Comparison of Fifth-OrderWENO Scheme and (2)

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efficient implementation of weighted ENO scheme

JOURNAL OF COMPUTATIONAL PHYSICS126,202–228(1996)ARTICLE NO.0130Efﬁcient Implementation of Weighted ENO Schemes*G UANG-S HAN J IANG†AND C HI-W ANG S HUDi v ision of Applied Mathematics,Brown Uni v ersity,Pro v idence,Rhode Island02912Received August14,1995;revised January3,1996or perhaps with a forcing term g(u,x,t)on the right-handside.Here uϭ(u1,...,u m),fϭ(f1,...,f d),xϭ(x1,...,x d) In this paper,we further analyze,test,modify,and improve thehigh order WENO(weighted essentially non-oscillatory)ﬁnite differ-and tϾ0.ence schemes of Liu,Osher,and Chan.It was shown by Liu et al.WENO schemes are based on ENO(essentially non-that WENO schemes constructed from the r th order(in L1norm)oscillatory)schemes,which wereﬁrst introduced by ENO schemes are(rϩ1)th order accurate.We propose a new wayHarten,Osher,Engquist,and Chakravarthy[5]in the form of measuring the smoothness of a numerical solution,emulatingthe idea of minimizing the total variation of the approximation,of cell averages.The key idea of ENO schemes is to use which results in aﬁfth-order WENO scheme for the case rϭ3,the‘‘smoothest’’stencil among several candidates to ap-instead of the fourth-order with the original smoothness measure-proximate theﬂuxes at cell boundaries to a high order ment by Liu et al.Thisﬁfth-order WENO scheme is as fast as theaccuracy and at the same time to avoid spurious oscillations fourth-order WENO scheme of Liu et al.and both schemes arenear shocks.The cell-averaged version of ENO schemes about twice as fast as the fourth-order ENO schemes on vectorsupercomputers and as fast on serial and parallel computers.For involves a procedure of reconstructing point values from Euler systems of gas dynamics,we suggest computing the weights cell averages and could become complicated and costly for from pressure and entropy instead of the characteristic values to multi-dimensional ter,Shu and Osher[14,15] simplify the costly characteristic procedure.The resulting WENOdeveloped theﬂux version of ENO schemes which do not schemes are about twice as fast as the WENO schemes using thecharacteristic decompositions to compute weights and work well require such a reconstruction procedure.We will formulate for problems which do not contain strong shocks or strong reﬂected the WENO schemes based on thisﬂux version of ENO waves.We also prove that,for conservation laws with smooth solu-schemes.The WENO schemes of Liu et al.[9]are basedtions,all WENO schemes are convergent.Many numerical tests,on the cell-averaged version of ENO schemes.including the1D steady state nozzleﬂow problem and2D shockFor applications involving shocks,second-order schemes entropy wave interaction problem,are presented to demonstratethe remarkable capability of the WENO schemes,especially the are usually adequate if only relatively simple structures WENO scheme using the new smoothness measurement in resolv-are present in the smooth part of the solution(e.g.,the ing complicated shock andﬂow structures.We have also applied shock tube problem).However,if a problem contains rich Yang’s artiﬁcial compression method to the WENO schemes tostructures as well as shocks(e.g.,the shock entropy wave sharpen contact discontinuities.ᮊ1996Academic Press,Inc.interaction problem in Example4,Section8.3),high ordershock capturing schemes(order of at least three)are more1.INTRODUCTION efﬁcient than low order schemes in terms of CPU time andmemory requirements.In this paper,we further analyze,test,modify,and im-ENO schemes are uniformly high order accurate rightprove the WENO(weighted essentially non-oscillatory)up to the shock and are very robust to use.However,theyﬁnite difference schemes of Liu,Osher,and Chan[9]for also have certain drawbacks.One problem is with the freelythe approximation of hyperbolic conservation laws of adaptive stencil,which could change even by a round-offthe type perturbation near zeroes of the solution and its derivatives.Also,this free adaptation of stencils is not necessary in u tϩdiv f(u)ϭ0,(1.1)regions where the solution is smooth.Another problem isthat ENO schemes are not cost effective on vector super-computers such as CRAY C-90because the stencil-choos-*Research supported by ARO Grant DAAH04-94-G-0205,NSFGrants ECS-9214488and DMS-9500814,NASA Langley Grant NAG-ing step involves heavy usage of logical statements,which 1-1145and Contract NAS1-19480while the second author was in resi-perform poorly on such machines.Theﬁrst problem could dence at ICASE,NASA Langley Research Center,Hampton,VA23681-reduce the accuracy of ENO schemes for certain functions 0001,and AFOSR Grant95-1-0074.[12];however,this can be remedied by embedding certain †Current address:Department of Mathematics,UCLA,Los Angeles,CA90024.parameters(e.g.,threshold and biasing factor)into the2020021-9991/96$18.00Copyright©1996by Academic Press,Inc.All rights of reproduction in any form reserved.WEIGHTED ENO SCHEMES203 stencil choosing step so that the preferred linearly stable the total variation of the approximations.This new mea-surement gives the optimalﬁfth-order accurate WENO stencil is used in regions away from discontinuities.See[1,3,13].scheme when rϭ3(the smoothness measurement in[9]gives a fourth-order accurate WENO scheme for rϭ3). The WENO scheme of Liu,Osher,and Chan[9]is an-other way to overcome these drawbacks while keeping the Although the WENO schemes are faster than ENOschemes on vector supercomputers,they are only as fast robustness and high order accuracy of ENO schemes.Theidea is the following:instead of approximating the numeri-as ENO schemes on serial computers.In Section4,wepresent a simpler way of computing the weights for the calﬂux using only one of the candidate stencils,one usesa con v ex combination of all the candidate stencils.Each approximation of Euler systems of gas dynamics.The sim-pliﬁcation is aimed at reducing theﬂoating point opera-of the candidate stencils is assigned a weight which deter-mines the contribution of this stencil to theﬁnal approxi-tions in the costly but necessary characteristic procedureand is motivated by the following observation:the only mation of the numericalﬂux.The weights can be deﬁnedin such a way that in smooth regions it approaches certain nonlinearity of a WENO scheme is in the computation ofthe weights.We suggest using pressure and entropy to optimal weights to achieve a higher order of accuracy(anr th-order ENO scheme leads to a(2rϪ1)th-order WENO compute the weights,instead of the local characteristicquantities.In this way one can exploit the linearity of the scheme in the optical case),while in regions near disconti-nuities,the stencils which contain the discontinuities are rest of the scheme.The resulting WENO schemes(rϭ3)is about twice as fast as the original WENO scheme which assigned a nearly zero weight.Thus essentially non-oscilla-tory property is achieved by emulating ENO schemes uses local characteristic quantities to compute the weights(see Section7).The same idea can also be applied to the around discontinuities and a higher order of accuracy isobtained by emulating upstream central schemes with the original ENO ly,we can use the undivideddifferences of pressure and entropy to replace the local optimal weights away from the discontinuities.WENOschemes completely remove the logical statements that characteristic quantities to choose the ENO stencil.Thishas been tested numerically but the results are not included appear in the ENO stencil choosing step.As a result,theWENO schemes run at least twice as fast as ENO schemes in this paper since the main topic here is the WENOschemes.(see Section7)on vector machines(e.g.,CRAY C-90)andare not sensitive to round-off errors that arise in actual WENO schemes have the same smearing at contact dis-continuities as ENO schemes.There are mainly two tech-computation.Atkins[1]also has a version of ENO schemesusing a different weighted average of stencils.niques for sharpening the contact discontinuities for ENOschemes.One is Harten’s subcell resolution[4]and the Another advantage of WENO schemes is that itsﬂux issmoother than that of the ENO schemes.This smoothness other is Yang’s artiﬁcial compression(slope modiﬁcation)[20].Both were introduced in the cell average context. enables us to prove convergence of WENO schemes forsmooth solutions using Strang’s technique[18];see Section Later,Shu and Osher[15]translated them into the pointvalue framework.In one-dimensional problems,the sub-6.According to our numerical tests,this smoothness alsohelps the steady state calculations,see Example4in Sec-cell resolution technique works slightly better than theartiﬁcial compression method.However,for two or higher tion8.2.In[9],the order of accuracy shown in the error tables dimensional problems,the latter is found to be more effec-tive and handy to use[15].We will highlight the key proce-(Table1–5in[9])seemed to suggest that the WENOschemes of Liu et al.are more accurate than what the dures of applying the artiﬁcial compression method to theWENO schemes in Section5.truncation error analysis indicated.In Section2,we carryout a more detailed error analysis for the WENO schemes In Section8,we test the WENO schemes(both theWENO schemes of Liu et al.and the modiﬁed WENO andﬁnd that this‘‘super-convergence’’is indeed superﬁ-cial:the‘‘higher’’order is caused by larger error on the schemes)on several1D and2D model problems and com-pare them with ENO schemes to examine their capability coarser grids instead of smaller error on theﬁner grids.Our error analysis also suggests that the WENO schemes in resolving shock and complicatedﬂow structures.We conclude this paper by a brief summary in Section can be made more accurate than those in[9].Since the weight on a candidate stencil has to vary ac-9.The time discretization of WENO schemes will be imple-mented by a class of high order TVD Runge–Kutta-type cording to the relative smoothness of this stencil to theother candidate stencils,the way of evaluating the smooth-methods developed by Shu and Osher[14].To solve theordinary differential equationness of a stencil is crucial in the deﬁnition of the weight.In Section3,we introduce a new way of measuring thesmoothness of the numerical solution which is based uponminimizing the L2norm of the derivatives of the recon-dudt ϭL(u),(1.2)struction polynomials,emulating the idea of minimizing204JIANG AND SHUTABLE I where L (u )is a discretization of the spatial operator,the third-order TVD Runge–Kutta is simplyCoefﬁcients a r k ,lrk l ϭ0l ϭ1l ϭ2u (1)ϭu n ϩ⌬tL (u n )u (2)ϭ u n ϩ u (1)ϩ ⌬tL (u (1))(1.3)20Ϫ1/23/211/21/2u n ϩ1ϭ u n ϩ u (2)ϩ ⌬tL (u (2)).301/3Ϫ7/611/61Ϫ1/65/61/3Another useful,although not TVD,fourth-order Runge–21/35/6Ϫ1/6Kutta scheme isu (1)ϭu n ϩ ⌬tL (u n )f (u )ϭf ϩ(u )ϩf Ϫ(u ),(2.4)u(2)ϭu nϩ ⌬tL (u (1))(1.4)where df (u )ϩ/du Ն0and df (u )Ϫ/du Յ0.For example,u (3)ϭu n ϩ⌬tL (u (2))one can deﬁneu n ϩ1ϭ (Ϫu n ϩu (1)ϩ2u (2)ϩu (3))ϩ ⌬tL (u (3)).f Ϯ(u )ϭ (f (u )ϮͰu ),(2.5)This fourth-order Runge–Kutta scheme can be made TVD by an increase of operation counts [14].We will mainly where Ͱϭmax ͉f Ј(u )͉and the maximum is taken over the use these two Runge–Kutta schemes in our numerical tests whole relevant range of u .This is the global Lax–Friedrichs in Section 8.The third-order TVD Runge–Kutta scheme (LF)ﬂux splitting.For other ﬂux splitting,especially the will be referred to as ‘‘RK-3’’while the fourth-order (non-Roe ﬂux splitting with entropy ﬁx (RF);see [15]for details.TVD)Runge–Kutta scheme will be referred to as ‘‘RK-4.’’Let f ˆϩj ϩ1/2and f Ϫj ϩ1/2be,resp.the numerical ﬂuxes obtained from the positive and negative parts of f (u ),we then have2.THE WENO SCHEMES OF LIU,OSHER,AND CHANf ˆj ϩ1/2ϭf ˆϩj ϩ1/2ϩf ˆϪj ϩ1/2.(2.6)In this section,we use the ﬂux version of ENO schemes Here we will only describe how f ˆϩj ϩ1/2is computed in [9]as our basis to formulate WENO schemes of Liu et al.and on the basis of ﬂux version of ENO schemes.For simplicity,analyze their accuracy in a different way from that used we will drop the ‘‘ϩ’’sign in the superscript.The formulas in [9].We use one-dimensional scalar conservation laws for the negative part of the split ﬂux are symmetric (with (i.e.,d ϭm ϭ1in (1.1))as an example:respect to x j ϩ1/2)and will not be shown.As we know,the r th-order (in L 1sense)ENO scheme u t ϩf (u )x ϭ0.(2.1)chooses one ‘‘smoothest’’stencil from r candidate stencils and uses only the chosen stencil to approximate the ﬂux Let us discretize the space into uniform intervals of sizeh j ϩ1/2.Let’s denote the r candidate stencils by S k ,k ϭ0,⌬x and denote x j ϭj ⌬x .Various quantities at x j will be 1,...,r Ϫ1,whereidentiﬁed by the subscript j .The spatial operator of the WENO schemes,which approximates Ϫf (u )x at x j ,will S k ϭ(x j ϩk Ϫr ϩ1,x j ϩk Ϫr ϩ2,...,x j ϩk ).take the conservative formIf the stencil S k happens to be chosen as the ENO interpola-L ϭϪ1⌬x(f ˆj ϩ1/2Ϫf ˆj Ϫ1/2),(2.2)tion stencil,then the r th-order ENO approximation of h j ϩ1/2iswhere the numerical ﬂux f ˆj ϩ1/2approximates h j ϩ1/2ϭf ˆj ϩ1/2ϭq r k (f j ϩk Ϫr ϩ1,...,f j ϩk ),(2.7)h (x j ϩ1/2)to a high order with h (x )implicitly deﬁned by [15]wheref (u (x ))ϭ1⌬x͵x ϩ⌬x /2x Ϫ⌬x /2h (␰)d ␰.(2.3)q r k (g 0,...,g r Ϫ1)ϭ͸r Ϫ1l ϭ0ark ,l g l .(2.8)Here a r k ,l ,0Յk ,l Յr Ϫ1,are constant coefﬁcients.For We can actually assume f Ј(u )Ն0for all u in the range of our interest.For a general ﬂux,i.e.,f Ј(u )Ն͞0,one can later use,we provide these coefﬁcients for r ϭ2,3,in Table I.split it into two parts either globally or locally,WEIGHTED ENO SCHEMES205TABLE II To just use the one smoothest stencil among the r candi-dates for the approximation of h j ϩ1/2,is very desirable nearOptimal Weights C r kdiscontinuities because it prohibits the usage of informa-C r k k ϭ0k ϭ1k ϭ2tion on discontinuous stencils.However,it is not so desir-able in smooth regions because all the candidate stencils r ϭ21/32/3—carry equally smooth information and thus can be used r ϭ31/106/103/10together to give a higher order (higher than r ,the order of the base ENO scheme)approximation to the ﬂux h j ϩ1/2.In fact,one could use all the r candidate stencils,which all together contain (2r Ϫ1)grid values of f to give f ˆj ϩ1/2ϭq 2r Ϫ1r Ϫ1(f j Ϫr ϩ1,...,f j ϩr Ϫ1)(2.12)a (2r Ϫ1)th-order approximation of h j ϩ1/2:ϩ͸r Ϫ1k ϭ0(ͶkϪC r k )q rk (f j ϩk Ϫr ϩ1,...,f j ϩk ).f ˆj ϩ1/2ϭq 2r Ϫ1r Ϫ1(f j Ϫr ϩ1,...,f j ϩr Ϫ1)(2.9)Recalling (2.9),we see that,the ﬁrst term on the right-which is just the numerical ﬂux of a (2r Ϫ1)th-order up-hand side of the above equation is a (2r Ϫ1)th-orderstream central scheme.As we know,high order upstreamapproximation of h j ϩ1/2.Since ͚r Ϫ1k ϭ0C r k ϭ1,if we require central schemes (in space),combined with high order ͚r Ϫ1k ϭ0Ͷk ϭ1,the last summation term can be written asRunge–Kutta methods (in time),are stable and dissipative under appropriate CFL numbers and thus are convergent,according to Strang’s convergence theory [18]when the ͸r Ϫ1k ϭ0(ͶkϪC r k)(q r k(fj ϩk Ϫr ϩ1,...,f j ϩk )Ϫh j ϩ1/2).(2.13)solution of (1.1)is smooth (see Section 6).The above facts suggest that one could use the (2r Ϫ1)th-order upstream central scheme in smooth regions and only use the r th-Each term in the last summation can be made O (h 2r Ϫ1)iforder ENO scheme near discontinuities.As in (2.7),each of the stencils can render an approxima-Ͷk ϭC r k ϩO (hr Ϫ1)(2.14)tion of h j ϩ1/2.If the stencil is smooth,this approximation is r th-order accurate;otherwise,it is less accurate or even for k ϭ0,1,...,r Ϫ1.Here,h ϭ⌬x .Thus C r k will bearnot accurate at all if the stencil contains a discontinuity.the name of optimal weight.One could assign a weight Ͷk to each candidate stencil S k ,The question now is how to deﬁne the weight such that k ϭ0,1,...,r Ϫ1,and use these weights to combine the (2.14)is satisﬁed in smooth regions while essentially non-r different approximations to obtain the ﬁnal approxima-oscillatory property is achieved.In [9],the weight Ͷk for tion of h j ϩ1/2asstencil S k is deﬁned byͶk ϭͰk0r Ϫ1,(2.15)fˆj ϩ1/2ϭ͸r Ϫ1k ϭ0Ͷk q r k (f j ϩk Ϫr ϩ1,...,f j ϩk ),(2.10)wherewhere q r k (f j ϩk Ϫr ϩ1,...,f j ϩk )is deﬁned in (2.8).To achieveessentially non-oscillatory property,one then requires the weight to adapt to the relative smoothness of f on each Ͱk ϭC r k(␧ϩIS k )p,k ϭ0,1,...,r Ϫ1.(2.16)candidate stencil such that any discontinuous stencil is ef-fectively assigned a zero weight.In smooth regions,one Here ␧is a positive real number number which is intro-can adjust the weight distribution such that the resultingduced to avoid the denominator becoming zero (in our approximation of the ﬂux fˆj ϩ1/2is as close as possible to later tests,we will take ␧ϭ10Ϫ6.Our numerical tests that given in (2.9).indicate that the result is not sensitive to the choice of ␧,Simple algebra gives the coefﬁcients C r k such thatas long as it is in the range of 10Ϫ5to 10Ϫ7);the power p will be discussed in a moment;IS k in (2.16)is a smoothness measurement of the ﬂux function on the k th candidate q 2r Ϫ1r Ϫ1(f j Ϫr ϩ1,...,f j ϩr Ϫ1)ϭ͸r Ϫ1k ϭ0C r k q rk(fj ϩk Ϫr ϩ1,...,f j ϩk )(2.11)stencil.It is easy to see that ͚r Ϫ1k ϭ0Ͷk ϭ1.To satisfy (2.14),it sufﬁces to have (through a Taylor expansion analysis)and ͚r Ϫ1k ϭ0C r k ϭ1for all r Ն2.For r ϭ2,3,these coefﬁcients are given in Table II.Comparing (2.11)with (2.10),we getIS k ϭD (1ϩO (h r Ϫ1))(2.17)206JIANG AND SHUfor k ϭ0,1,...,r Ϫ1,where D is some non-zero quantity IS 1ϭ ((f Јh Ϫ f Љh 2)2ϩ(f Јh ϩ f Љh 2)2)independent of k .As we know,an ENO scheme chooses the ‘‘smoothest’’ϩ(f Љh 2)2ϩO (h 5)(2.24)ENO stencil by comparing a hierarchy of undivided differ-IS 2ϭ ((f Јh ϩ f Љh 2)2ϩ(f Јh ϩ f Љh 2)2)ences.This is because these undivided differences can be used to measure the smoothness of the numerical ﬂux on ϩ(f Љh 2)2ϩO (h 5).(2.25)a stencil.In [9],IS k is deﬁned asWe can see that the second-order terms are different from stencil to stencil.Thus (2.22)is no longer valid at critical IS k ϭ͸r Ϫ1l ϭ1͸r Ϫli ϭ1(f [j ϩk ϩi Ϫr ,l ])2r Ϫl,(2.18)points of f (u (x ))which implies that the WENO scheme of Liu et al.for r ϭ3is only third-order accurate at these points.In fact,the weights computed from the smoothness where f [и,и]is the l th undivided difference:measurement (2.18)diverge far away from the optimal weights near critical points (see Fig.1in the next section)f [j ,0]ϭf jon coarse grids (10to 80grid points per wave).But on f [j ,l ]ϭf [j ϩ1,l Ϫ1]Ϫf [j ,l Ϫ1],k ϭ1,...,r Ϫ1.ﬁne grids,since the smoothness measurements IS k for all k are relatively smaller than the non-zero constant ␧in For example,when r ϭ2,we have(2.16),the weights become close to the optimal weights.Therefore the ‘‘super-convergence’’phenomena appeared IS k ϭ(f [j ϩk Ϫ1,1])2,k ϭ0,1.(2.19)in Tables 1–5in [9]are caused by large error commitment on coarse grids and less error commitment on ﬁner grids When r ϭ3,(2.18)giveswhen using the errors of the ﬁfth-order central scheme as reference (see Tables III and IV).IS k ϭ ((f [j ϩk Ϫ2,1])2ϩ(f [j ϩk Ϫ1,1])2)(2.20)At discontinuities,it is typical that one or more of the r candidate stencils reside in smooth regions of the numerical ϩ(f [j ϩk Ϫ2,2])2,k ϭ0,1,2.solution while other stencils contain the discontinuities.The size of the discontinuities is always O (1)and does not In smooth regions,Taylor expansion analysis of (2.18)change when the grid is reﬁned.So we have for a smooth givesstencil S k ,IS k ϭ(f Јh )2(1ϩO (h )),k ϭ0,1,...,r Ϫ1,(2.21)IS k ϭO (h 2p )(2.26)where f Јϭf Ј(u j ).Note the O (h )term is not O (h r Ϫ1)that we and for a non-smooth stencil S l ,would want to have (see (2.17)).Thus in smooth monotone regions,i.e.,f Ј϶0,we haveIS l ϭO (1).(2.27)Ͷk ϭC r k ϩO (h ),k ϭ0,1,...,r Ϫ1.(2.22)From the deﬁnition of the weights (2.15),we can see that,for this non-smooth stencil S l ,the corresponding weight Recalling (2.12),we see that the WENO schemes with theͶl satisﬁessmoothness measurement given by (2.18)is (r ϩ1)th-order accurate in smooth monotone regions of f (u (x )).This re-Ͷl ϭO (h 2p ).(2.28)sult was proven in [9]using a different approach.For r ϭ2,this is optimal in the sense that the third-order upstream Therefore for small h and any positive integer power p ,central scheme is approximated in most smooth regions.the weight assigned to the non-smooth stencil vanishes as However,this is not optimal for r ϭ3,for which this h Ǟ0.Note,if there is more than one smooth stencils in measurement can only give fourth-order accuracy while the r candidates,from the deﬁnition of the weights in (2.15),the optimal upstream central scheme is ﬁfth-order accu-we expect each of the smooth stencils will get a weight rate.We will introduce a new measurement in the next which is O (1).In this case,the weights do not exactly section which will result in an optimal order accurate resemble the ‘‘ENO digital weights.’’However,if a stencil WENO scheme for the r ϭ3case.is smooth,the information that it contains is useful and When r ϭ3,Taylor expansion of (2.20)givesshould be utilized.In fact,in our extensive numerical ex-periments,we ﬁnd the WENO schemes in [9]work very IS 0ϭ ((f Јh Ϫ f Љh 2)2ϩ(f Јh Ϫ f Љh 2)2)well at shocks.We also ﬁnd that p ϭ2is adequate to obtain essentially non-oscillatory approximations at leastϩ(f Љh 2)2ϩO (h 5)(2.23)WEIGHTED ENO SCHEMES207for r ϭ2,3,although it is suggested in [9]that p should IS 1ϭ (f j Ϫ1Ϫ2f j ϩf j ϩ1)2ϩ (f j Ϫ1Ϫf j ϩ1)2(3.3)be taken as r ,the order of the base ENO schemes.We will use p ϭ2for all our numerical tests.Notice that,IS 2ϭ(f j Ϫ2f j ϩ1ϩf j ϩ2)2ϩ (3f j Ϫ4f j ϩ1ϩf j ϩ2)2.(3.4)in discussing accuracy near discontinuities,we are simply concerned with spatial approximation error.The error due In smooth regions.Taylor expansion of (3.2)–(3.4)gives,to time evolution is not considered.respectively,In summary,WENO schemes of Liu et al.deﬁned by (2.10),(2.15),and (2.18)have the following properties:IS 0ϭ (f Љh 2)2ϩ (2f Јh Ϫ f ٞh 3)2ϩO (h 6)(3.5)1.They involve no logical statements which appear in IS 1ϭ(f Љh 2)2ϩ (2f Јh ϩ f ٞh 3)2ϩO (h 6)(3.6)the base ENO schemes.IS 2ϭ (f Љh 2)2ϩ (2f Јh Ϫ f ٞh 3)2ϩO (h 6),(3.7)2.The WENO scheme based on the r th-order ENO scheme is (r ϩ1)th-order accurate in smooth monotone where f ٞϭf ٞ(u j ).If f Ј϶0,thenregions,although this is still not as good as the optimal order (2r Ϫ1).IS k ϭ(f Јh )2(1ϩO (h 2)),k ϭ0,1,2,(3.8)3.They achieve essentially non-oscillatory property by emulating ENO schemes at discontinuities.which means the weights (see (2.15))resulting from thismeasurement satisfy (2.17)for r ϭ3;thus we obtain a 4.They are smooth in the sense that the numerical ﬂux ﬁfth-order (the optimal order for r ϭ3)accurate f ˆj ϩ1/2is a smooth function of all its arguments (for a general WENO scheme.ﬂux,this is also true if a smooth ﬂux splitting method is Moreover,this measurement is also more accurate at used,e.g.,global Lax–Friedrichs ﬂux splitting).critical points of f (u (x )).When f Јϭ0,we have3.A NEW SMOOTHNESS MEASUREMENTIS k ϭ (f Љh 2)2(1ϩO (h 2)),k ϭ0,1,2,(3.9)In this section,we present a new way of measuring thesmoothness of the numerical solution on a stencil which which implies that the weights resulting from the measure-can be used to replace (2.18)to form a new weight.As we ment (3.1)is also ﬁfth-order accurate at critical points.know,on each stencil S k ,we can construct a (r Ϫ1)th-To illustrate the different behavior of the two measure-order interpolation polynomial,which if evaluated at x ϭments (i.e.,(2.18)and (3.1))for r ϭ3in smooth monotone x j ϩ1/2,renders the approximation of h j ϩ1/2given in (2.7).regions,near critical points or near discontinuities,we com-Since total variation is a good measurement for smooth-pute the weights Ͷ0,Ͷ1,and Ͷ2for the functionness,it would be desirable to minimize the total variation for the approximation.Consideration for a smooth ﬂux and for the role of higher order variations leads us to the f j ϭͭsin 2ȏx j if 0Յx j Յ0.5,1Ϫsin 2ȏx j if 0.5Ͻx j Յ1.(3.10)following measurement for smoothness:let the interpola-tion polynomial on stencil S k be q k (x );we deﬁneat all half grid points x j ϩ1/2,where x j ϭj ⌬x ,x j ϩ1/2ϭx j ϩ⌬x /2,and ⌬x ϭ .We display the weights Ͷ0and Ͷ1in IS k ϭ͸r Ϫ1l ϭ1͵x j ϩ1/2x j Ϫ1/2h 2l Ϫ1(q (l )k )2dx ,(3.1)Fig.1(Ͷ2ϭ1ϪͶ0ϪͶ1is omitted in the picture).Notethe optimal weight for Ͷ0is C 30ϭ0.1and for Ͷ1is C 31ϭ0.6.We can see that the weights computed with (2.18)where q (l )kis the l th-derivative of q k (x ).The right-hand side (referred to as the original measurement in Fig.1)are of (3.1)is just a sum of the L 2norms of all the derivatives far less optimal than those with the new measurement,of the interpolation polynomial q k (x )over the interval especially around the critical points x ϭ , .However,near (x j Ϫ1/2,x j ϩ1/2).The term h 2l Ϫ1is to remove h -dependent the discontinuity x ϭ ,the two measurements behave factors in the derivatives of the polynomials.This is similar similarly;the discontinuous stencil always gets an almost-to,but smoother than,the total variation measurement zero weight.Moreover,for the grid point immediately left based on the L 1norm.It also renders a more accurate to the discontinuity,Ͷ0Ȃ and Ͷ1Ȃ ,which means,when WENO scheme for the case r ϭ3,when used with (2.15)only one of the three stencils is non-smooth,the other two and (2.16).stencils get O (1)weights.Unfortunately,these weights do When r ϭ2,(3.1)gives the same measurement as (2.18).not approximate a fourth-order scheme at this point.A However,they become different for r Ն3.For r ϭ3,similar situation happens to the point just to the right of (3.1)givesthe discontinuity.For simplicity of notation,we use WENO-X-3to standIS 0ϭ (f j Ϫ2Ϫ2f j Ϫ1ϩf j )2ϩ (f j Ϫ2Ϫ4f j Ϫ1ϩ3f j )2(3.2)208JIANG AND SHUFIG.1.A comparison of the two smoothness measurements.for the third-order WENO scheme (i.e.,r ϭ2,for which u 0(x )ϭsin 4(ȏx ).Again we see that WENO-RF-4is more accurate than WENO-RF-5on the coarsest grid (N ϭ20)the original and new smoothness measurement coincide)but becomes less accurate than WENO-RF-5on ﬁner grids.(where X ϭLF,Roe,RF refer,respectively,to the global The order of accuracy for WENO settles down later than Lax–Friedrichs ﬂux splitting,Roe’s ﬂux splitting,and Roe’s in the previous example.Notice that this is the example ﬂux splitting with entropy ﬁx).The accuracy of this scheme for which ENO schemes lose their accuracy [12].has been tested in [9].We will use WENO-X-4to represent the fourth-order WENO scheme of Liu et al.(i.e.,r ϭ3 4.A SIMPLE WAY FOR COMPUTING WEIGHTS FORwith the original smoothness measurement of Liu et al .)EULER SYSTEMSand WENO-X-5to stand for the ﬁfth-order WENO scheme resulting from the new smoothness measurement.In later For system (1.1)with d Ͼ1,the derivatives d f i /dx i ,i ϭsections,we will also use ENO-X-Y to denote conventional 1,...,d ,are approximated dimension by dimension;forENO schemes of ‘‘Y’’th order with ‘‘X’’ﬂux splitting.We caution the reader that the orders here are in L 1sense.So TABLE IIIENO-RF-4in our notation refers to the same scheme as ENO-RF-3in [15].Accuracy on u t ϩu x ϭ0with u 0(x )ϭsin(ȏx )In the following we test the accuracy of WENO schemes Method N L ȍerror L ȍorder L 1error L 1order on the linear equation:WENO-RF-410 1.31e-2—7.93e-3—u t ϩu x ϭ0,Ϫ1Յx Յ1,(3.11)20 3.00e-3 2.13 1.32e-3 2.5940 4.27e-4 2.81 1.56e-4 3.08u (x ,0)ϭu 0(x ),periodic.(3.12)80 5.17e-5 3.05 1.13e-5 3.79160 4.99e-6 3.37 6.88e-7 4.04320 3.44e-7 3.86 2.74e-8 4.65In Table III,we show the errors of the two schemes at t ϭ1for the initial condition u 0(x )ϭsin(ȏx )and compare WENO-RF-510 2.98e-2— 1.60e-2—them with the errors of the ﬁfth-order upstream central 20 1.45e-3 4.367.41e-4 4.4340 4.58e-5 4.99 2.22e-5 5.06scheme (referred to as CENTRAL-5in the following ta-80 1.48e-6 4.95 6.91e-7 5.01bles).We can see that WENO-RF-4is more accurate than 160 4.41e-8 5.07 2.17e-8 4.99WENO-RF-5on the coarsest grid (N ϭ10)but becomes 320 1.35e-9 5.03 6.79e-10 5.00less accurate than WENO-RF-5on the ﬁner grids.More-over,WENO-RF-5gives the expected order of accuracy CENTRAL-510 4.98e-3— 3.07e-3—20 1.60e-4 4.969.92e-5 4.95starting at about 40grid points.In this example and the 40 5.03e-6 4.99 3.14e-6 4.98one for Table IV,we have adjusted the time step to ⌬t ȁ80 1.57e-7 5.009.90e-8 4.99(⌬x )5/4so that the fourth-order Runge–Kutta in time is 160 4.91e-9 5.00 3.11e-9 4.99effectively ﬁfth-order.3201.53e-105.009.73e-115.00In Table IV,we show errors for the initial condition。

WENO 格式权重分析与改进

WENO 格式权重分析与改进柴得林;李雨润;孙中国;席光【摘要】为了优化 WENO 格式计算性能，在对 Jiang 和 Shu 的经典 WENO 格式（记为 WENO-JS）加权方法分析的基础上，通过引入间接光滑指数，构造出一种新的 WENO 格式———WENO-E 格式，取得减小间断区耗散的效果。

理论分析表明，该格式与WENO-JS 格式计算效率基本相同，可达到相同阶的计算精度；但在相同网格下，较之 WENO-JS 格式，该格式对光滑区域的求解有更小的截断误差，对间断的捕捉有更高的分辨率。

与 WENO-JS 格式相比，采用 WENO-E 格式进行线性迁移方程、非线性 Burgers 方程、欧拉方程等相关问题的数值实验，均能取得更好的数值结果。

%In order to improve the computational performance of the WENO scheme,a new WENO scheme,namely WENO-E scheme was constructed,which reduces dissipation close to discontinuities.Based on the analysis of the algorithm for weighted factors in the classical WENO scheme (namely WENO-JS)proposed by Jiang and Shu,the new scheme was constructed by introducing indirect smooth indicator.Theoretical analysis shows that the WENO-E scheme can reachthe same convergence order of WENO-JS with the same computational efficiency;while it can obtain smaller truncation errors at smooth parts of the solution and higher resolution close to the discontinuities with the same grids than the WENO-JS.Subsequently,compared with the classical WENO scheme,when numerical experiments with the linear transport equation,the nonlinear Burgers equation and the one dimensional Eulersystem of equations are conducted,the WENO-E scheme achieves better numerical solutions.【期刊名称】《国防科技大学学报》【年(卷),期】2016(038)004【总页数】7页(P39-45)【关键词】加权本质无震荡格式;光滑因子;高精度;高分辨率【作者】柴得林;李雨润;孙中国;席光【作者单位】西安交通大学能源与动力工程学院，陕西西安 710049;西安交通大学能源与动力工程学院，陕西西安 710049;西安交通大学能源与动力工程学院，陕西西安 710049;西安交通大学能源与动力工程学院，陕西西安 710049【正文语种】中文【中图分类】V211.1;O241合理、高效、精确地模拟非定常、含间断、多尺度复杂流场是计算流体动力学研究领域的一大难题。

Financial Ratios as Predictors of Failure

The emphasis upon financial ratios does not imply that ratios are the o72nly pErMePdIiRcItCoArLs oREfSEAfaRiClHureI.N TAhCeCOUpNrTimINaGr:y SEcLoEnCcTeErDn SiTsUnDoIEtSw, ith1966predictors of failure per se but rather with financial ratios as predictors of important events—one of which is failure of the firm. Further, the primary concern is not with the ratios as a form of presenting financial-statement data but rather with the underlying predictive ability of the financial statements themselves. The ultimate motivation is to provide an empirical verification of the usefulness (i.e., the predictive ability) of accounting data (i.e., financial statements).
71
bers consist of financial statement items. A third term, predictive ability, also reqiiires explanation but cannot be defined briefly. The various dimensions of predictive ability will be explored later.

英语教学法原著选读48：克拉申二语习得五假说之二——自然顺序假说（NaturalOrder）

英语教学法原著选读48：克拉申⼆语习得五假说之⼆——⾃然顺序假说（NaturalOrder）导读：本篇是⼆语习得泰⽃Stephen D. Krashen的著作《⼆语习得原则与实践（Principles andPractice of Second Language Acquisition）》第⼆章“第⼆语⾔习得理论”A节“有关第⼆语⾔习得的五个假说”中的第⼆个假说，探讨的是语⾔习得过程中各学习项⽬的⾃然顺序。

按照这⼀假说，不管教的⼀⽅多么努⼒，学的⼈总是按照⼀定顺序学会语⾔项⽬，从⽐较简单的ing逐步⾛向稍为复杂的s/es、's等。

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祝朋友们学习进步！------------------------原⽂One of the most exciting discoveries in language acquisition research in recent years has beenthe finding that the acquisition of grammatical structures proceeds in a predictable order.Acquirers of a given language tend to acquire certain grammatical structures early, and otherslater. The agreement among individual acquirers is not always 100%, but there are clear,statistically significant, similarities.English is perhaps the most studied language as far as the natural order hypothesis isconcerned, and of all structures of English, morphology is the most studied. Brown (1973)reported that children acquiring English as a first language tended to acquire certaingrammatical morphemes, or functions words, earlier than others. For example, the progressivemarker ing (as in "He is playing baseball".) and the plural marker /s/ ("two dogs") were amongthe first morphemes acquired, while the third person singular marker /s/ (as in "He lives in NewYork") and the possessive /s/ ("John's hat") were typically acquired much later, cominganywhere from six months to one year later. de Villiers and de Villiers (1973) confirmedBrown's longitudinal results cross-sectionally, showing that items that Brown found to beacquired earliest in time were also the ones that children tended to get right more often. In otherwords, for those morphemes studied, the difficulty order was similar to the acquisition order.Shortly after Brown's results were published, Dulay and Burt (1974, 1975) reported thatchildren acquiring English as a second language also show a "natural order" for grammaticalmorphemes, regardless of their first language. The child second language order of acquisitionwas different from the first language order, but different groups of second language acquirersshowed striking similarities. Dulay and Burt's results have been confirmed by a number ofinvestigators (Kessler and Idar, 1977; Fabris, 1978; Makino, 1980). Dulay and Burt used asubset of the 14 morphemes Brown originally investigated. Fathman (1975) confirmed thereality of the natural order in child second language acquisition with her test of oral production,the SLOPE test, which probed 20 different structures.Following Dulay and Burt's work, Bailey, Madden, and Krashen (1974) reported a natural orderfor adult subjects, an order quite similar to that seen in child second language acquisition. Aswe shall see later, this natural order appears only under certain conditions (or rather, itdisappears only under certain conditions!). Some of the studies confirming the natural order inadults for grammatical morphemes include Andersen (1976), who used composition, Krashen,Houck, Giunchi, Bode, Birnbaum, and Strei (1977), using free speech, and Christison (1979),also using free speech. Adult research using the SLOPE test also confirms the natural orderand widens the data base. Krashen, Sferlazza, Feldman, and Fathman (1976) found an ordersimilar to Fathman's (1975) child second language order, and Kayfetz-Fuller (1978) alsoreported a natural order using the SLOPE test.As noted above, the order of acquisition for second language is not the same as the order ofacquisition for first language, but there are some similarities. Table 2.1, from Krashen (1977),presents an average order for second language, and shows how the first language orderdiffers. This average order is the result of a comparison of many empirical studies ofgrammatical morpheme acquisition.TABLE 2.1. "Average" order of acquisition of grammatical morphemes for EnglishWhile English is the best studied language, it is not the only one studied. Research in order of acquisition for other language is beginning to emerge. As yet unpublished papers by Bruce (1979), dealing with Russian as a foreign language, and van Naerssen (1981), for Spanish as a foreign language, confirm the validity of the natural order hypothesis for other languages. We will deal with the pedagogical implications of the natural order hypothesis later, I should point out here, however, that the implication of the natural order hypothesis is not that our syllabi should be based on the order found in the studies discussed here, that is, I do not recommend teaching ing early and the third person singular /s/ late. We will, in fact, find reason to reject grammatical sequencing in all cases where our goal is language acquisition. We will deal with this later, however, after we have finished laying the theoretical groundwork.(a) Transitional formsStudies supporting the natural order hypothesis show only the order in which mature, or well-formed structures emerge. Other studies reveal the path acquirers take en route to mastery. (For a review, see Dulay, Burt, and Krashen, in press. Ravem, 1974; Milon, 1974; Gillis and Weber, 1976; Cancino, Rosansky, and Schumann, 1974; Wode, 1978 and Nelson, 1980 are some second language studies in this area.) There is surprising uniformity here as well--acquirers make very similar errors, termed developmental errors, while they are acquiring. For example, in acquiring English negation, many first and second language acquirers pass through a stage in which they place the negative marker outside the sentence, as in:No Mom sharpen it. (from Klima and Bellugi's (1966)study of child L1 acquisition)and Not like it now. (from Ravem's (1974) study of childL2 acquisition)A typical later stage is to place the negative marker between the subject and the verb, as in:I no like this one. (Cancino et al. (1975) study of childL2 acquisition)and This no have calendar. (from Schumann's (1978a) study of adult L2 acquisition)before reaching the correct form.Predictable stages in the acquisition of wh-questions in English include an early stage in which the wh-word appears before the rest of the sentence, which is otherwise left in its normal uninverted form, as in:How he can be a doctor? (Klima and Bellugi, 1966, child L1acquisition)and What she is doing? (Ravem, 1974, child L2 acquisition)Only later do acquirers begin to invert the subject and verb of the sentence. (A detailed review can be found in Dulay et al., in press.)Transitional forms have been described for other languages and for other structures. The stages for a given target language appear to be strikingly similar despite the first language of the acquirer (although particular first languages may influence the duration of certain stages; see Schumann, 1979). This uniformity is thought to reflect the operation of the natural language acquisition process that is part of all of us. (For a discussion of some of the current issues and controversies concerning the natural order hypothesis, see Krashen, 1981.)------------------------读后感教学者有⾃⼰的⽇程，学习者有⾃⼰的⾃然顺序，克拉申⼜说不⼀定要完全遵循这个⾃然顺序，问题来了：咱们到底该怎么办呢？这让我想起了《西⾏漫记》⾥埃德加·斯诺提到的⼀件趣事：红军招募新战⼠，⼊伍的往往都是中国西部、西北部农村⽬不识丁的青壮年乃⾄少年⼈。

基于WENO格式的高精度高分辨台风风暴潮数值模式

基于WENO格式的高精度高分辨台风风暴潮数值模式王如云;汪天;吴楚敏;羌丹丹;周钧;张鑫;张彬;占飞【摘要】考虑到台风风暴潮在近岸浅水地区的非线性效应,基于无结构网格,通过采用有限体积法和高精度高分辨率的WENO数值格式对二维浅水方程进行空间离散,并利用三阶的Runge-Kutta格式进行时间离散,最后利用Rogers方法解决复杂海底地形造成的通量梯度项与源项数值离散后的不平衡问题,从而建立了二维台风风暴潮数值模式.模式中的风场和气压场分别采用宫崎正卫风场模式和藤田气压场模式.最后通过对江苏沿海的风暴增水的模拟和验证,表明了该数值模式对台风风暴潮模拟的有效性和可行性.%Considering the non-linear effects of typhoon storm surge in coastal areas,based on the unstructured meshes,the numerical model of two-dimensional typhoon storm surge was established by using the finite volume method,high-order high-resolution WENO scheme to complete the space discretization of the two-dimensional typhoon storm surge equation,the third-order Runge-Kutta scheme for the time discretization,and the Rogers method to solve the problem of the imbalance between the flux gradient and the discrete source terms,which was caused by the complex submarine topography.Fujita formula and Veno Takeo formula are used to simulate pressure and wind in the model,respectively.At last,through the simulation and verification of the typhoon storm surge along Jiangsu coastal areas,it's proved that the simulation of typhoon storm surge by this numerical model is validity and feasibility.【期刊名称】《海洋预报》【年(卷),期】2017(034)002【总页数】6页(P21-26)【关键词】WENO格式;无结构网格;台风风暴潮;数值模式;高精度高分辨【作者】王如云;汪天;吴楚敏;羌丹丹;周钧;张鑫;张彬;占飞【作者单位】河海大学海洋学院,江苏南京210098;河海大学港航学院,江苏南京210098;河海大学海洋学院,江苏南京210098;南通市生产力促进中心,江苏南通226019;河海大学水文水资源学院,江苏南京210098;河海大学海洋学院,江苏南京210098;河海大学海洋学院,江苏南京210098;河海大学海洋学院,江苏南京210098【正文语种】中文【中图分类】P731.23台风风暴潮是由台风引起的海水水位异常升降的现象[1]，对其进行准确、快速数值模拟方法方面的研究，一直是人们关注的问题。

对比论证英文作文怎么写

对比论证英文作文怎么写英文回答：A comparative essay is a type of academic writing that compares and contrasts two or more subjects. It is acritical analysis of similarities and differences between the subjects, which can be anything from literary works to historical events to scientific theories.The first step in writing a comparative essay is to choose your subjects. The subjects should be similar enough to make a meaningful comparison, but they should also be different enough to make the comparison interesting. Once you have chosen your subjects, you need to develop a thesis statement. The thesis statement is a one-sentence summary of your argument, and it should state the main similarities and differences between your subjects.The next step is to write the body of your essay. The body of the essay should be divided into paragraphs, eachof which focuses on a different similarity or difference between your subjects. Each paragraph should begin with a topic sentence that states the main point of the paragraph, and it should provide evidence to support the topic sentence. The evidence can come from your own research, from the sources you have read, or from your own experiences.The conclusion of your essay should restate your thesis statement and summarize the main points of your essay. It should also provide a final thought or reflection on the comparison.中文回答：对比论证英文作文是一种学术写作类型，用来比较和对比两个或多个主题。

Reducing phase error in long numerical binary black hole evolutions with sixth order finite

a r X i v :0706.0740v 1 [g r -q c ] 5 J u n 2007Reducing phase error in long numerical binary black hole evolutions with sixth order ﬁnite diﬀerencingSascha Husa,Jos´e A.Gonz´a lez,Mark Hannam,Bernd Br¨u gmann,Ulrich Sperhake Friedrich Schiller University Jena,Max-Wien-Platz 1,07743Jena,Germany E-mail:sascha.husa@uni-jena.de Abstract.We describe a modiﬁcation of a fourth-order accurate “moving puncture”evolution code,where by replacing spatial fourth-order accurate diﬀerencing operators in the bulk of the grid by a speciﬁc choice of sixth-order accurate stencils we gain signiﬁcant improvements in accuracy.We illustrate the performance of the modiﬁed algorithm with an equal-mass simulation covering nine orbits.PACS numbers:04.25.Dm,04.30.Db,95.30.Sf 1.Introduction In the last two years the numerical solution of the general relativistic two-body problem has made a giant leap forward with a series of breakthroughs in 2005[1,2,3].More than forty years after Hahn and Lindquist started the numerical investigation of colliding black holes [4],the ﬁeld has now passed several crucial milestones toward simulating general inspiral situations,such as simulations of unequal-mass binaries and calculations of the gravitational recoil eﬀect and the evolution of black-hole binaries with spin [5,6,7,8,9,10].The latter have recently lead to the spectacular ﬁnding that extremely large recoils are possible for spinning black holes [11,12,13].In order to fulﬁll numerical relativity’s promise of providing useful information to the gravitational wave data analysis community,it is desirable to perform long numerical inspiral evolutions that allow us to cleanly match fully general relativisticand post-Newtonian waveforms with error bars,and thus produce “complete”waveforms,which contain large numbers of gravitational-wave cycles from the inspiral phase,as well as simulating the merger and ringdown phases.Such simulations will be necessary for a suﬃciently dense sample of the black-hole binary conﬁguration parisons with post-Newtonian results have already started,and several groups have published results showing good agreement of various aspects of non-spinning simulations with post-Newtonian predictions (see e.g.[14,15,16,17,18,19,20]).Precise error estimates and detailed coverage of the unequal-mass and spinning cases are however missing.One serious technical problem is that performing such long evolutions with good accuracy in the phase is still computationally expensive —at least for standard “moving puncture”ﬁnite-diﬀerence codes [21,22,23,16,13,10].In order to overcome phase inaccuracies in long evolutions,an alternative route isprovided by spectral methods,and signiﬁcant progress has been made in this direction by the Caltech-Cornell group[24,25].The initial data we evolve diﬀer from the choice in[26,25],and we cannot make a direct comparison of the accuracy of the results since puncture evolutions cannot start with excision initial data.It would certainly be worthwhile in the future to conduct direct comparisons regarding eﬃciency for diﬀerent codes.We only quote some preliminary numbers here to illustrate the fact that while ultimately the convergence behavior of spectral codes is superior,for the purpose of“gravitational wave astrophysics”,ﬁnite diﬀerence codes may still be competitive:The performance of the Caltech/Cornell spectral code has been quoted for long time medium resolution runs as roughly10(for a“643”grid conﬁguration)and27(“763”grid)CPU hours per M[26](we will quote time and length units of the total black-hole mass M,see [16]).The highest resolution run presented here performs at5.5CPU hours per M on an Intel Woodcrest2.66GHz dual core processor.Both codes show satisfactory accuracy for long evolutions,and we certainly expect both codes to undergo further optimizations.Long evolutions that show fourth order convergence for most of the simulation have also been presented by the Goddard group[27],using impressively low spatial resolution,but no details are given on code timing.We have previously reported accurate evolutions for approximately two orbits (initial separation of D=6.45M)in[16],with an error in the merger time of0.2%at a computational cost of505CPU hours(1.44CPU hours/M),and we have reported a merger time error of0.5%for D=8M simulations[28].At larger initial separations the number of orbits is a steep function of the separation,and phase accuracy rapidly decreases.Our fourth order code would thus require resolutions which weﬁnd hard to tolerate for performing large parameter studies in the style of[6].In the context ofﬁnite diﬀerencing it is natural to consider higher order methods.For example,it already turned out to be important to move from second-orderﬁnite diﬀerencing to fourth-orderﬁnite diﬀerencing as the feasible evolution time for puncture evolutions increased.However,using higher orderﬁnite diﬀerencing for the types of codes used to simulate black-hole binary inspiral is not entirely trivial:for the moving-puncture method,which is currently employed in the majority of the current codes,the continuum equations become singular at the location of the“puncture”,and one might worry about the robustness ofﬁnite-diﬀerence schemes.Furthermore,current mesh reﬁnement algorithms in theﬁeld are based on the use of buﬀer zones,whose number depends on the stencil width of theﬁnite diﬀerencing scheme.In three spatial dimensions high-orderﬁnite-diﬀerence schemes with wide stencils easily lead to a drastic decrease in performance caused by the additional computational load due to the extra buﬀer points.This gain in accuracy pertains in particular to the phase of the evolution.In the present paper we report on aﬁrst step to signiﬁcantly improve the accuracy of currentﬁnite-diﬀerence codes to evolve black-hole binaries by using sixth order accurateﬁnite diﬀerencing operators in the bulk of the grid.We combine the sixth order accurate derivative operators with fourth-order accurate dissipation operators [setting r=3in Eq.(3)]and Runge-Kutta time integrators,and aggressively reduce the number of AMR buﬀer zones compared to the number we would theoretically require for sixth order convergence.The penalty in computational cost is a rather moderate30%compared with our fourth-order code.(We have compared the average speed over100M of evolution time for our largest grid conﬁguration,which producesa29%increase in computational cost;in our experience this is typical for our current code).2.Summary of the“moving-puncture method”as implemented in the BAM codeThere is a large freedom in writing the Einstein equations as a system of partial diﬀerential equations,and much research has gone intoﬁnding optimal choices.In this work we employ the currently most popular choice,the BSSN system[29,30,31,32], which so far is the only system for which results have been reported of long-time black-hole binary simulations that do not rely on black hole excision as has been used for example in[1,24,25].We use the BSSN system together with the1+log and gamma freezing coordinate gauges[33,34,31]as described in[16](choosing in particular the parameterηin the gamma freezing shift condition asη=2as we have done previously).These gauge conditions allow the“punctures”to move across the grid(“moving puncture”approach[2,3])and allow an eﬀective softening of the singularity in the metric associated with an internal asymptotic region[35,36,37],which had been prohibited by the traditional“ﬁxed punctures”approach.The BSSN system is based on a conformal decomposition of the spatial geometry,writing the physical spatial metric as g ij=χ−1˜g ij(following[2]).The blowup of the metric at the“punctures”is absorbed into the conformal factorχ,which vanishes at the“puncture”.For our numerical evolutions we use the BAM[38,39]code,which is designed to solve partial diﬀerential equations on structured meshes,in particular a coupled system of(typically hyperbolic)evolution equations and elliptic equations.The complexity of the equations is addressed by using a Mathematica package integrated into the code,which produces C-code from Mathematica expressions in tensor notation. Using such a system as we do in BAM,or as has been discussed in detail for the Cactus environment in[40]drastically simpliﬁes the modiﬁcation of complex codes for black-hole binary simulations,as was required to adapt codes from the“ﬁxed puncture”to the“moving puncture”paradigm,or in the present case to implement the improved numerical algorithms discussed here.The structure of the BAM code has also made it straightforward to implement higher orderﬁnite diﬀerencing methods The computational domain is decomposed into rectangular boxes,following standard domain-decomposition algorithms,and is parallelized with MPI[41].Our mesh reﬁnement algorithm is based on the standard Berger-Oliger algorithm,but with additional buﬀer zones,along the lines of[42,43]as described in[16]and summarized in the next section.We essentially use aﬁxed-mesh-reﬁnement strategy,with inner level reﬁnement boxes following the motion of the black holes.Typically we use about 10reﬁnement levels(reﬁning the grid spacing by factors of2),roughly half of which follow the movement of the black holes.In order to represent black holes in the initial data,we use the so-called“puncture method”[44].For these data it is well understood how to write the constraint equations in a form suitable for numerical solution[45,46].Following the approach of[44]our initial data sets are chosen to be conformallyﬂat with Bowen-York extrinsic curvature.The momentum parameter in the Bowen-York extrinsic curvature is determined from a quasi-equilibrium condition at third post-Newtonian order as described in[16].The elliptic constraint equations are solved in BAM with the pseudo-spectral collocation code described in[47].AMR data are then obtained by barycentricinterpolation,typically with eighth-order polynomials for both the fourth-and sixth-orderﬁnite diﬀerencing methods.The eﬃciency of the spectral solver is suﬃcient to solve the initial data problem on a single processor.3.Sixth orderﬁnite diﬀerencing3.1.Mesh reﬁnement in the BAM codeOur numerical evolution algorithm is based on a method-of-lines approach usingﬁnite diﬀerencing in space and explicit fourth-order accurate Runge-Kutta time stepping (with aﬁxed time step).We apply sixth-order accurate polynomial interpolation in space between diﬀerent reﬁnement levels so that all spatial operations of the AMR method(i.e.restriction and prolongation)are sixth-order accurate,such that the second derivatives of interpolated values are at least fourth-order accurate.Although the time stepping used for evolution is also fourth-order accurate through the Runge-Kutta integrator,there arises the additional issue of how to provide boundary values for the intermediate time-levels of the Berger-Oliger algorithm that are not aligned in time with a coarser level(otherwise spatial interpolation can be used).Using higher than third-order interpolation has lead to spurious noise at mesh reﬁnement boundaries as described in[16].We therefore use third-order interpolation in time,which introduces a second-order error within the Berger-Oliger time-stepping scheme[16],which however is not noticeable in typical runs as we have checked by running with uniform(as opposed to Berger-Oliger non-uniform)time-stepping.In summary,if the outer boundary is placed suﬃciently far away and if time-interpolation errors at reﬁnement boundaries are small,then fourth-order convergence can be observed.A relatively straightforward modiﬁcation of the standard Berger-Oliger scheme is to replace the single-point reﬁnement boundary by a buﬀer zone consisting of several points,e.g.,[42,43,48].For a suﬃciently large number of buﬀer zones(the product of number of points in the stencil toward the mesh reﬁnement boundary and the number of source evaluations during a full time step of the coarse grid),no time interpolation is required and excellent results have been reported for this scheme[43](note that special methods like[49]seem to achieve similar performance).For example,our fourth-order Runge Kutta scheme requires four source evaluations,and if the lop-sided stencil with three points in one direction,f′(x)=−3f−1−10f0+18f1−6f2+f320f(5)(x)h4+O(h5)(1)is used,then the numerical domain of dependence for a given point has a radius of12 points.Here and in the following we use the notation f j=f(x+j∆x).Therefore,it is possible to provide12buﬀer points at the reﬁnement boundary and to perform one RK4time step with size three stencils that does not require any boundary updates. Only after the time step is completed,the buﬀer zones have to be repopulated.In the context of Berger-Oliger AMR,the buﬀer update is based on interpolation from the coarser levels.Since every second time step at level l coincides in time with level l−1,one can provide24buﬀer points,perform two time steps,and then update the buﬀer by interpolation in space.With12buﬀer points,one can interpolate in time to obtain data for the buﬀer points at intermediate time levels.To use fewer than12buﬀer points,we can interpolate into all buﬀer points before starting an RK4update as described,and then evolve all points except theoutermost points located exactly on the boundary,which are keptﬁxed at their initial interpolated value.The inner points next to the boundary are updated using second-orderﬁnite diﬀerencing for the centered derivatives and shifted advection stencils for the advection derivatives.Even though for large grids the number of buﬀer zones becomes negligible,for the grid sizes that we use,the buﬀer points aﬀect the size of the grids signiﬁcantly.For example,even for our largest inner box size of80points in one direction,adding six points on both sides instead of12or24points leads to92,104, and128points,respectively,which corresponds to a signiﬁcant saving in the number of points in3d since1043/923≈1.44and1283/923≈2.69.For clarity,we always quote grid sizes without buﬀer points,because this is the number of points owned by a particular grid.Experimentally we have found that for the fourth-order case using just six buﬀer points leads to very small diﬀerences compared to12buﬀer points,but even smaller buﬀer zones lead to noticeable diﬀerences.For simulations with fourth-order accurate derivative operators,we have therefore chosen a standard setup of RK4with dissipation and lop-sided advection stencils,6buﬀer points,quadratic interpolation in time,and Berger-Oliger time-stepping on all but the outermost grids following[16].As is common in numerical relativity,we use symmetricﬁnite-diﬀerence stencils for all spatial derivatives but the advection terms associated with the shift vector, where we use lop-sided upwind stencils,see e.g.[50]for the fourth-order accurate case.3.2.Artiﬁcial DissipationInﬁnite-diﬀerence codes targeted at smooth solutions of nonlinear hyperbolic equations,it is common practice to add artiﬁcial dissipation terms to all right-hand-sides of the time evolution equations,schematically written as∂t u→∂t u+Q u.(2) Such dissipation terms are very eﬃcient at suppressing very high-frequency waves, which are not part of the physical solution.This may be necessary for numerical stability[51],but also to reduce numerical noise generated at mesh-reﬁnement boundaries.As has become rather common in numerical relativity,we follow[51] and choose an operator(Q)of order2r asQ=σ(−h)2r−1(D+)r(D−)r/22r,(3) for consistency with a2r−2accurate scheme,withσa parameter regulating the strength of the dissipation.As we have done in the past with our fourth-order code we choose the factorσasσ=0.1in the inner levels andσ=0.5in the outer levels (where the waves are extracted).For high orders,these dissipation stencils become rather large(seven points for the fourth-order case and nine points for the sixth-order case).We therefore do not add dissipation terms where these stencils would“cross”mesh reﬁnement boundaries. Also,adding dissipation terms with large stencils can lead to a loss of performance. We have therefore attempted to combine the use of sixth-order accurate stencils for the derivative operators with a fourth-order accurate dissipation operator and time integrator and second-order time interpolation at mesh reﬁnement boundaries with an aggressively small number(6)of buﬀer points.3.3.Sixth order accurateﬁnite diﬀerence operatorsFor the sixth-order case weﬁnd that several choices for the advection term stencils yield stable evolutions,but the lop-sided upwind stencil which is closest to the symmetric case yields(probably not surprisingly)by far the best accuracy,i.e.we usef′(x)=2f−2−24f−1−35f0+80f1−30f2+8f3−f4105d7f(x)60h−1dx7h6+O(h7),f′(x)=−147f0+360f1−450f2+400f3−225f4+72f5−10f6)7d7f(x)60h−1dx7h6+O(h8),but does not show equally robust results,as is common for solving advection equations. For non-advection derivative terms we again use the standard symmetric stencil, similarly for second derivatives in one direction we use the symmetric stencilf′′(x)=−490f0+270(f1+f−1)−27(f2+f−2)+2(f3+f−3))560f(8)(x)h6+O(h8).For mixed derivatives,we use the stencils which result from a product of the symmetric sixth order accurateﬁrst derivative operators.4.Results for long equal-mass evolutionsAll runs are carried out with the symmetry(x,y,z)→(−x,−y,z)and(x,y,z)→(x,y,−z),reducing the computational cost by a factor of four.The Courant factor C=∆t/h l is kept constant,and is set to C=1/2for the inner grids,while for the outer grids at levels0–4the time step is kept constant at the value of level3, following our previous work[16].All runs presented here use six AMR buﬀer points, the same number that we have used for our fourth-order accurate code[16].We stress that this is less than required to isolate theﬁne level“half”timestep from time interpolation errors at the mesh-reﬁnement boundary,and in particular also less than required for a fully sixth-order scheme following the approach of[43].The grid setups we have used for our simulations are displayed in table1(using the naming convention introduced in[16]).All the runs listed here have been performed on the Kepler cluster at the University of Jena(using Intel dual Woodcrest CPUs running at2.66GHz and an Inﬁniband interconnect),additional runs have been performed at LRZ Munich and the Doppler cluster at the University of Jena.We will denote the individual simulations by the inner-box size,i.e.,48,56,64,72or80,as indicated in bold in table1.Our initial data are chosen as follows:the initial coordinate separation of the punctures is chosen as D=12M,the horizon mass for each individual hole is chosenTable1.Grid setups used for convergence test simulations.The notation in the“Run”column is the same as we have used in[16].The quantities h min and h max(rounded to three digits)denote theﬁnest and coarsest grid spacings,and r max isthe location of the outer boundary(rounded to4digits),and all are in units of M.Also speciﬁed are the numbers of processors used,maximal memory requirementin GByte(to be precise,we quote the resident size of the program,i.e.,the physicalmemory a task has used),and average speed in M/hour for the Kepler cluster atthe University of Jena(using Intel dual Woodcrest CPUs running at2.66GHz).The number in bold are used to indicate individual simulations throughout thispaper.h min r max mem.(GByte)χη=2[5×48:5×96:6]16815.6χη=2[5×56:5×112:6]96/71211.6χη=2[5×64:5×128:6]12128.4χη=2[5×72:5×144:6]32/316 3.5χη=2[5×80:5×160:6]48/524 4.402505007501000125015001750t M 0123456D M 60080010001200140016001800t M 255075100125150175200Φ r a d i a n 806448,6th ord.48,4th ord.Figure 2.Left panel:coordinate distance of the black holes for the fourth orderversion of the 48-conﬁguration and sixth-order simulations {48,64,80}in the orderof increasing merger time.Right panel:the gravitational wave phase for the sameruns.The 72simulation would not be distinguishable from the 80simulation onthe scale shown here.1500 1000 5000t M 0.000010.00010.0010.01r 422 M 1 1750 1500 1250 1000 750 500 2500t M 0.10.20.30.40.50.6ΩM 8064Figure 4.The l =2,m =2mode of the wave signal is split into the absolute valueof Ψ4,22(left panel)and the wave frequency ω(right panel).Both panels showthe simulations {64,72,80},aligned in time to coincide at the peak of |Ψ4|.Thecurves are clipped at early times,where they are very noisy due to the smallnessof the signal and ﬁnite diﬀerencing error in computing the wave frequency fromthe phase.170017501800185019001950t M 0.01 0.00500.0050.01r 422 M 172 8064 72 2.1926080100120140160180ΦGW radian 0.0010.00050.00050.0010.0015 r 422 M 1 72 80 64 72 2.192Figure 5.Convergence plot for the wave amplitude |ψ422|in the l =2,m =2mode.Both panels show the diﬀerence between the 72and 80runs and diﬀerencesbetween the 64and 72runs rescaled for sixth-order convergence.In the left paneldata at diﬀerent resolutions are compared at the same coordinate time,whichleads to a seeming loss of convergence near the radiation peak,which is due tothe relatively large phase error.In the right panel the data are compared atthe same value of the gravitational wave phase,which restores clean sixth orderconvergence.Richardson extrapolation with convergence order 5.5yields an error estimate of ≈4M for both times.Note that oscillations,which are probably mainly due to eccentricity,can clearly be seen in the black hole distance,and also in the wave amplitude shown in ﬁgure 4.A method to reduce eccentricity will be discussed in [52].We have obtained roughly sixth order convergence for the gravitational wave phase between about t =1000M and t =1800M ,as shown in ﬁgure 3.At earlier times the convergence factor becomes very noisy due to the smallness of the signal.Shortly before the merger the convergence factor “glitches”to a value of roughly 7.This problem can also clearly be seen in the convergence of the radiation frequency and amplitude as shown in ﬁgures 6and 5.This “glitch”appears when the frequency and phase increase very sharply,and small phase errors have a large eﬀect.The logarithmic scaling version of the convergence plot in ﬁgure 3shows a slow and rather clean exponential growth for the phase error at intermediate times,a nonlinear ﬁt for 300M ≤t ≤1400M yields δϕ=0.0117exp(0.003t/M )for the phase error.This80859095100105110ΦGW radian 0.10.080.06 0.04 0.02s e p a r a t i o n M 72 80 64 72 2.192100150200300500700100015002000t M0.00010.0010.010.11∆Φ w a v e c y c l e s Figure 6.Left panel:Reparametrisation of the black hole distance bygravitational wave phase yields clean sixth order convergence until the time ofmerger (which occurs roughly at the peak of the error)–the diﬀerence betweenthe 72and 80runs and diﬀerences between the 64and 72runs rescaled for sixth-order convergence lie essentially on top of each other.Right panel:The diﬀerencein wave phase between the 4th and 6th order versions of the lowest resolution 48conﬁguration shows the fourth-order algorithm “falling behind”.observation provides one way to optimize numerical methods in the inspiral phase,without evolving all the way to the merger.In the last stage of the inspiral the phase error grows very rapidly.We have noted this previously in a diﬀerent context in [16],where we have compared diﬀerent methods to provide quasicircular inspiral data.Small changes on the order of 1%of the initial momenta have lead to drastic changes of ≈40M in merger time.In order to clarify the convergence behaviour of our code,we have applied a new technique to check for convergence in situations where the numerical error is dominated by phase shift:We ﬁrst perform a convergence analysis for the dependence of the gravitational (or orbital)phase on code time,and then perform a standard convergence test on a quantity like the puncture separation (Fig.(6))or wave amplitude (Fig.(5))regarding the functional dependence of this quantity on the phase.An important question aside from convergence is how the sixth order and fourth order algorithms compare in absolute numbers.For this purpose we have re-run the 48conﬁguration with our standard fourth order algorithm.We ﬁnd that already at this low resolution the sixth order algorithm is superior,as shown in ﬁgures 2and 6,while at higher resolutions the larger convergence factor increases the gain in accuracy.5.ConclusionsWe have described a minimal extension of the fourth-order accurate evolution algorithm described in [16],where by replacing spatial fourth-order accurate diﬀerencing operators in the bulk of the grid by sixth-order accurate stencils,we gain drastic improvements in accuracy for the phase in long simulations of equal-mass inspiral.The crucial technical point regarding the choice of sixth-order accurate ﬁnite diﬀerence operators has been a speciﬁc choice for the advection stencil,which is used to discretize Lie derivative terms with respect to the shift vector in the Einstein ing this method we have demonstrated evolutions of about nine orbits or 1800M with a phase error of approximately 4M in the merger time,requiring ≈11100CPU hours on our in-house cluster.We emphasize that our code has several lower-order accurate ingredients,which however do not seem to contribute signiﬁcantly tothe numerical error at the resolutions we employ.Our emphasis here has been on boosting the current generation of“moving puncture”codes regarding their eﬃciency to analyse physical situations that require long evolutions,such as an accurate comparison with post-Newtonian results(see[53]), rather than on numerical analysis.Some technical questions certainly remain,such as the reduction of the numerical error at mesh reﬁnement boundaries,the optimization for diﬀerent architectures,and a rigorous mathematical analysis of numerical stability, e.g.,by extending[54]to the BSSN system and the complications arising in the context of mesh-reﬁnement.In future work we will present applications of our algorithm to other situations such as unequal masses and evolutions of spinning black holes. AcknowledgmentsThis work was supported in part by DFG grant SFB/Transregio7“Gravitational Wave Astronomy”.We thank the DEISA Consortium(co-funded by the EU, FP6project508830),for support within the DEISA Extreme Computing Initiative ();computations were performed at LRZ Munich and the Doppler and Kepler clusters at the Institute of Theoretical Physics of the University of Jena. 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比较排序算法英语

比较排序算法英语Comparative sorting algorithms are a collection of algorithms used to sort a list of elements in a specific order. These algorithms compare pairs of elements and make decisions based on the comparison results to rearrange the elements until the entire list is sorted.One commonly used comparative sorting algorithm is the Bubble Sort. This algorithm compares adjacent pairs of elements and swaps them if they are in the wrong order. The process is repeated until the entire list is sorted. Bubble Sort has a time complexity of O(n^2), where n is the number of elements in the list.Another popular comparative sorting algorithm is the Insertion Sort. It starts with an empty left portion and a sorted right portion of the list. It sequentially takes an element from the unsorted portion and inserts it into the correct position in the sorted portion. This process is repeated until the entire list is sorted. The time complexity of Insertion Sort is also O(n^2).Merge Sort is a more efficient comparative sorting algorithm. It follows the divide-and-conquer approach, where the list is recursively dividedinto smaller sublists until they are sorted individually. Then, these sorted sublists are merged to produce a final sorted list. Merge Sort has a time complexity of O(n log n), which makes it more efficient than Bubble Sort and Insertion Sort for larger lists.Quick Sort is another popular comparative sorting algorithm known for its efficiency. It also uses the divide-and-conquer approach by selecting a pivot element and partitioning the list into two sublists, one with elements smaller than the pivot and the other with elements larger than the pivot. This process is recursively applied to the sublists until the entire list is sorted. Quick Sort has an average time complexity of O(n log n), but it can degrade to O(n^2) in the worst case scenario.These are just a few examples of comparative sorting algorithms. There are many other algorithms, each with its unique characteristics and trade-offs in terms of time complexity and space complexity. The choice of sorting algorithm depends on the specific requirements and constraints of the problem at hand.。

英文作文对比写作格式模板

英文作文对比写作格式模板Comparison of Writing Formats: Essay Title。

When it comes to writing essays, there are various formats that writers can choose from based on the type of essay and the specific requirements of the assignment. In this article, we will compare and contrast two commonly used writing formats: the MLA format and the APA format.MLA Format。

The MLA (Modern Language Association) format is commonly used for writing papers and citing sources in the humanities and liberal arts. This format is known for its emphasis on authorship and page numbers, as well as its use of in-text citations and a Works Cited page.When writing in MLA format, the paper should be double-spaced and use a legible font such as Times New Roman in12-point size. The first page should include the writer'sname, the instructor's name, the course title, and the date, all aligned to the left. The title of the essay should be centered, and there should be a header with the writer'slast name and page number in the top right corner of each page.In-text citations in MLA format typically include the author's last name and the page number of the source, if available. The Works Cited page should list all the sources used in the paper, arranged alphabetically by the author's last name. Each entry should include the author's name, the title of the source, the publication information, and the medium of publication.APA Format。

Comparison of Influence of Socrates and

Comparison of Influence of Socrates and ConfuciusSocrates (BC 469---BC 399), was the most famous ancient Greek philosopher and educator. He and Confucius(BC 551--BC 479) are basically at the same time. Same with Confucius, Socrates is not only a Philosopher full of wisdom, but also a educator was tireless in teaching others. They are both philosophers in the same era who have many same ideological points in educational thoughts.I In the object of education, they have advocated “make no social distinctions in teaching”, to promote the popularization of education and culture.In the spring and Autumn Period , education is still followed the "study in the government", the traditional education had been monopolized by the children of the nobility, ordinary people in general that have no chance to receive education. Confucius strongly in folk held private schools, students regardless of the status hierarchy, refusing nobody. In this way, his disciples from all classes of society. Not only that, there were father and son as Confucius disciples, such as the "seventy-two Yin" of Yan Hui, and his father Yan Lu, were involved Confucius students.Confucius students ran Yong, his father "base and evil", Yong was very virtuous, Confucius thought, such a man of virtue is God won't give up, nobody can deprive him of the opportunity of education. This is the Confucius to "make no social distinctions in teaching" principle.Same with Confucius,Socrates also adhere to the principle of"make no social distinctions in teaching".The age of Socrates lived,is the age of Athenian democracy popularized. With the development of the polis democracy, highlights on the role of people. When both humanism origin meaning "Sophists" produced, but they give lectures is to charge. Unlike the Sophists, Socrates does not charge tuition fees, refuse nobody, teach without fixed, square, workshop, and even on the market are the places he taught. Socrates said: "I would like to also answer questions raised by the rich and the poor, as long as any person willing to listen to me talk and answer my questions, I would like to company with," "not only no reward, and be willing to listen to me, I am willing to pay back." Socrates has cultivated many outstanding disciples, such as Platon, Isocrates, Xenophon etc..Platon had carried forward Socrates's objective idealism system, become the precious resources in the world history of philosophy;Isocrates established a special Professor eloquent schools in 392 BC, which is believed to be the first Western so impart professional knowledge for the purpose of educational institutions of higher education, has the nature of. Xenophon is a famous ancient Greek historians, military strategist and statesman, its history is the study of the important information of ancient Greece.Confucius and Socrates this "make no social distinctions in teaching" theory, with a broad mind only belong to the original aristocratic education free to civilian open, spread to the popularization and culture at the time of their national education has played an indelible contribution.II Comparison of the thought of rule of virtueThe connotation of Confucius moral refers to the kernel.----Ren, is both interpersonal relationship, the benevolent have to do with love, loyalty filial piety, gentle, modest and courteous quality. Benevolence is the Confucius to the troubled times out of a medicine, if everyone can achieve. Ren standard, to live a moral life, then the society will be harmonious and beautiful.Socrates's virtue is more complex. He thought virtue is the nature of man, and everyone has the virtue. But everyone has virtue, not realistic to have, but potentially. In other words, people are not born with human nature, only their own moral awareness under the guidance of the rationality, in order to achieve the virtue. Therefore, he put the virtue and knowledge are equated, the 'knowledge is virtue, ignorance is evil', 'no harm' conclusions. Including justice, temperance, courage, piety, etc.On the relationship between rule of virtue and rule of law, two people have different views. Confucius view is roughly de above the law.This sentence not only the moral force is greater than the law, also shows the connotation de from the relation between person and person of Chinese front; in the statute law are reduced to ordinary people, equality of the distinguished and lose their dignity, this is not conducive to the maintenance of high and low, upper and lower order. Compared with the legal system, Confucius tend rule.Socrates's view is roughly law is the moral embodiment, the two are equivalent. Socrates believed that the laws of the city's citizens consistent agreement, should go to the executive not to move or retreat, only obey the law, in order to enable people to unite in a concerted effort, the city is powerful, obey the law is the people's fundamental guarantee happiness, city strong, its value is much higher than the individual life.By the simple comparison of Confucius and Socrates thought above, we discover not hard, although the East and the West was then still in isolation from each other, but the two sages should agree without prior without previous consultation in thought, perhaps this is the natural law of education ". Confucius and Socrates the two East West philosophers thought in today still shines the light of wisdomAs the two land mark thinker,educator,they voluntarily give up through his writings to the education of future generation so opportunities, they are adhering to a thought. The master said, "a transmitter and not a maker, believing in and loving the ancients, I venture to compare myself with our old p'ang." Even the "Analects of Confucius", also written his disciples. Socrates is also without leaving any works, for his thoughts, we can only be obtained from his disciple Platon and Kseno letter writings.Why didn't they want to use their own words to leave something? This is a puzzle, a mysterious beautiful mystery, perhaps only this point defect was perfect.。

2019-3-83-JWL 状态方程及其等效多方状态方程在内爆炸计算中的应用分析

Abstract：［Objectives］ In order to study the influence rule of JWL Equation of State（JWL-EOS）and its equivalent polytropic EOS on internal explosion parameters.［Methods］An independently developed procedure was used to carry out the numerical calculation of explosive blasting process in the cabin. Firstly，the influence rule of different values of parameter ω in the JWL-EOS and the specific heat ratio of ambient gas on the calculated explosion parameters was discussed. Secondly，based on the JWL-EOS， the way to determine parameters in the equivalent polytropic EOS of gas was proposed，and the feasibility and reliability of this method is verified through the algorithm case.［Results］The results show that the increase of parameter ω and specific heat ratio of ambient gas can increase the specific heat ratio of the final mixture，the peak values of shock wave and make the initial arrival time of shock wave to the cabin wall ahead of schedule. The increase of parameter ω can accelerate the expansion rate of detonation products at the initial stage of explosion and increase the peak of quasi-static pressure. While，the increase of specific heat ratio of ambient gas slows down the expansion rate of detonation products at the initial stage of explosion and has little effect on the final quasi-static pressure peak.［Conclusions］It is concluded that the proposed equivalent EOS is of some reliability and the study result in this paper can provide certain reference and basis for the design of anti-explosion structure and damage assessment. Key words： explosion in confined space； fifth-order WENO scheme； detonation products； JWL Equation of State（JWL-EOS）；variable index polytropic gas state equation；numerical calculation

A Brief Comparison of Chinese and Western Festivals

A Brief Comparison of Chinese and Western Festivals作者：添加：08/10/10 访问量：I. IntroductionFestival is defined as a special occasion on which people may give thanks for a harvest, commemorate an honored person or event, pay respect to the dead, or celebrate a culture. Festival, as a cultural phenomenon, comes into being during the course of human development and historical evolution. This unique cultural phenomenon embodies human cognition and attention to the natural environments and peripheral surroundings. As we know, a diverse multitude of festivals are enjoyed worldwide. Although the different regions, nations, religions and cultures bring about multifarious festivals, and the means of celebration vary from one place to another, most of people bear common feelings for the festivals, that is praying for happiness and health, fulfilling the good will, and showing the love of life. The festival celebration also expresses people’s reminiscent mood for traditions and convey their aspiration to pursue the dreams.Festival is considered as “ a carrier of culture”, or an embodiment of the patterns of behavior and thinking that people learn, create and share. Through this unique and specific phenomenon, it is more convenient and direct for the researcher to probe into the deeper layer of human culture. Moreover, the festival provides us a shortcut to explore the divergence and similarity of two cultures. As we know, it is impossible for us to make research on all aspects of festival customs since culture itself is a very complicated phenomenon. Thus, I just choose some representative and well-known festivals--- Spring Festival & Christmas Day to make a brief comparison.II. Comparison of Spring Festival & Christmas Day1. Origin of the HolidaysIn China, Spring Festival is the first and foremost one that should be mentioned. Spring Festival is also known as the Chinese New Year. Spring Festival starts with the new moon on the first day of the new year and ends on the full moon 15 days later. So the timing of the holiday varies from late January to early February. The Spring festival celebrates the earth coming back to life, and the start of ploughing and sowing. In the past, feudal rulers of dynasties placed great importance on this occasion, and ceremonies to usher in the season were performed.The first day of lunar years called “nian chu yi” (the first day of nian). There is folklore about the monster called “nian”. In fact, the festival stems from the agricultural production. China is an agricultural country and our ancestors have known how to crop in the fields and formed a farming industry thousands of years ago. After accumulating experience during the long-time cultivation, the ancient people find out that the proper seeds planting in springtime leads to a good harvest in autumn. Later, a famous saying comes into being “ the whole year’s work depends on a good starting in spring(一年之际在于春)”. Therefore, the springtime becomes the prime time for the Chinese. Because the Chinese regard springtime as a start of the whole year, no wonder the Chinese place so much emphasis on the springtime, and SpringFestival is treated as such an important festival. Meanwhile, this festival mirrors the Chinese’ attitudes to the relationship between nature and human---keeping harmony. In daily life and work, people find a certain rule in nature; following it, people set the production schedule to help plant in a reasonable way. In nature, people make development and enjoy life.Christmas to the westerns is compared as Spring Festival to Chinese. It is on December 25th every year and usually lasts for a week, until a new year. Christmas originated from Europe. How Christmas comes into being must date back to 1A.D.when Jesus came into world. The name Christmas is short for “Christ’s Mass”. A Mass is a kind of church service. Christmas is the feast of the nativity of Jesus, is on 25th, December every year. It is the day we celebrate as the birthday of Jesus. In Bible, there is a vivid description of the story about Jesus birth: in a city of Galilee, the virgin’s name was Mary was betrothed to Joseph. Before they came tighter, she was found with a child of the Holy Spirit. While Joseph her husband wanted to put her away secretly, Gabriel, an angel of the Lord appeared to him in a dream and told him not be afraid to take Mary as wife. And Mary would bring forth a son, and he should call him, Jesus, for he would save his people from their sins. Finally, on December 25th, Mary brought forth her firstborn son---Jesus.2. Preparation for the HolidaysPreparations for the New Year festival start during the last few days of the last moon. Houses are thoroughly cleaned, debts repaid, hair cut and new clothes bought. Doors are decorated with vertical scrolls of characters on red paper whose texts seek good luck and praise nature, this practice stemming from the hanging of peach-wood charms to keep away ghosts and evil spirits. In many homes incense is burned, and also in the temples as a mark of respect to ancestors. In addition, symbolic flowers and fruits were used to decorate the house, and colorful new year pictures (NIAN HUA) were placed on the walls (for more descriptions of the symbolism of the flowers and fruits.Preparation for Christmas starts in early December when decorations are put up, including door wreaths, artificial trees and nativity scenes. A common tradition is that of decorating and blessing their Christmas tree. Often the Sunday before Christmas is set aside for this task. Decorations include colored lights (which replace an earlier era''s candles), balls (originally reflecting the candlelight in a dazzling way), tinsel (resembling the glittering icicles found on fir trees in colder lands), chrismons (wooden or foam symbols or monograms for Christ), and on top, a star. The roots of the use of trees and decorations are definitely in Europe''s pre-Christian religions, but the symbols were transformed by the early missionaries in order to express some aspect of Christ''s life. Sometimes, the meaning was much the same as the pagans treasured, but drawn through Christ. In other cases, the old meaning was deliberately turned inside-out to bring further honor to God and more cause for the people to celebrate. Christmas cards are sent out to family and friends. In many homes, Christmas cakes and puddings will be baked ready to eat on Christmas Day. In the cities and towns, many shopping centres and stores have their own ''Santa'' for children to meet.3. Activities of the HolidaysOn New Year’s Eve houses are brightly lit and a la rge family dinner is served. The New Year’s Eve and New Year’s Day celebrations were strickly family affairs. All members of the family would gather for the important family meal on the evening of the New year’s Eve. Even if a family member could not atten d, an empty seat would be kept to symbolize that person’s presence at the banquet. At midnight following the banquet, the younger members of the family would bow and pay their respects to their parents and elders. In the south China sticky-sweet glutinous rice pudding called nian gao is served, while in the north the steamed dumpling jiaozi is popular. Most celebrating the festival stay up till midnight, when fireworks are lit, to drive away evil spirits. New Years day is often spent visiting neighbours, family and friends. The children are given Red Lai-See Envelopes , good luck money wrapped in little red envelopes. On New Year’s day, everyone had on new clothes, and would put on his best behavior. It was considered improper to tell a lie, raise one’s voic e, use indecent language, or break anything on the first day of the year. Starting from the second day, people began going out to visit friends and relatives, taking with them gifts andLai-See for the children. Visitors would be greeted with traditional New year delicacies, such as melon seeds, flowers, fruits, tray of togetherness, and NIANGAO, New Year cakes.On Christmas Eve, most people who celebrate Christmas also participate I n special holiday rituals in their homes. Families often decorate evergreen Christmas trees and place colorfully wrapped presents beneath them. It is also a popular tradition that decorate the beauty of the starlit for trees with blown-glass ornaments, paper chains, candles and other decorations, which made the simple evergreen tree into a beautiful Christmas tree. On Christmas Day, there are many activities for observation. Exchanging gifts and sending Christmas cards to friends and relatives are the customs of celebrating the Christmas in the world. The card often depicts a family celebration and its caption reads, “A Merry Christmas and a Happy New Year to You.” The Santa Claus---from the Dutch Sinter Klaas---was depicted as a tall, dignified, religious figure riding a white horse through the air, delivered toys to all good children. Children often hang stockings, they awake in the morning to find the stocking filled with gifts from Santa Claus.III. The Differences &The Causes of the DifferencesSpring Festival, is formed based on the ancient calendar. According to solar terms and harvest, the date for certain agricultural activities has been fixed. Therefore, the celebration also has close ties with agriculture. Though religions such as Buddhism, Taoism and Confucianism play very important role in social life, they never match the far-reaching impact of agriculture on people’s tendency and thoughts. In China, agriculture is always placed in the first position. Through one year’s toil, people expect a joyful moment to have relaxation, after that they have to start a new round of hard working again. Up to now, in countryside, many antithetical couplets on the farmers’ doors read something like a prayer for a good harvest in the next year.As mentioned above, people take December 25th as “ Christ” birthday---Christmas Day. Therefore Christmas is a religious festival. Combined with our common sense,we can draw a brief conclusion that religion must play a dominant role in western society. As a matter of fact, the religion is a center of culture and society. People response their mi nd on religion, meanwhile religion controls people through people’s belief in God or spirit. It is not exaggerated to say that religion is ubiquitous in all walks of life. And just like the influence of agriculture in China, religion in west is connected with daily life and its impact is obviously to be witnessed in the other festivals, like Easter Day.Expect the above differences, Spring Festival reflects the feature of the strong localization, whether in its origin or in the way of observation or celebration. The origin is closely related to agriculture. Therefore, the way of observation or celebration is remarked by a lot of agricultural imprint. They are very unique local celebration way of letting off firecrackers, putting on the paper scrolls and having a get-together banquet. On the contrary, Christmas Day is a festival originated from Europe. For the historical reason, the European immigrants moved to America at the same time they brought a batch of European customs, traditions and festivals, including Christmas Day. Consequently, Christmas Day is world-wide, all westerners have this festival.The effect of history, which in broadest sense, is the totality of all past events. From the long history of China, we can see Chinese culture is deeply rooted in agriculture and the festivals are localized by the agricultural activities. It is definitely decided by the state situation and people’s living habits in the long history. Spring Festival has close relation with the agricultural activities such as planting and harvest. In fact, other festivals such as Ching Ming and Chong Yang festivals which have not been mentioned in this article are also settled in line with the solar terms and also connected with agriculture. Western culture is structured on the foundation of religion, thus the festivals have strong ties with Christianity. Apart from Christmas Day, the Easter Day is also connected with religion. The religion have an intensive impact on people’s life.IV. SimilaritySpring Festival and Christmas Day customs reflect the variety of cultures that celebrate the holiday. For some people, Spring Festival is a day related to agriculture, while Christmas Day is primarily a holy day marked by religious services, because of its root of Christianity. However, for most people, gift giving, feasting, and good times are far more prominently than their origins. Perhaps, celebrations in both culture backgrounds enjoy the same harmonious atmosphere of the family reunion. There is one common purpose for people’s activiti es, that is, to express their pleasure in hearts. V. ConclusionFestival is a wonderful part of daily life. During the festivity, the particular and colourful activities preserve the most delicate and representative aspect of national culture. Even though the forms of festivals are various form one another, they all stand for wisdom, experience or a treasure handed down from the ancient people. So we can find the feature of culture through the understanding of the festivals in this country or nation. Festivals belong to a certain nation, and also belong to all the humanbeing. Despite the different origins, backgrounds and social structures, humanbeing always have the same beautiful feeling in pursuit for love and happiness.Bibliography1. “ Festival of the World” Microsoft Encarta Encyclopedia99.1993-19982. 陈立浩，《春节民俗杂说》，《琼州大学学报》，vol8:no.1 （2001.3.28）3. 许树安，郑春苗，王秀芳，《中国文化知识》，北京语言出版社.19874. 秦桂英，陈泽春，《文化礼仪应用写作》，中国人民大学出版社.19945. c.uvic.ca/faculty/mroth/438/CHINA/chinese_new_year.html6. /chineseculture/festival/newyear/newyear.html7. /pages/culture/festivals.html8. /culture/spring/9./Angier/DimSum/china__dim_sum__spring_fes.html10. /christmas/11. /cathen/03724b.htm12. /wiki/Christmas_Day13. /Article_Show.asp?ArticleID=54814.http://202.95.18.169/education/modernchina/9813052_Christmas/Religion%20Fest ival.htm15. /christmasholida_rfju.htm16. /web/01bkswartz/xmaspub.html17. /artholiday/publish/article_52.shtml18/connie/christmas/world/world2.htm19./advent.html20. /christmaseve.html21. /catholic/customschristmas2.html22. http://www.lfcc.lt/publ/thelt/node17.html。

声母顺序的重要性英语作文

声母顺序的重要性英语作文The importance of consonant order in English is often overlooked. Many people focus on vowels and forget aboutthe significance of consonants. However, the order of consonants in a word can completely change its meaning. For example, the word "cat" becomes "act" with just a simple switch of the consonants.Consonant order also affects the pronunciation of words. The way consonants are arranged in a word can determine how the word is pronounced. For example, the word "strength"has a unique consonant order that affects its pronunciation, making it different from other words with similar letters.Furthermore, consonant order plays a crucial role in understanding and learning new words. When learning a new word, the order of consonants helps us to recognize and remember the word more easily. It provides a structure and pattern for our brains to process and store the information.In addition, consonant order is essential for language learners and non-native speakers. Understanding the orderof consonants helps them to improve their pronunciation and fluency in English. It enables them to differentiate between similar words and avoid misunderstandings in communication.Moreover, the order of consonants in English reflects the history and evolution of the language. It shows the influence of different languages and cultures on the development of English. By studying the order of consonants, linguists and language enthusiasts can gain insights into the rich and diverse history of the English language.In conclusion, the importance of consonant order in English should not be underestimated. It affects the meaning, pronunciation, learning, communication, and historical context of the language. By paying attention to the order of consonants, we can gain a deeper understanding and appreciation of the complexity and beauty of theEnglish language.。

说明顺序的手法的英语作文

说明顺序的手法的英语作文Sequential Writing: A Narrative Approach to Effective Communication.Sequential writing, also known as chronological writing, is a narrative technique that narrates events in a logical, time-based order. It allows writers to present informationin a coherent and sequential manner, guiding readersthrough a series of events or stages in a structured way. This technique is widely employed in various forms of writing, including historical accounts, biographies, memoirs, and certain types of scientific and technical writing.Characteristics of Sequential Writing:Linear Progression: Sequential writing follows alinear progression, with events presented in the order they occurred or unfolded.Chronological Structure: Time serves as the primary organizing principle, with events arranged in a chronological sequence.Clear Transitions: Smooth transitions connectdifferent events or stages, providing coherence and guiding readers through the narrative.Cause-and-Effect Relationships: Sequential writing often emphasizes cause-and-effect relationships, showing how events lead to subsequent outcomes.Focus on Details: Chronological writing often incorporates specific details and descriptions to provide a vivid and detailed account of events.Benefits of Sequential Writing:Enhances Comprehension: The logical and orderly presentation of information facilitates reader comprehension, making it easier for them to follow the sequence of events.Builds Suspense: In storytelling or historical accounts, sequential writing can create suspense bygradually revealing information and building tension.Establishes Context: By presenting events in chronological order, sequential writing helps readers understand the context and background for subsequent events.Encourages Critical Analysis: The clear structure and logical flow of sequential writing allows readers tocritically analyze the progression of events and draw conclusions.Provides a Sense of Progression: Sequential writing creates a sense of progression and movement, guidingreaders through the narrative and revealing the interconnectedness of events.Types of Sequential Writing:Historical Narratives: Chronological accounts ofhistorical events, often focusing on key moments, turning points, and significant figures in history.Biographical Sketches: Narratives that tell the life story of an individual, typically following the order of their birth, childhood, education, career, and accomplishments.Memoirs: Personal accounts written from the perspective of the author, narrating their own experiences, recollections, and reflections.Scientific Reports: Certain scientific and technical writing follows a sequential structure, presenting experimental procedures, results, and conclusions in a chronological order.Process Descriptions: Sequential writing can be used to describe a process, such as manufacturing procedures, scientific experiments, or instructions for completing a task.Writing Effective Sequential Narratives:Establish a Clear Timeline: Determine the starting and ending points of your narrative and establish a clear chronological framework.Identify Key Events: Select the key events or stages that are essential for understanding the progression of the story.Use Transitions: Employ effective transitions to smoothly connect different events or stages, ensuring the narrative flows coherently.Provide Supporting Details: Include specific details and descriptions to enhance the vividness and credibility of your account.Maintain Focus: Keep the narrative focused on the main events or stages, avoiding unnecessary digressions or irrelevant information.Sequential writing is a powerful tool for presenting information in a clear, logical, and engaging manner. By following the principles of chronological progression, cause-and-effect relationships, and clear transitions, writers can effectively guide readers through a series of events or stages, enhancing comprehension, building suspense, establishing context, and providing a sense of progression.。

二维激波与剪切层相互作用的直接数值模拟研究

二维激波与剪切层相互作用的直接数值模拟研究刘旭亮;张树海【期刊名称】《力学学报》【年(卷),期】2013(45)1【摘要】采用五阶weighed esseritially non-oscillatory (WENO)格式和三阶total variation diminishing (TVD) Runge-Kutta格式,通过求解二维非定常Navier-Stokes方程,直接数值模拟了激波与剪切层相互作用,目的在于揭示激波与剪切层相互作用过程中噪声产生的机理.研究发现:(1)当入射激波穿过剪切层时,剪切层中心位置向下层区域偏移；(2)入射激波穿过剪切层产生小激波,在小激波与剪切层接触点处产生声波并向外辐射；(3)反射激波穿过剪切层后形成了分段弧状激波；(4)当反射激波穿过剪切层时,激波在鞍点处泄漏并向外辐射声波,这是一种激波泄漏机制.%Direct numerical simulation (DNS) of the interaction of shock wave and shear layer was performed. The compressible unsteady two-dimensional Navier-Stokes equations were solved using the fifth-order WENO scheme combined with the third-order TVD Runge-Kutta scheme. The purpose of this paper is to reveal the mechanism of sound generation in the interaction of shock wave and shear layer. The results show that: (1) When incident shock wave is passing through the shear layer, the center of the vortex cores is shifted towards the lower side; (2) The interaction of incident shock wave and shear layer generates shocklet, and then acoustic wave is generated and radiated at the locus of contact of shocklet and shear layer; (3) Several arc-shocks are formed after reflected shock wavepassing through shear layer; (4) When reflected shock wave is passing through shear layer, shock wave is leaking in the braid region and shock-associated noise is generated at the saddle points between vortices. This is a form of shock leakage mechanism.【总页数】15页(P61-75)【作者】刘旭亮;张树海【作者单位】中国空气动力研究与发展中心,空气动力学国家重点实验室,四川绵阳621000;中国空气动力研究与发展中心,空气动力学国家重点实验室,四川绵阳621000【正文语种】中文【中图分类】V211.3【相关文献】1.二维湍流与弱激波相互干扰的数值模拟研究 [J], 刘洪伟;王健平2.高超声速激波湍流边界层干扰直接数值模拟研究 [J], 童福林;李欣;于长平;李新亮3.激波/边界层相互作用诱导的激波风洞气体污染问题 [J], 李进平;冯珩;姜宗林4.二维高超声速进气道内激波-边界层相互作用 [J], 黄舶;李祝飞;贾立超;杨基明;罗喜胜5.用STC格式求解二维激波─边界层相互作用问题 [J], 黄修乾;徐建中因版权原因，仅展示原文概要，查看原文内容请购买。

五分位数英语

五分位数英语Quintiles, also known as percentiles, are a statistical measure used to divide a dataset into five equal parts. This method of data analysis is particularly useful in a variety of fields, from healthcare to finance, as it provides valuable insights into the distribution and characteristics of a given set of data. By understanding the quintiles of a dataset, researchers and decision-makers can gain a better understanding of the underlying trends and patterns, allowing them to make more informed decisions.One of the primary applications of quintiles is in the healthcare industry. Healthcare providers often use quintiles to analyze patient data and identify at-risk populations. For example, a hospital may use quintile analysis to examine the distribution of patient ages within a specific treatment program. By dividing the patient population into five equal groups based on age, the hospital can identify the youngest and oldest quintiles, who may require specialized care or monitoring. This information can then be used to develop targeted interventions and improve patient outcomes.Similarly, in the field of finance, quintiles are used to analyze the performance of investment portfolios. Fund managers may use quintile analysis to compare the returns of their funds to the broader market, or to identify the best-performing and worst-performing investment sectors. By understanding the quintile distribution of their portfolio's returns, fund managers can make more informed decisions about asset allocation and risk management.Another important application of quintiles is in the field of education. Educators can use quintile analysis to assess student performance and identify areas for improvement. For instance, a school district may use quintiles to analyze the standardized test scores of its students, with the lowest quintile representing the students who are struggling the most and the highest quintile representing the top-performing students. This information can then be used to allocate resources and develop targeted interventions to support students in need.In the realm of social sciences, quintiles are often used to study income inequality and wealth distribution. Researchers may use quintile analysis to examine the distribution of income or wealth within a population, identifying the wealthiest and poorest quintiles. This information can then be used to inform policy decisions and develop strategies to address economic disparities.One of the key advantages of using quintiles is their ability to provide a more nuanced understanding of data distributions than other statistical measures, such as the mean or median. By dividing a dataset into five equal parts, quintiles can reveal information about the tails of the distribution, which can be particularly important in identifying outliers or extreme values. This information can be crucial in fields where understanding the full range of a dataset is essential, such as in risk management or public health.Moreover, quintiles are relatively easy to calculate and interpret, making them a popular tool for data analysis across a wide range of disciplines. The process of calculating quintiles involves sorting the data in ascending or descending order and then dividing the dataset into five equal groups. The resulting quintiles can then be used to compare the relative performance or characteristics of different subgroups within the dataset.Despite the many benefits of using quintiles, it is important to note that the interpretation of quintile data can be complex and may require a good understanding of statistical principles. Additionally, the choice of quintile cutoff points can have a significant impact on the results of the analysis, and researchers must be careful to select appropriate thresholds that align with the specific goals of their study.In conclusion, quintiles are a powerful tool for data analysis that can provide valuable insights across a wide range of fields. From healthcare to finance to education, the use of quintiles can help researchers and decision-makers better understand the underlying trends and patterns within their data, enabling them to make more informed and effective decisions. As the volume and complexity of data continue to grow, the importance of quintile analysis is likely to only increase, making it an essential skill for anyone working with data-driven decision-making.。

双曲守恒律方程的Lax-Wendroff时间离散WENO格式

双曲守恒律方程的Lax-Wendroff时间离散WENO格式李兴华;孙阳;艾晓辉【摘要】The research of high accuracy and high resolution schemes have been a hot topic in computational mathematics.According to low resolution and large amount of calculation of the original WENO-JS scheme , we propose a simple new limiter fifth order upwind WENO scheme to reconstruct the numerical flux of the simple structure to improve the computational efficiency .Compared with other efficient high accuracy schemes such as ENO and WENO, it is shown that the computational cost of this scheme is less than that of WENO -JS in the same accuracy .By use of MATLAB software , we compared and analyzed computational efficiencies and computational accuracies of Lax-Wendroff WENO-JS scheme , Lax-Wendroff simple limiter WENO scheme , Runge-Kutta simple limiter WENO scheme and Runge-Kutta WENO-JSscheme .The numerical results show that the new Lax-Wendroff simple limiter WENO scheme can improve the computing speed and reduce the computing time by 20% while maintaining the original WENO resolution .%双曲守恒型方程的高精度、高分辨率计算格式的研究一直是计算流体力学的热点问题.针对原WENO-JS格式分辨率较低和计算量偏大的不足问题,提出利用简单的重构数值通量的方法以提高计算效率,构造了新的简单限制器的5阶迎风型WENO格式.通过MATLAB软件的仿真对Lax-Wendroff WENO-JS格式、Lax-Wendroff简单限制器WENO格式、Runge-Kutta WENO-JS格式、Runge-Kutta简单限制器的WENO格式的实验结果进行了分析,并比较了这四种计算格式的计算效率和计算精度.数值实验表明:新格式Lax-Wendroff简单限制器WENO格式在保持原WENO分辨率的前提下,计算速度有明显提高,减少了20%的计算时间.【期刊名称】《哈尔滨理工大学学报》【年(卷),期】2017(022)006【总页数】6页(P134-139)【关键词】高精度;WENO;Runge-Kutta;Lax-Wendroff;时间离散【作者】李兴华;孙阳;艾晓辉【作者单位】哈尔滨理工大学理学院,黑龙江哈尔滨150080;哈尔滨理工大学理学院,黑龙江哈尔滨150080;东北林业大学理学院,黑龙江哈尔滨150040【正文语种】中文【中图分类】O175双曲守恒律方程(组)为科学理论和工程应用研究中一类非常重要的偏微分方程(组)。

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put.Phys.doi:10.4208/cicp.110212.021112a Vol.14,No.3,pp.599-620 September2013Comparison of Fifth-Order WENO Scheme andFinite Volume WENO-Gas-Kinetic Scheme forInviscid and Viscous Flow SimulationJun Luo,Lijun Xuan and Kun Xu∗Mathematics Department,Hong Kong University of Science and Technology,Clear Water Bay,Kowloon,Hong Kong.Received11February2012;Accepted(in revised version)2November2012Available online25January2013Abstract.The development of high-order schemes has been mostly concentrated onthe limiters and high-order reconstruction techniques.In this paper,the effect of theﬂux functions on the performance of high-order schemes will be studied.Based on thesame WENO reconstruction,two schemes with differentﬂux functions,i.e.,theﬁfth-order WENO method and the WENO-Gas-kinetic scheme(WENO-GKS),will be com-pared.Theﬁfth-orderﬁnite difference WENO-SW scheme is a characteristic variablereconstruction based method which uses the Steger-Warmingﬂux splitting for invis-cid terms,the sixth-order central difference for viscous terms,and three stages Runge-Kutta time stepping for the time integration.On the other hand,theﬁnite volumeWENO-GKS is a conservative variable reconstruction based method with the sameWENO reconstruction.But,it evaluates a time dependent gas distribution functionalong a cell interface,and updates theﬂow variables inside each control volume byintegrating theﬂux function along the boundary of the control volume in both spaceand time.In order to validate the robustness and accuracy of the schemes,both meth-ods are tested under a wide range ofﬂow conditions:vortex propagation,Mach3step problem,and the cavityﬂow at Reynolds number3200.Our study shows thatboth WENO-SW and WENO-GKS yield quantitatively similar results and agree witheach other very well provided a sufﬁcient grid resolution is used.With the reduc-tion of mesh points,the WENO-GKS behaves to have less numerical dissipation andpresent more accurate solutions than those from the WENO-SW in all test cases.Forthe Navier-Stokes equations,since the WENO-GKS couples inviscid and viscous termsin a singleﬂux evaluation,and the WENO-SW uses an operator splitting technique,itappears that the WENO-SW is more sensitive to the WENO reconstruction and bound-ary treatment.In terms of efﬁciency,theﬁnite volume WENO-GKS is about4timesslower than theﬁnite difference WENO-SW in two dimensional simulations.The cur-rent study clearly shows that besides high-order reconstruction,an accurate gas evolu-tion model orﬂux function in a high-order scheme is also important in the capturing of600J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620 physical solutions.In a physicalﬂow,the transport,stress deformation,heat conduc-tion,and viscous heating are all coupled in a single gas evolution process.Therefore,it is preferred to develop such a scheme with multi-dimensionality,and uniﬁed treat-ment of inviscid and dissipative terms.A high-order scheme does prefer a high-ordergas evolution model.Even with the rapid advances of high-order reconstruction tech-niques,theﬁrst-order dynamics of the Riemann solution becomes the bottleneck forthe further development of high-order schemes.In order to avoid the weakness of thelow orderﬂux function,the development of high-order schemes relies heavily on theweak solution of the original governing equations for the update of additional degreeof freedom,such as the non-conservative gradients ofﬂow variables,which cannot bephysically valid in discontinuous regions.PACS:02.60Cb,47.11.Df,47.45.AbKey words:WENO scheme,gas-kinetic scheme,Euler equations,Navier-Stokes equations,high-order methods.J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620601 smoothness-dependent weight to each stencil and results in a(2r−1)th order accuracy. The WENO reconstruction is more accurate,efﬁcient and stable than the ENO reconstruc-tion.So far,the WENO schemes have found wide applications.The general impression is that the WENO scheme is not sensitive to theﬂuxes used,such as Lax-Friedrichs or Steger-Warming.The full accuracy of the scheme mainly depends on the order of the re-construction.One of the purpose of the current paper is to test the effect ofﬂux functions on the performance of high-order schemes.It turns out that besides the high-order ini-tial reconstruction theﬂux modeling also plays an essential role to capture accurateﬂow evolution,and to reduce the sensitive dependence of the solution on the initial recon-struction,especially in a barely resolvedﬂow region.Allﬂux functions are equivalent in a well-resolvedﬂow region,because theﬂux consistency plays a dominant role here.But their performance may deviate from each other in not well-resolved cases,such as3or4 points inside a boundary layer.Currently,in order to increase the accuracy of the WENO scheme,many attempts have been tried to develop hybrid schemes,where the WENO is used in the discontinuous region and high-order compact scheme is used in the smooth region[13,18].The aim of the hybrid scheme is to basically develop a method which could make a smooth transition between the upwind and central difference method,be-cause the fundamental basis of the1st-order Riemann solver for wave decomposition and theﬂux splitting technique contradicts with theﬂow physics in the smooth region,where the high-order spatial and temporal evolution are fully coupled.In the smooth region,the traditional central difference approximation with Cauchy-Kowalevskaya technique is far more appropriate to describeﬂow evolution than upwind concept.The Riemann solu-tion is a low order dynamic model,which is needed to model discontinuousﬂow in order to introduce enough dissipation.And this amount of dissipation is closely related to the initial jump at the cell interface[26].Without modifying the1st-order Riemann dynamic model,the development of high-order schemes becomes a game of reconstruction,since the interface jump is the only freedom many high-order schemes are able to control.Un-fortunately,there is no principle to design such an optimal and universal interface jump, and this kind of research will be endless.In order to get out of this dilemma,the use of a high-order dynamic evolution model,which could make a smooth transition between the upwind and central difference scheme,is necessary.For a second order scheme for the inviscidﬂow,we have such a high-order dynamic model,which is the generalized Riemann solver and the gas-kinetic scheme[1,11].In the past decades,a gas-kinetic scheme(GKS)based on the kinetic equation has been developed for the modeling of gas evolution process starting from a discontinu-ity[12,16,26].Theoretically,the GKS does not target to solve accurately the gas kinetic BGK model[2],but uses the kinetic equation to do the modeling around a cell interface. In the GKS evolution,the whole process from particle free transport to the Navier-Stokes (NS)or Euler solution formation has been recovered[9,28].Theﬂow regime of the gas evolution depends on the ratio of time step to the particle collision time.In the smooth ﬂow region,based on the Chapman-Enskog expansion,a time dependent gas distribu-602J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620 tion function corresponding to the NS or Euler solutions can be obtained accurately by the GKS.This limiting formulation is basically the central difference method.In the dis-continuity region,where the physical solution cannot be well resolved by the numerical cell size,theoretically it is not necessary to know the precise”macroscopic”governing equations here,because it is not needed to incorporate a precise amount of numerical dissipation.But,in such a region the gas evolution model should follow a path which is consistent with the physical one,such as keeping a non-equilibrium gas distribution function at a cell interface to cope with the dissipative mechanism of a physical shock layer[11,29].This limiting case is theﬂux vector splitting upwind method.The advan-tage of the GKS is that theﬂux function makes a smooth transition from a upwind(kinetic scale)to a central difference(hydrodynamic scale)method.Recently,based on the WENO reconstruction and high-order gas-kinetic solution [12],aﬁnite volume high-order WENO-gas-kinetic scheme(WENO-GKS)has been de-signed and tested for multi-dimensionalﬂows[15].Different from the traditional WENO scheme,the WENO technique is only used in the data reconstruction of the conservative ﬂow variables at the beginning of each time step.Starting from the high-order recon-struction,a space and time dependent gas distribution function is obtained along the tangential direction of a cell interface,from which the numericalﬂux is evaluated and used in aﬁnite volume scheme.In GKS,there is no need to use the Runge-Kutta time stepping and Gaussian point integration along a cell interface for the update ofﬂow variables inside each control volume.Among all WENO schemes we tested,theﬁnite difference WENO method with Steger-Warmingﬂux splitting gives the best results.The detailed formulation of this WENO scheme is presented in Section3.In order to show the effect of aﬂux function on the per-formance of high-order schemes,the numerical solutions of theﬁnite difference WENO-Steger-Warming scheme(WENO-SW)will be compared with that of theﬁnite volume WENO-GKS.The test cases are carefully chosen in order to test the accuracy,the shock-capturing ability,the robustness,and the stability of the schemes.In order to eliminate the numerical error due to the complicated geometry,all tests are conducted in rectangu-lar meshes.The5th-order WENO reconstruction is used on the characteristic variables in the WENO-SW,and the same WENO reconstruction is used on the conservative vari-ables in the WENO-GKS.The reason we use conservative variables for the WENO-GKS is that the scheme is not sensitive to the variable used in the reconstruction,because the evolution of the whole reconstructed curves will be followed dynamically in theﬂux evaluation.2The5th order WENO reconstructionThe5th order WENO reconstruction on a uniform rectangular mesh is a standard recon-struction method[20].Assume that Q is the variable to be reconstructed.¯Q i is the cell averaged value inJ.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620603 the ith cell.Q l i and Q r i are the two pointwise values reconstructed at the left and right interfaces of the ith cell.The5th WENO reconstruction is deﬁned as,Q r i=2∑s=0w s q(s)i,Q l i=2∑s=0˜w s˜q(s)i,whereq(0)i=16¯Qi+1−16¯Qi−1+53¯Qi+1,q(2)i=16¯Qi−1+116¯Qi−73¯Qi+2,˜q(1)i=16¯Qi−16¯Qi−2+53¯Qi,w s=αs(ǫ+βs)2,˜w s=˜αs(ǫ+βs)2,s=0,1,2,β0=134(3¯Q i−4¯Q i+1+¯Q i+2)2,β1=134(¯Q i−1−¯Q i+1)2,β2=134(¯Q i−2−4¯Q i−1+3¯Q i)2,d0=˜d2=35,d2=˜d0=1604J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620 The BGK equation in2-D isf t+ u·∇f=g−f2(u2+v2+ξ2) T,dξ=dξ1dξ2···dξK,and K is the number of degrees of internal freedom,i.e.,K=(4−2γ)/(γ−1)for2-Dﬂow andγis the speciﬁc heat ratio.Since the mass,momentum, and energy are conserved during particle collisions,f and g satisfy the conservation con-straint, (g−f)ψαd u d v dξ=0,α=1,2,3,4,(3.3)at any point in space and time.The integral solution of(3.1)isf( x,t, u,ξ)=12g l,r0((a l,r1)2+d l,r11)x2+1J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620605Here g l0and g r0are two Maxwellian distribution functions which correspond to the left and right macroscopic variables respectively.For example,g l0corresponding to W l= (ρl,(ρl U l),(ρl V l),(ρl E l))T has the formg l0=ρl λl2eλl((u−U l)2+(v−V l)2+ξ2),(3.7)whereλl equals to m/2kT l,m is the molecular mass,k is the Boltzmann constant,and T l is the temperature.For the modeling of the local equilibrium distribution function g,we can use the Taylor expansion of the equilibrium state and getg( x,t, u,ξ)=¯g+¯g¯a1x+¯g¯a2y+¯g¯At+12¯g(¯a22+¯d22)y2+¯g(¯a1¯a2+¯d12)xy+1τ t0g(−ut′,y−vt′,t′, u,ξ)e−(t−t′)/τdt′=C1¯g+C2¯g¯a1u+C1¯g¯a2y+C2¯g¯a2v+C3¯g¯A+12C1¯g(¯a22+¯d22)y2+C2¯g(¯a22+¯d22)vy+12C5¯g(¯A2+¯B)+C6¯g(¯A¯a1+¯b1)u+C3¯g(¯A¯a2+¯b2)y+C6¯g(¯A¯a2+¯b2)v,(3.10) ande−t/τf0(−ut,y−vt, u,ξ)= C7f l0(−ut,y−vt, u,ξ),u>0,C7f r0(−ut,y−vt, u,ξ),u<0,(3.11) whereC1=1−e−t/τn,C2=(t+τ)e−t/τn−τ,C3=t−τ+τe−t/τn,(3.12a) C4=(−t2−2τt)e−t/τn,C5=t2−2τt,C6=−τt(1+e−t/τn),C7=e−t/τn.(3.12b) In any numerical simulation,numerical dissipation is necessary in order to cope with the limited cell resolution.In order to add numerical dissipation but not change theﬂow606J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620 property,in(3.12),we distinguish two particle collision times.One is the physical one (τ)which corresponds to the real collision time in the Chapman-Enskog expansion for the Navier-Stokes solution,and another one is the numerical one(τn)which takes into account the effect of initial pressure jump at the cell interface.As considered in[15],the BGK-NS solver uses√τ=µ/¯p,τn=µ/¯p+β∆x√¯λ+β∆xJ.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-620607 becomesW n+1ij =W n ij+12∆x i−1∆x∆y t n+1t n12∆y j[F i−1/2(t,y)−F i+1/2(t,y)]dydt,(3.16)where F j−1/2(t,x),F j+1/2(t,x),F i−1/2(t,y),F i+1/2(t,y)are theﬂuxes along the four cell in-terfaces respectively.Because the time-dependentﬂuxes can be explicitly integrated along a cell interface in the GKS,theﬁnal scheme presents a high-orderﬂow transport through the interface within a time step without using Runge-Kutta time stepping and ﬂux construction at Gauss points.3.2WENO-Steger-Warming schemeThe WENO-SW solves the hydrodynamic equationsW t+F(W)x+G(W)y=F v(W,W x,W y)x+G v(W,W x,W y)y,(3.17) where W is the conservative variables,F and G are the inviscidﬂuxes,and F v and G v are the viscousﬂuxes.For aﬁnite-difference scheme,we need to construct both inviscid and viscousﬂuxes at the cell interface.The following WENO scheme is the5th orderﬁnite-difference WENO-Steger-Warming scheme,where the Steger-Warming splitting[23]is used to obtain the inviscidﬂuxes at the cell interface.3.2.1Inviscidﬂux reconstructionThe x-directionﬂux F can be decomposed asF=RΛLW,(3.18) whereΛ=diag[λ1,λ2,···,λn]is a diagonal matrix with eigenvalues of∂F/∂W,R is the right eigenvector matrix and L is the left one,n is the number of the equations.Then,F can be split according toF=F++F−.(3.19) In the Steger-Warming splitting,F±=RΛ±LW,Λ=Λ++Λ−.(3.20) andΛ±=diag[λ±1,λ±2,···,λ±n]withλ±i=λi± 2(i=1,···,n),(3.21)whereεis a small constant.In our simulation,ε=1.0e−3.For the construction of the numerical x-directionﬂuxˆF i+1/2,j at the cell interface (x i+1/2,y j)of the cell(i,j),we use the following steps.608J.Luo,L.Xuan and K.Xu/put.Phys.,14(2013),pp.599-6201.Split theﬂuxes F i+l,j(l=−2,···,3)in the surrounding cells to get F±i+l,j(l=−2,···,3).e the variable W i+1/2,j=(W i,j+W i+1,j)/2to calculate the left and right eigenvector matrixL i+1/2,j and R i+1/2,j at the cell interface.3.Change the conservativeﬂuxes F±i+l,j(l=−2,...,3)to the characteristicﬂuxes˜F±i+l,j(l=−2, (3)by˜F±i+l,j=L i+l/2,j F±i+l,j(l=−2,···,3).e the5th order WENO reconstruction(see Section2)on the characteristicﬂuxes˜F±i+l,j(l=−2,···,3)to get the characteristicﬂux˜F±i+1/2,j,and calculate theﬁnal characteristicﬂux˜Fi+1/2,j =˜F+i+1/2,j+˜F−i+1/2,j.5.Construct theﬂuxes for the conservative variables at the cell interface byˆFi+1/2,j=R i+1/2,j˜F i+1/2,j.In a similar way,we can get the y-directional numericalﬂuxˆG i,j+1/2at the cell inter-face(x i,y j+1/2).3.2.2Viscousﬂux reconstructionThe viscousﬂuxes F v and G v are related to the derivatives ofﬂow variables.For the NS equations,we need to calculate T x,U x,U y,V x and V y in order to get the x-direction viscousﬂux F v,where T is the temperature,U is the x-direction velocity,and V is the y-direction velocity.So,the basic step is to construct the derivatives of a variable q.Firstly,in each cell(i,j),we use a6th-order central difference to calculate the deriva-tive,i.e., ∂q60∆x.(3.22)Then,with all the derivatives,we can calculate the x-directional and y-directional viscous ﬂuxes F v i,j and G v i,j in each cell.Finally,the numerical viscousﬂuxes at the cell interfaces can be calculated by the6th order central interpolationˆF vi+1/2,j =37(F v i,j+F v i+1,j)−8(F v i−1,j+F v i+2,j)+F v i−2,j+F v i+3,j60.(3.23b)3.2.3Time evolutionTheﬁnal numericalﬂuxes at the cell interface are denoted byˆF i+1/2,j=ˆF i+1/2,j−ˆF v i+1/2,j,(3.24a)ˆG i,j+1/2=ˆG i,j+1/2−ˆG v i,j+1/2.(3.24b)With the above numericalﬂuxes,we can get the increment∆W i,j=ˆF i+1/2,j−ˆF i−1/2,j∆y.(3.25)A3rd-order TVD Runge-Kutta method is used toﬁnish the time evolution,W(1)i,j=W n i,j−∆t∆W i,j(W n),(3.26a)W(2)i,j =34[W(1)i,j−∆t∆W i,j(W(1))],(3.26b)W(n+1)i,j =13[W(2)i,j−∆t∆W i,j(W(2))].(3.26c)4Numerical examplesFor all tests in this paper,the time step is determined by the CFL condition with CFL number0.5.4.1Mach3step problemThe Mach3step problem wasﬁrst proposed by Woodward and Colella in[25].The com-putational domain is[0,3]×[0,1].A step with height0.2is located at x=0.6.The upstream velocity is(U,V)=(3,0).The adiabatic slip Euler boundary condition is implemented at all boundaries.As explained in[25],the corner of the step is the center of a rarefaction fan and itis a singular point of theﬂow.Theﬂow will be much affected by the numerical errorsgenerated just in the neighborhood of this singular point.Almost for all WENO schemes,in order to get better results,a special treatment introduced in[25]has been applied.Or,a reﬁned mesh is used around the corner.In order to avoid confusion and compare theresults of the WENO-GKS and the WENO-SW truthfully,there is no special treatmentanywhere in the current study.The parameterǫ=10−6in the5th order WENO recon-struction is used.In the WENO-GKS,α=0andβ=100for numerical collision time in (3.14).Figs.1,2and3present the numerical solutions from both WENO-SW and WENO-GKS with different number of mesh points,i.e.,120×40,60×20,and30×10.All these results are consistent with theﬂow structures in[25].Based on the theseﬁgures,the WENO-SW gives a large tilted Mach stem above the up surface of the step,and the WENO-GKS presents a much straighter and shorter one.After a few shock reﬂections, the shock front of the WENO-SW gets smeared and disappeared in the case with30×10 mesh points case.But,for the WENO-GKS,the shock reﬂection can be observed clearly in the coarse mesh case.This illustrates that WENO-SW has large numerical dissipation in comparison with WENO-GKS.In order to understand why the WENO-SW has a tiltedFigure1:Mach3the WENO-GKS5.(upper one)and the WENO-SW(lower one).In eachﬁgure,there are50contours from0.5toFigure2:Mach3the WENO-GKS (upper one)and the WENO-SW(lower one).In eachﬁgure,there are50contours from0.5to5.Mach stem,the WENO-GKS is tested for the NS solution with Reynolds number1000, but with the Euler boundary condition.Fig.4shows the comparison of WENO-SW for the inviscidﬂow and WENO-GKS for the viscousﬂow.It shows that the solution given by the WENO-SW is more close to a viscous solution.This indicates again that a high level of numerical dissipation exists in the WENO-SW.Figure3:Mach3the WENO-GKS (upper one)and the WENO-SW(lower one).In eachﬁgure,there are50contours from0.5to5.Figure4:Mach3Upper one:NS solution with Reynolds number1000by the WENO-GKS.Lower one:Euler solution by the WENO-SW.In each ﬁgure,there are50contours from0.5to5.4.2Isentropic periodic vortex propagationThis is a test for the accuracy of the Euler solutions(see[13,20]).The initial condition is given byκ(U(x,x,0),V(x,y,0))=(1,1)+e1−r2,S(x,y,0)=1,8γπ2Figure5:Isentropic periodic vortex propagation:density distribution at t=10(1period)by the WENO-GKSFigure6:Isentropic periodic vortex propagation:density distribution at t=100(10periods)by the WENO-GKS (WGKS)and the WENO-SW(WSW).where the temperature T and the entropy S are related to the densityρand the pressurep byT=pργ,and(¯x,¯y)=(x−5,y−5),r2=¯x2+¯y2,and the vortex strengthκ=5.The computational domain is[0,10]×[0,10].The periodic boundary condition is used in both directions.The numerical results with80×80and40×40cells at t=10(1period)and t=100 (10periods)are shown in Figs.5and6for both WENO-SW and WENO-GKS.In order to show the relationship between the parameterǫin the WENO reconstruction and the accuracy of the numerical solution,we test this case with two differentǫ,which are EPS1 (ǫ=10−6)and EPS2(ǫ=10−2).Theoretically,a large value ofǫpresents a reconstruction with more equally weights for different stencils.For a smoothﬂow,a smallǫmakes thereconstruction to reduce to a3rd order interpretation.A largerǫbalances the weights of different stencils and make the reconstruction to be close to the5th order interpolation. As shown in Figs.5and6,for such a smoothﬂow,in general,the reconstruction with EPS2will give more accurate and less dissipative results than that from EPS1for both WENO-SW and WENO-GKS.At the same time,for both meshes the results from WENO-GKS are more accurate than these from WENO-SW,especially at a mesh size40×40and t=100,see Fig.6.At time t=10and40×40mesh points,see Fig.5,EPS1introduces numerical dissipation and presents undershoot for both WENO-SW and WENO-GKS. However,with EPS2,the WENO reconstruction introduces overshoot for both schemes. But,in both cases due to the time accurate gas evolution model in the gas-kinetic scheme, the WENO-GKS could increase the undershoot and decrease the overshoot in compari-son with WENO-SW.In other words,the WENO-GKS has a better capacity to drive the solution in the correct direction due to the participation of the subcellﬂow distribution in the construction of the dynamicalﬂux function.More speciﬁcally,the gas-kineticﬂux function does not only depend on theﬂow variables at the cell interface,but also takes into account the whole curve evolution around the interface.Based on the above test,we can clearly realize that even though the gas-kinetic scheme is not solving the inviscid Euler equations directly,it gives accurate inviscid solution. In general,the numerical dissipation of the gas-kinetic scheme is less than the schemes based on the Riemann solvers,since the GKS can make a smooth transition from the up-wind to the central difference.This is one of the reason for the reduction of numerical dissipation in GKS in the smoothﬂow region.4.3CavityﬂowThe cavityﬂow at low Mach number is a standard test case for validating incompress-ible or low speed NSﬂow solvers.This is also a good test case for the shock capturing scheme in validating its capacity in capturing the Navier-Stokes solutions,especially with the non-linear limiters involved.For a directional splittingﬁnite volume scheme,it will be difﬁcult to present an accurate viscous solution with strong vortex structure.Fortu-nately,theﬁnite difference WENO-SW is an intrinsic multi-dimensional scheme due to its simultaneous evaluation of∂x and∂y terms of the NS equations at a grid point.At the same time,theﬁnite volume WENO-GKS has a multi-dimensionalﬂux function as well, where both x-and y-directionﬂow derivatives contributes to theﬂux in the interface normal direction.Theﬂow is bounded by a unit square and is driven by a uniform translation of the top boundary.In the simulation,the diatomic gasγ=1.4is considered.The boundary is isothermal and nonslip,and theﬂow condition is Ma=0.3and Re=3200,where Ma is the Mach number and Re is the Reynolds number.Since the benchmark solution is from incompressible NS equations,in order to avoid kinematic dissipation[6,27],most simu-lations in the past are based on the numerical methods for the incompressible equations or the artiﬁcial compressibility ones,where a continuous initial reconstruction across acell interface is assumed.However,here we are going to use the same shock capturing WENO reconstruction in the cavity simulation.This is challenge for any shock captur-ing NSﬂow solver,because the cell interface discontinuity may generate large numerical dissipation.In order to get the best results for the WENO-SW,we tested many boundary condi-tion treatments andﬁnally choose the following reconstruction.The special treatment is the following.The temperature and velocities are given directly since the boundary is isothermal and nonslip.Other data at the boundary are extrapolated from theﬂow.For the WENO-SW,the three layer interfaces close to the cavity wall but inside the compu-tational domain are specially treated.For theﬁrst layer cell interface,a3rd-order extrap-olation is used to reconstruct the conservative variableﬂuxes.For the second interface, a4th-order interpolation is used to reconstruct the conservative variableﬂuxes.For the third interface,a5th-order upwind interpolation is used to reconstruct the characteristic ﬂuxes.We use a4th-order interpolation to reconstruct the viscousﬂuxes at all of the three interfaces.For the WENO-GKS,the boundary treatment is relatively simple.A5th-order extrapolation for the conservative variables is used at the boundary.Firstly,we compare the efﬁciency in the cavity simulation for both schemes.The results are shown in Table1.As shown in the table,the WENO-GKS is about4times slower than the WENO-SW.This shows that the speed of WENO-GKS is similar to aﬁnite volume WENO scheme.As shown in[21],the computational cost of theﬁnite volume WENO scheme is indeed at least4times more expensive than theﬁnite difference one in a two-dimensional simulation.Table1:Average computational time for one time-step.WENO-GKS33×33cells 4.3276e-0037.4369e-002For the cavityﬂow simulations,we use three sets of meshes,which are101×101, 65×65,and33×33for both schemes.Fig.7shows the distributions of streamline for both schemes with a mesh of65×65points.Except around the up left corner,the streamlines from both WENO-SW and WENO-GKS are close to each other.The results of U-velocity along the vertical symmetric line at x=0.5and V-velocity along the horizontal symmetric line at y=0.5with different mesh sizes will be presented in detail.Fig.8shows U and V velocity distributions for a mesh with101×101points. The benchmark solution is from[5,24].As shown in thisﬁgure,the results from both WENO-SW and WENO-GKS are consistent with the reference solutions.The reconstruc-tion with EPS1presenting a better shock capturing property turns out to be more dissipa-tive than those with EPS2.This is clearly observed in the V-velocity for the WENO-SW. The WENO-GKS solutions are less sensitive to the values ofǫ.At such a reﬁned mesh, it is hard to distinguish the two solutions from WENO-GKS.The overall solutions from WENO-GKS are closer to the benchmark ones than the WENO-SW results.。