)=0 ;(7.19) U2 = U1 � ~��tDF (U1)jR(U1)=0 ;(7.20) Un+1 = �Un + �U2 ;(7.21)by ignoring the O(") terms. In (7.19) and (7.20), DF (U )jR(U)=0 are the limiting spa-tial discretization given in (7.3) and (7.4) for the model euler equation (2.9). Clearly(7.21) is a second order (both spatially and temporally) discretization of the model Eu-ler equation (2.9). This shows that the new splitting scheme combined with MUSCL ux indeed has the correct uid limit in the coarse regime.If the upwind scheme is used in the convection term then the limiting uxDF (U )jR(U)=0 is given in (7.2) which is a �rst order spatial discretization of themodel Euler equation.

HYPERBOLIC SYSTEMS WITH RELAXATION 177.2. The Intermediate Regime. The intermediate regime is de�ned as " �1;�t� 1 but "=�t = O(1). We now show that the splitting scheme (4.2)-(4.7) alsohas the correct uid limit in this intermediate regime. The key is to demonstrate that(7.10) are valid also in this regime.First, since � = 0, after the initial layer �x (the assumption on (7.9)), (7.11) isvalid. Thus (4.4) impliesU3 � U2 = ���t" R(U3) +O(")= ���t" (R(U3) �R(U2)) � ��t" R(U2) +O(")= ���t" R0(U�)(U3 � U2)� ��t" (R(U1) + O(�t)) +O(")= ���t" R0(U�)(U3 � U2) + O(�t)Therefore U3 � U2 = (1 + ��t" R0(U�))�1O(�t) = O(�t) :Applying this to (4.4) givesR(U3) = O(�t)O( "�t) = O(") :Similarly one can also show R(Un+1) = O(") :Thus, one always hasR(U3) = O(") ; R(Un) = O(") ; for all n � 1 :independent of the initial data. So the solution is always a local Maxwellian. More-over, as "! 0 (�t! 0 as well) one gets (2.9). The limiting fully discrete discretiza-tion is, in fact, the same as in (7.19) and (7.20). Thus in the intermediate regime thisscheme also has the correct uid limit.Remark: In [24] Pember found that in the limit � ! 0 a second order spacediscretization may reduced to �rst order in a fractional step method. A high ordermethod needs to incorporate the e�ect of the source term into the convective ux. Jin[16] also showed such a phenomena in a Strang's splitting. However, as indicated in[16], a properly designed fractional step method may maintain a second order accuracyeven if the convective ux is based solely on the Riemann problem of the homogeneoussystem. The result in this section con�rms this for the new splitting scheme NSPIF,and our numerical results in Section 9 seem to verify this point.8. Theoretical Results.In this section we shall provide some analytic analyses on a simple splitting scheme(abbreviated as SP1):~�j = �nj ; ~mj = mnj ; ~zj = znj + �t2" ((�nj )2 + (mnj )2 � 2�nj ~zj) ;(8.1) fn+1j = ~fnj�1 ; hn+1j = ~hnj ; gn+1j = ~gj+1 :(8.2)

18 R. E. CAFLISCH, S. JIN and G. RUSSOThis simple �rst order method, although signi�cantly simpler compared to NSPIF,does possess some key properties of the NSPIF, such as the A-stability, L-stability, andthe correct uid limit as � ! 0, and thus it serves an analytic model here. Analyticstudy of the second order method NSPIF would require more advanced techniquesthat are out of reach for this work.In SP1, the collision step is the backward Euler method that has a numericalstability independent of ". In this step it is very convenient to use the uid variables.In the convection step, the characteristic method can be applied for the free stream.The overall Courant-Friedrichs-Lewy (CFL) number then will be solely determinedby the convection step, which is 1. In (8.2) by a tilde we denote the intermediateresult after the collision step. Between the collision and convection steps the relation(2.2) and (2.3) will be used. This method is going to have the correct uid limit when"! 0 since the �rst step always yield the correct local Maxwellian z = (�2+m2)=(2�).Applying it to the next step, one gets a �rst order approximation to the model Eulerequations. However, the overall accuracy is �rst order, uniformly in ", as will beshown later.In this section we derive several properties of the SP1, including positivity, theentropy inequality and a bound in the consistency error which is independent on themean free path. To the best of our knowledge erned this is the �rst uniform (in �)result for the numerical discretization of the Broadwell equation.8.1. Positivity. The continuous Broadwell equations maintain positivity of thedensities f , h, and g. This property is important because of the physical meaning off , h, and g as particle density. Such property is maintained by the discrete scheme.The velocity densities (f; g; h) are positive if and only if� > z > jmj ;by (2.2) and (2.3). In the collision step ~� = �n and ~m = mn are unchanged, while~z = (1 + 2��n)�1(zn + �((mn)2 + (�n)2))in which � = �t=(2"). Since ~� = �n > zn > jmnj = j ~mj, it follows easily that~� > ~z > j ~mj:In fact, it is ~z � j ~mj = ~z � jmj = z � jmj+ �(� � jmj)21 + 2�� > 0and ~� � ~z = � � ~z = �� z + �(�2 �m2)1 + 2�� > 0:So ( ~f ; ~g; ~h) is positive. Positivity of ( ~f ; ~g;~h) clearly implies positivity of fn+1, gn+1and hn+1, the result of the convective step in (8.2).8.2. Entropy Inequality. For the system of Broadwell equations (with peri-odic boundary conditions or in unbounded domain) there exist a function that ismonotonically increasing, and which is constant only at equilibrium:H(t) = Z H(x; t) dx;

HYPERBOLIC SYSTEMS WITH RELAXATION 19with H(x; t) = f(x; t) logf(x; t) + 2h(x; t) logh(x; t) + g(x; t) logg(x; t):The proof of this property (the so called H-theorem) is very simple in this case, andwe show it here (in the case of periodic boundary conditions) for completeness.Making use of the Broadwell equations and of the periodic boundary conditionsone has: dHdt = Z h2�1� fgh2� log fgh2 dxThe right hand side is negative, since (1� x) logx � 0; 8x > 0, and it is zero if andonly if fg � h2 � 0.The entropy inequality has the meaning of irreversibility in time, and shows thatthe system relaxes to an equilibrium given by a Maxwellian distribution.We shall prove that such property is maintained by the simple splitting scheme(8.2). This property is relevant, �rst because it shows that the discrete model relaxesto equilibrium, and second because of the connection between the H-function and theentropy of the uid dynamic limit.Before proving the H-theorem for the discrete scheme, we shall exploit this con-nection in detail.The kinetic model is described by the Broadwell equations, which constitute a3� 3 semilinear system of partial di�erential equations, while the uid dynamic limitis described by a 2 � 2 quasilinear hyperbolic system of conservation laws. In bothcases the entropy inequality has the physical meaning of irreversibility in time. Forthe quasilinear system, the entropy principle is used to select the unique weak solutionin the presence of shocks.Now let us consider a n� n system of conservation laws of the form@U@t + @F (U )@x = 0(8.3)An entropy for this system is a convex function s = s(U ) which, in case of regularsolutions, satis�es an additional conservation law of the form [11]:@s(U )@t + @q(U )@x = 0(8.4)where q(U ) is the entropy ux. By comparing (8.3) and (8.4), the following compati-bility condition hold: @q@u� = nX�=1 @s@u� @F�@u� ; � = 1; : : : ; nWe now show that the function H, when computed at a local Maxwellian (denotedby the su�x M ), i.e. if fg = h2, and expressed in terms of the uid variables � andu, is indeed an entropy function for the 2� 2 system of uid dynamic limit.At equilibrium one hasfM = �4(1 + u)2; gM = �4 (1� u)2; hM = �4(1 � u2):

20 R. E. CAFLISCH, S. JIN and G. RUSSOSubstituting in the expression of H, after some simpli�cations one has:HM = �(log � + S(u));(8.5)with S(u) � (1 + u) log(1 + u) + (1� u) log(1� u)� log 4:(8.6)Let us compute @HM=@t: @HM@t = @HM@� @�@t + @HM@u @u@t :Making use of expression (8.5) and of the equations (2.9) for � and u one has:@HM@t = ��[log � + S(u) + 1]@u@x � [(log� + S(u) + 1)u+ 12S0(u)(1� u2)]@�@x:Compatibility with (8.4) requires that@q@u = f1(�; u) � �[log �+ S(u) + 1];@q@� = f2(�; u) � (log � + S(u) + 1)u+ 12S0(u)(1� u2):These conditions can be satis�ed by a function q(�; u) provided@f1@� = @f2@u :This last condition becomes 1� 12S00(u)(1 � u2) = 0;which is satis�ed by S(u) de�ned in Eq. (8.6). This proves that HM(�; u) is indeedan entropy function for the system of equations describing the uid dynamic limit.It is evident that a scheme that has the property of preserving the entropy in-equality for the Broadwell model for any value of the relaxation parameter ", will alsopreserve the entropy inequality in the uid dynamic limit. This property is essentialin approximating the correct weak solution of the uid dynamic equations.We prove next that the H function, given byH = f log f + 2h logh+ g log g;decreases in the collision step at each grid point j. This result is valid both for periodicboundary conditions and for unbounded domain.Let (~�; ~m; ~z) be the state after the collision step, and (�;m; z) = (~�; ~m; ~z + 2�)be the state before the collision step, in which� = 2�(~h2 � ~f~g):Since f = (z +m)=2; g = (z �m)=2 and h = (� � z)=2, thenf = ~f �� ; g = ~g �� ; h = ~h+� :(8.7)

HYPERBOLIC SYSTEMS WITH RELAXATION 21Moreover since (f; g; h) and ( ~f; ~g; ~h) are all positive, then1 > max��~f ; �~g ;��~h � :Before the collision the H function isH = f log f + 2h logh+ g log g:First rewrite the f term in H asf log f = ( ~f ��) log( ~f ��)= ~f �1� �~f � log�1� �~f �+ � ~f ��� log ~f� ��+ ( ~f ��) log ~f :The last bound uses the elementary inequality(1� a) log(1� a) � �afor 1 > a, in which a = �= ~f . Similarlyg logg � ��+ (~g ��) log ~g ; h logh � �+ (~h+�) log~h :(8.8)Add these together to obtainH � ( ~f ��) log ~f + 2(~h+�) log~h+ (~g ��) log ~g= ~H +� log(~h2= ~f~g)= ~H + 2�(~h2 � ~f ~g) log(~h2= ~f~g)� ~H:Finally note that the total \entropy"Hn = �xXj Hnjis preserved by the convection step of the fractional step method (8.2), since the valuesof f; g and h do not change but only move between j points.The results of these two subsections are summarized by the following proposition.Proposition 8.1. Let (fnj ; gnj ; hnj ) be the solution of the fractional step method(8.2). Assume that f0j > 0; g0j > 0 and h0j > 0 for all j. It follows that:(i) fnj > 0; gnj > 0; hnj > 0 for all n > 0 and all j.(ii) De�ne Hn =Xj (fnj logfnj + 2hnj loghnj + gnj log gnj )and suppose that H0 <1. Then for all n � 0Hn+1 � Hn:

22 R. E. CAFLISCH, S. JIN and G. RUSSO8.3. Formal Analysis of Convergence: Uniform Bounds on ConsistencyError. Finally we present a formal analysis of the splitting method which shows thatit is �rst order accurate uniformly in ", if the underlying Broadwell solution is smooth.This analysis does not apply for the solution in a boundary layer, initial layer or shock,in which numerical results indicate that a half order of accuracy may be lost. Theanalysis here is only of the consistency error, which is shown to be of size O(�t)uniformly in ". Analysis of the stability error has not yet been successful.Because of the di�culty of the problem, no result is available for higher orderschemes. To our knowledge, this is the �rst analytic result showing a bound on theconsistency error, uniformly in the relaxation parameter, for such a problem.The estimate will be performed by writing the Broadwell equations and the dis-cretized Broadwell equations for the uid dynamic variables � and m, and for thedi�erence from equilibrium w, de�ned byw = z � �2 +m22�(8.9)De�ne also zE (�;m) = �2 +m22� :(8.10)Whereas the equation for z involves a forcing term of size "�1, in the w equation thefactor "�1 appears only in a decay term.The Broadwell equations (2.1) can be written in terms of �;m and w as�t = �mx(8.11) mt = �(w + zE(�;m))x(8.12) wt = �"�1�w �mx � zE (�;m)t(8.13)The equation for w can be written in integral form, for each x, asw(t) = A(t)w0 + Z t0 A(t)A(s) g(s)ds(8.14)in which A(t) = exp(�"�1 Z t0 �(t0; x)dt0)g(t) = �(mx + zE(�;m)t):De�ne the discretized variables ~�; ~m and ~w as in (3.2), (3.3) to be the state after acollision step. The discrete Broadwell equations (with �t = �x) can then be written,with the convection and collision steps combined, for each spatial value j, as~�n+1 � ~�n = �12D0 ~mn + 12D2( ~wn + ~znE)(8.15) ~mn+1 � ~mn = �12D0( ~wn + ~znE ) + 12D2 ~mn(8.16) ~wn+1 = (1 + �t" ~�n+1)�1( ~wn +�t~gn+1)(8.17)

HYPERBOLIC SYSTEMS WITH RELAXATION 23in which ~gn+1 = ��t�1(~zn+1E � ~znE + 12D0 ~mn+1 � 12D2( ~wn+1 + ~zn+1E )):The di�erence operators D0 and D2 are de�ned as(D0f)j = fj+1 � fj�1 ; (D2f)j = fj+1 � 2fj + fj�1 :(8.18)The equation for ~wn+1 can be considered a linear di�erence equation in ~wn. Thesolution can be written as ~wn = ~Bnw0 +�t nXk=1 ~Bn~Bk�1 ~gk(8.19)in which ~Bn = nYj=1(1 + �t" ~�j)�1 ; ~B0 = 1 :(8.20)Assume that �;m and w are smooth and bounded uniformly in " (this excludesshocks, boundary layers and initial layers). De�ne the continuous solution at discretetimes as (�n;mn; wn) = (�(n�t);m(n�t); w(n�t)). This is an abuse of notation,since (�n;mn; wn) was used in Section 3 to denote the discretized solution after theconvection step. Nevertheless, it will be used for simplicity.The consistency error is de�ned is the error formed by substituting the continuoussolution (�n;mn; wn) = (�(n�t);m(n�t); w(n�t)) into the discrete equations. De�neE1 = (�n+1 � �n)=�t���12D0mn + 12D2(wn + znE )� =�t(8.21) E2 = (mn+1 �mn)=�t���12D0(wn + znE) + 12D2mn� =�t(8.22) E3 = wn �(Bnw0 +�t nXk=1 BnBk�1 gk)(8.23)in which Bn and gn are de�ned as above in terms of the solution (�n;mn; wn). Notethat the consistency error for the w equations is de�ned here in terms of the \inte-grated solution" rather than the �nite di�erence equation. In fact the error due to theimplicit collision operator is better behaved over many time steps than over a singlestep. This is also the reason that the factor 1=�t does not appear in the de�nition ofE3. The consistency error serves as a forcing term in a �nite di�erence equation forthe total error. De�ne the di�erence between the numerical solution (~�n; ~mn; ~wn) andthe continuous solution (�n;mn; wn) = (�;m;w)(n�t) asan = ~�n � �n ; bn = ~mn �mn ; cn = ~wn �wn :(8.24)Denote A(�n;mn; wn) and B(�n;mn; wn) to be the bracketed convection terms in the� and m equations (8.21) and (8.22). Also denote C[�;m;w] to be the bracketed term

24 R. E. CAFLISCH, S. JIN and G. RUSSOin the w equation (8.23). Then the error quantities (an; bn; cn) satisfyan+1 � an = (A(�n + an;mn + bn; wn + cn) �A(�n;mn; wn)) + �tE1bn+1 � bn = (B(�n + an;mn + bn; wn + cn)� B(�n;mn; wn)) + �tE2cn+1 � cn = (C[�+ a;m+ b; w+ c]�C[�;m;w]) +E3(8.25)The �rst term on the right hand sides of these equations is the stability error, andis approximately a linear operator in (a; b; c). If the di�erence scheme is uniformlystable, which we have not proved, then the error will be of the size of the consistencyerror E1; E2; E3:Under the assumption that the continuous solution �;m;w is smooth and thatthe density � is uniformly bounded above and below, i.e.�� < � < �c��(8.26)we prove the following uniform bound on the consistency error.Proposition 8.2. Suppose that �;m;w is smooth and that the density � satis�es(8.26) for some constants �� and �c. Then the consistency error E1; E2; E3, de�ned by(8.21), (8.22) and (8.23), for the discretized Broadwell equations (8.15), (8.16) and(8.19) with �t = �x satis�es jE1j+ jE2j+ jE3j � c�t(8.27)for some constant c that is independent of ". In other words, the consistency error isuniformly �rst order.The estimates on E1 and E2 are straightforward; i.e. for some constant c,jE1j+ jE2j � c�t:(8.28)In order to compare the integral (8.14) to the sum (8.23), we �rst replace theintegrand by a piecewise-constant function. De�ne�A(s) = A((m � 1)�t) ; �g(s) = g((m � 1)�t) :(8.29)for (m� 1)�t � s < m�t. For t = n�t de�ne�w(t) = A(t)w0 + Z t0 A(t)�A(s) �g(s)ds= A(t)w0 +�t nXk=1 A(n�t)A((k � 1)�t)g(k�t)(8.30)Now estimate, for (m � 1)�t � s < m�t, m � n,jg(s) � �g(s)j < c�t(8.31) ����A(t)A(s) � A(t)�A(s) ���� = ����A(t)A(s) ���� � ����1� A(s)�A(s) ����� exp�� Z ts �(s0)ds0="������1� exp � Z s(m�1)�t �(s0)ds0="!������ min(1; �c�t��=") exp(�(n�m)�t��="):(8.32)

HYPERBOLIC SYSTEMS WITH RELAXATION 25since je�� � 1j � min(1; �) for any �.Next estimate the di�erence between A(t) and Bn. First make a simple generalestimate (for t = n�t) A(t) � e�t��=" � (1 + �t" ��)�n ;(8.33) Bn � �1 + �t" ����n ;(8.34)so that jA(t)� Bnj � 2�1 + �t" ����n :(8.35)If �t��=" is small, a more re�ned estimate is needed. In this caselog�1 + �t" �k� = �t" �k +O��t" ���2(8.36)Then A(t) �Bn = A(t)(1� exp Z t0 �(s)ds=" � nXk=1 log�1 + �t" �k�!)= A(t)(1� exp nO��t" ���2!)(8.37)in which we ignore a term of size nO(�t2��=") in the exponential.Now n(�t��=")2 = (t��=")(�t��="). If this is larger than 2, we use the estimate(8.35); while if it is less than 2, thenjA(t)� Bnj � A(t)O� t��" ��t" ���� :(8.38)Combining these estimates, we obtain the boundjA(t)�Bnj � min�2; t�t"2 ��2��1 + �t" ����n :(8.39)Finally the estimate on A(t)=A((k � 1)�t)� Bn=Bk�1 is found in a similar wayto be ���� A(t)A((k � 1)�t) � BnBk�1 ���� � min�2; "�2(n� k + 1)�t2��2��1 + �t" ����(n�k) :(8.40)Now these basic estimates are combined to estimate E3. Firstw(t)� �w(t) = Z t0 �A(t)A(s) g(s) � A(t)�A(s) �g(s)� ds� Z t0 �����A(t)A(s) � A(t)�A(s) ���� j�g(s)j+ ����A(t)A(s) ���� jg(s) � �g(s)j�ds

26 R. E. CAFLISCH, S. JIN and G. RUSSO� c�t nXm=1min(1; �c�t��"�1) exp(�(n �m)�t��=")+c�t Z t0 e���(t�s)="ds� c�tmin(1;�t��"�1)(1� e��t��=")�1 + c"�t=��� ~c�t(8.41)since maxy>0 � min(1; y)1� exp(�y)� = ee� 1 :Next, making use of (8.39),jE3 + �w(t) �wnj = ����(A(t)� Bn)w0+ �t nXk=1� A(t)A((k � 1)�t) �g(k�t)� BnBk�1 gk������ jw0jmin�2; t�t"2 ��2��1 + �t" ����n+ c�t nXk=1�1 + �t" ����(n�k+1)�t+ c�t nXk=1min�1; �t2��2"2 (n� k + 1)��1 + ���t" ��n+k�1 :(8.42)Denote the three terms on the right side of (8.42) as D1; D2; D3 respectively.The �rst term D1 is bounded bycmin�2; n�t" ���t��" ��1 + �t" ����n � 2c�t :(8.43)In fact, if �t��=" > 1=2, then D1 < 2c(3=2)�n < 2c�t(8.44)while if �t��=" < 1=2 thenD1 < c(1=n)(n�t��=")2(1 +�t��=")�n < c=n < c�t ;(8.45)since x2(1 + x=n)�n is bounded by 1 uniformly in n > 2 and x � 0.Now we consider the second term D2. Let q = (1 + ���t=")�1. ThenD2 = c�t2 1Xk=1 qn�k+1 = c�t2q1� qn1� q� c�t2 q1� q = c�t"��(8.46)Finally consider the third term D3.

HYPERBOLIC SYSTEMS WITH RELAXATION 27If �t��=" > 2, then D3 � c�t nXk=1 qn�k+1= c�tq1� qn1� q � c�t q1� q= c�t "���t � c�t2If �t��=" < 2, then D3 � c�t��t��" �2 1X0 mqm= c�t��t��" �2 q(1� q)2= c�t(1 +�t��=") � 3c�t(8.47)since 1X0 mxm�1 = (1� x)�2:This shows that jE3 + �w(t) �wnj � c�t:(8.48)Combine this with (8.41) to obtain jE3j � c�t:(8.49)Together with (8.28), this implies thatjE1j+ jE2j+ jE3j � c�t:(8.50)This concludes the formal demonstration that consistency error in the discrete methodis �rst order accurate, uniformly in ".9. Numerical Results.In this section we present some numerical examples in order to compare some ex-isting schemes with the new splitting scheme NSPIF developed in this article. Rewritethe Broadwell equations (2.4)-(2.6) as@tU + A@xU = Q(U )(9.1)Schemes that will be compared are:1. NSP (introduced in Section 4).2. NSPIF (introduced in Section 6).3. SP1 (introduced in Section 8).4. SCN. Free streaming (8.2) for the convection @tU + A@xU = 0, followed bya half-step Crank-Nicolson method for the collision @tU = Q(U ), and then followedby another step of free streaming (8.2) on @tU + A@xU = 0. Strang's splitting [27] isused here.

28 R. E. CAFLISCH, S. JIN and G. RUSSO5. SCNvL. Same as SCN except the two free streaming steps on the convection@tU +A@xU = 0 be replaced van Leer's MUSCL spatially and the second order TVDRunge-Kutta method [26] temporally. Strang's splitting [27] is used here.6. SCNvLIF. Same as SCNvL except in the �rst time step we use the initiallayer �x introduced in Section 6.7. SP1vL. Same as SP1 except that the free streaming step (8.2) on @tU +A@xU = 0 is replaced by MUSCL spatially and the second order TVD Runge-Kuttamethod temporally.Note that the free streaming (8.2) is the characteristic method for the convection,corresponds to the upwind scheme with the CFL number equals one. Thus we willalways use CFL= 1 for SP1 and SCN. For other methods we will use CFL=0:5, unlessotherwise stated.First we solve the Broadwell equation with the following initial data� = 2 ; m = 1 ; z = 1 ; for x < 0:2 ;(9.2) � = 1 ; m = 0:13962 ; z = 1 ; for x > 0:2 ;(9.3)We integrate over domain [�1; 1] with re ecting boundary conditions. We take �x =0:01 and �t = O(�x). We test six di�erent schemes. The exact solution is obtainedusing �ne grids with �x = 0:0005.First we take " = 1. This is in the rare�ed regime. The numerical solutions of�;m and z are depicted with the \exact" solution in Figure 1. In this regime, SP1yields the best resolution, SCN is very di�usive, SCNvL is slightly better than SCN,while SP1vL, NSP and NSPIF give comparable results, which is more di�usive thanSP1 but better than SCN and SCNvL. While SP1vL, NSP and NSPIF show standardsecond order behavior, in the rare�ed regime it seems that the free stream is a nicething to use, for SP1 gives the best result, and SCN is better then SCNvL.Next we take " = 10�8 and compare the behavior of NSP and NSPIF and showshow the initial layer �x works. This is the regime when the mean free path is verysmall and the limiting Euler equation has a shock wave moving right with a speeds = 0:86038 determined by the Rankine-Hugoniot jump condition. Note the initialdata for z is not a local Maxwellian, which yields an initial layer. The results aredisplayed in Figure 2. The NSP and NSPIF are comparable with respect to theshock, however, without the initial layer �x the NSP creates a hump near the initialdiscontinuity x = 0, solely caused by the kinetic e�ect. In the next run we see thatfor a larger � this hump exists in the exact solution. This shows that NSP is draggedby the kinetic behavior even when � is much smaller and the solution is in the uidregime.We then choose " = 0:02. We still use �x = 0:02. This is in the intermediateregime where ";�x and �t are of the same order. The results are depicted in Figure3. The small hump located at x = 0 is part of the exact solution. It is due to the factthat the initial condition represents an exact traveling shock for the relaxed system,i.e. for the system with " = 0, while in this case it is " = 0:02. It seems that SCNvL,NSP and NSPIF give comparable and the best results, while SP1vL is slightly moredi�usive, SP1 and SCN are much more smeared out and exhibited a typical �rst orderbehavior.We then choose another initial data� = 1 ; m = 0 ; z = 1 ; for x < 0:5 ;(9.4) � = 0:2 ; m = 0 ; z = 1 ; for x > 0:5 ;(9.5)

HYPERBOLIC SYSTEMS WITH RELAXATION 29We integrate over domain [0; 1] with re ecting boundary conditions. We take �x =0:01 and " = 10�8 so the solution is close to that of the model Euler equation. Bysolving the model Euler equation one obtains a left moving rarefaction wave and a rightmoving shock wave. The initial data are not in the local Maxwellian. The numericalsolutions are plotted in Figure 4. In this uid limit we observe that NSPIF givesthe best resolution for both the shock and rarefaction waves, and NSP is comparablealthough it smears a tiny bit more than NSPIF. All other schemes smear a lot more.Both SCN and SCNvL give z far from the local Maxwellian, which is not surprisingsince the Crank-Nicolson does not project to the local Maxwellian if the initial dataare not a local Maxwellian. This can be �xed rather easily if these schemes arecombined with the initial layer �x introduced in Section 6. Such a phenomenon alsodisappear if the initial data are in the local Maxwellian. Very interesting is the resultof SCNvL, which gives completely spurious wave structures.To examine the spurious waves produced by the SCNvL for this problem wedepict the results of SCNvL and SCNvLIF for the same problem in Figure 5 withCFL=0:5 and CFL=0:005 respectively. When CFL= 0:5 both SCNvL and SCNvLIFgive spurious but di�erent waves! By taking �t 100 times smaller such spurious wavesdisappear and the schemes give qualitatively correct waves, although the waves aresmeared more due to the smallness of �t. This example shows that large temporalspacing in SCNvL may produce numerical results which look completely reasonablebut are totally unphysical! The reason is simple: The SCNvL does not have thecorrect uid limit in the coarse regime.In summary, NSPIF seems to produce satisfactory results and exhibit a typicalsecond order behavior in all these regimes. Other methods may give unsatisfactory(even wrong) results in one or another regime.Next we perform the numerical convergence study. We consider an initial valueproblem with periodic boundary conditions, such that the solution is smooth in atime interval [0; T ] for any value of the parameter ". We compute the error at time Tby di�erencing, i.e. by comparing the result obtained with a given grid (�x;�t) withthe one obtained with the grid (�x=2;�t=2).In Sec. 9 we proved that the consistency error for SP1 is uniformly �rst order.By truncation analysis we know that the scheme NSP and NSPIF are second orderboth in space and time, if �t� ", and they are second order also in the uid regime(for smooth solutions).The goal of the test is to perform a numerical study of the convergence ratefor a wide range of ", and check whether the convergence is uniform in " also inthe intermediate regime. The test problem is given by equations (2.1) with periodicboundary conditions: s(x + L; t) = s(x; t) with s = f; g; h. The initial data is givenby: �(x; 0) = 1 + a� sin 2�xL ; u(x; 0) = 12 + au sin 2�xL ;m(x; 0) = �(x; 0)u(x; 0); z(x; 0) = zE (�(x; 0)u(x; 0))�M ;where �M is a real parameter. If �M = 1 then the initial condition is a localMaxwellian, otherwise it is not. If �M 6= 1, " � 1, there is an initial layer. Thesystem is integrated for t 2 [0; T ]. The values of the parameters used in the compu-tations are: L = 20; T = 30; a� = 0:3; au = 0:1

30 R. E. CAFLISCH, S. JIN and G. RUSSOThe values of �x used in the computations are:�x = 0:4; 0:2; 0:1;0:05; 0:025for the �rst order scheme SP1 and�x = 1; 0:5; 0:25; 0:125; 0:0625for the second order schemes. The time step is chosen in such a way that CFLcondition is satis�ed: �t = �x for scheme SP1 and �t = �x=2 for the second orderschemes. The convergence rate is computed from the error according to the formula:convergence ratei = log(errori=errori+1)log(�xi=�xi+1)where errori is obtained by comparing the solution obtained with �xi to that obtainedwith �xi+1. The errors and convergence rate are computed and plotted as functionof ". For each value of ", �ve runs have been done for �ve di�erent values of �x,resulting in four error curves and three curves of convergence rate.Several measures of the error have been used, namely, L1, L2, and L1 relativenorm of the error. The di�erent norms give essentially the same results, therefore weshall show only the L1 norm.First we consider the simple splitting scheme SP1. In Figure 6 we show therelative discrete norm of the error in � and in m as function of " (left column) andthe corresponding convergence rate (right column).The initial state is a local Maxwellian in the �rst two cases and it is not in the last.The convergence rate increases when the mesh becomes �ner, and seems to con�rmthat the scheme is �rst order, uniformly in ", for a �ne enough mesh.Next we consider second order schemes. In Figure 7a-b the result of scheme NSPis shown. The initial condition is a local Maxwellian.As it is evident from the �gures, the scheme is second order accurate for small andlarge values of ", and there is a slight deterioration of the accuracy in the intermediateregime.Figure 7c shows the e�ect of the initial layer if scheme NSP is used, withoutusing Richardson extrapolation for the �rst step. The accuracy of the scheme ofcourse degrades due to the initial layer. In Figure 7d the convergence rate is shown.The problem of the initial layer can be overcome by using Richardson extrapola-tion for the �rst step (scheme NSPIF, Figure 7e-f). A similar result is obtained byusing a scheme entirely based on Richardson extrapolation, in which every step is ofthe form (6.5). AcknowledgmentS. Jin and G. Russo would like to thank the Mathematics Department at UCLAfor its hospitality during their visits there. S. Jin is grateful for Prof. George Papan-icolaou for his constant encouragement and support for this work. We also thank thetwo unknown referees for their critical remarks on the �rst draft of the paper.

HYPERBOLIC SYSTEMS WITH RELAXATION 31REFERENCES[1] A. V. Bobylev, E. Gabetta and L. Pareschi, On a boundary value problem for theplane Broadwell model. Exact solutions and numerical simulation, Mathematical Models& Methods in Applied Sciences, to appear.[2] J. E. Broadwell, Phys. Fluids, 7 (1964), pp. 1013{1037.[3] R. E. Caflisch and G. Papanicolau, Comm. Pure. Appl. Math., 22 (1979), pp. 586{616.[4] C. Cercignani, The Boltzmann Equation and its Applications, Springer-Verlag, New York,1988.[5] G. Q. Chen, C. D. Levermore, and T. P. Liu, Hyperbolic conservation laws with sti�relaxation terms and entropy, Comm. Pure. Appl. Math., 47 (1994), pp.787-830.[6] P. Colella, A. Majda, and V. Roytburd, SIAM J. Sci. Statist. Comp., 7 (1986), pp. 1059{1080.[7] F. Coron, and B. Perthame, SIAM J. Numer. Anal., 28 (1991), pp.26{42.[8] S. Deshpande, A second order accurate, kinetic-theory based method for inviscid compressible ow, NASA Langley Tech. paper No. 2613, 1986.[9] B. Engquist, and B. Sjogreen, Robust Di�erence Approximations of Sti� Inviscid Detona-tion Waves, UCLA CAM Report 91-03, 1991.[10] H. Emamirad, C. R. Acad. Sc. Paris, t.304, S�eries II (1987), pp. 487{490.[11] K. O. Friedrichs, and P. D. Lax, System of conservation laws with a convex extension,Proc. Nat. Acad. Sci. USA, 68 (1971), pp. 1686{1688.[12] E. Gabetta, and L. Pareschi Mathematical Models and Methods in Applied Sciences, 2(1992), pp.1{19.[13] R. Gatignol, Th�eorie cin�etique des gas �a r�epartition discr�ete de vitesses, Lecture Notes inPhysics, vol. 36, Springer-Verlag, Heidelberg, 1975.[14] F. Golse, S. Jin, and C. D. Levermore, The convergence of numerical transfer schemesin di�usive regimes I: the discrete-ordinate method, SIAM. J. Num. Anal., submitted.[15] A. Harten, P. D. Lax, and B. van Leer, SIAM Rev. 25 (1983), pp. 35{61.[16] S. Jin, it Runge-Kutta methods for hyperbolic systems with sti� relaxation terms, J. Comp.Phys., to appear.[17] S. Jin, and C. D. Levermore, Numerical schemes for hyperbolic systems with sti� relaxationterms, J. Comp. Phys., submitted.[18] S. Jin, and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitraryspace dimensions, Comm. Pure. Appl. Math., 1995, to appear.[19] E. W. Larsen, J. E. Morel and W. F. Miller, Jr., J. Comp. Phys. 69 (1987), pp. 283{324.[20] R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel (1992).[21] R. J. LeVeque, and H. C. Yee, J. Comp. Phys. 86 (1990), pp. 187{210.[22] T. P. Liu, Comm. Math. Phys. 108 (1987), pp. 153{175.[23] R. B. Pember, SIAM J. Appl. Math. 53 (1993), pp. 1293{1330.[24] , SIAM J. Sci. Stat. Comp. 14 (1993), pp. 824-859.[25] B. Perthame, SIAM J. Numer. Anal. 29 (1992), pp. 1{19.[26] C. W. Shu and S. Osher, J. Comp. Phys. 77 (1988), pp. 439{471.[27] G. Strang, SIAM J. Num. Anal. 5 (1968), pp. 506{517.[28] B. van Leer, J. Comp. Phys. 32 (1979), pp. 101{136.[29] G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.[30] Z. P. Xin, Comm. Pure Appl. Math., 44 (1991), pp. 679{713.

32 R. E. CAFLISCH, S. JIN and G. RUSSO-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

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HYPERBOLIC SYSTEMS WITH RELAXATION 33-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

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Fig. 9.2. The numerical solutions of � ('+'), m ('o') and z ('*') at t = 0:5 in x 2 [�1; 1] forinitial data (9.2) and (9.3) by NSP and NSPIF. " = 10�8, �x = 0:01 and CFL= 0:5.

34 R. E. CAFLISCH, S. JIN and G. RUSSO-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

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Fig. 9.3. The numerical solutions of � ('+'), m ('o') and z ('*') at t = 0:5 in x 2 [�1; 1] forinitial data (9.2) and (9.3) by (from left to right, then top to bottom) SP1, SCN, SP1vL, SCNvL,NSP and NSPIF. " = 0:02, �x = 0:02. CFL= 1 for SP1 and SCN , CFL= 0:5 for others.

HYPERBOLIC SYSTEMS WITH RELAXATION 350 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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36 R. E. CAFLISCH, S. JIN and G. RUSSO0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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HYPERBOLIC SYSTEMS WITH RELAXATION 3710

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Fig. 9.6. Uniform convergence of the simple splitting scheme SP1. Relative error (left) andconvergence rate (right) vs " for various values of the grid step size �x. Continuous line: coarsestmesh; dashed line: �nest mesh. (a-b): L1 error in �. Initial condition: local Maxwellian. (c-d): L1error in m. Initial condition: local Maxwellian. (e-f): L1 error in �. Initial condition: not localMaxwellian.

38 R. E. CAFLISCH, S. JIN and G. RUSSO10

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Fig. 9.7. Relative L1 norm of the error in density and convergence rate vs " for various valuesof the grid step size �x. (a) and (b): Scheme NSP. Initial state: local Maxwellian. (c) and (d):Scheme NSP. The initial state is not a local Maxwellian. (e) and (f): Scheme NSPIF. Same initialstate of (c-d).