Magneto-driven Shock Waves in Core-Collapse Supernova

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。

磁概念永磁材料：永磁材料被外加磁场磁化后磁性不消失，可对外部空间提供稳定磁场。

钕铁硼永磁体常用的衡量指标有以下四种：剩磁（Br）单位为特斯拉（T）和高斯（Gs） 1Gs =0.0001T将一个磁体在闭路环境下被外磁场充磁到技术饱和后撤消外磁场，此时磁体表现的磁感应强度我们称之为剩磁。

它表示磁体所能提供的最大的磁通值。

从退磁曲线上可见，它对应于气隙为零时的情况，故在实际磁路中磁体的磁感应强度都小于剩磁。

钕铁硼是现今发现的Br最高的实用永磁材料。

磁感矫顽力（Hcb）单位是安/米（A/m）和奥斯特（Oe）或1 Oe≈79.6A/m处于技术饱和磁化后的磁体在被反向充磁时，使磁感应强度降为零所需反向磁场强度的值称之为磁感矫顽力（Hcb）。

但此时磁体的磁化强度并不为零，只是所加的反向磁场与磁体的磁化强度作用相互抵消。

（对外磁感应强度表现为零）此时若撤消外磁场，磁体仍具有一定的磁性能。

钕铁硼的矫顽力一般是11000Oe以上。

内禀矫顽力（Hcj）单位是安/米（A/m）和奥斯特（Oe）1 Oe≈79.6A/m使磁体的磁化强度降为零所需施加的反向磁场强度，我们称之为内禀矫顽力。

内禀矫顽力是衡量磁体抗退磁能力的一个物理量，如果外加的磁场等于磁体的内禀矫顽力，磁体的磁性将会基本消除。

钕铁硼的Hcj会随着温度的升高而降低所以需要工作在高温环境下时应该选择高Hcj的牌号。

磁能积(BH)单位为焦/米3（J/m3）或高•奥（GOe） 1 MGOe≈7. 96k J/m3退磁曲线上任何一点的B和H的乘积既BH我们称为磁能积,而B×H的最大值称之为最大磁能积(BH)max。

磁能积是恒量磁体所储存能量大小的重要参数之一，(BH)max越大说明磁体蕴含的磁能量越大。

设计磁路时要尽可能使磁体的工作点处在最大磁能积所对应的B和H附近。

各向同性磁体：任何方向磁性能都相同的磁体。

各向异性磁体：不同方向上磁性能会有不同；且存在一个方向，在该方向取向时所得磁性能最高的磁体。

珊瑚礁地形上波浪传播变形的非静压数值模拟张其一;史宏达;高伟;李金峰【摘要】珊瑚礁广泛分布于我国南海等海域,具有重要的生态、旅游、港口和军事价值.波浪是珊瑚礁环境最重要的动力因素之一.基于非静压浅水方程波浪模型SWASH,模拟了典型岸礁地形上的波浪传播过程,并采用Demirbilek等的水槽实验数据分别对有效波高、波浪增水与爬高进行了验证,模拟结果与实测数据符合良好,展示了非静压模型在珊瑚礁地形上良好的适用性.通过波浪频率谱和波面小波谱,分析了波浪谱在该珊瑚礁地形上的演化规律,并揭示了礁坪上波浪能量从高频向低频转移的现象.本研究结果可为珊瑚礁环境动力研究以及相关的工程建设提供有益的参考依据.【期刊名称】《海岸工程》【年(卷),期】2017(036)004【总页数】9页(P1-9)【关键词】珊瑚礁;波浪传播;非静压模型;SWASH;小波分析【作者】张其一;史宏达;高伟;李金峰【作者单位】中国海洋大学工程学院 ,山东青岛266100;中国海洋大学工程学院 ,山东青岛266100;中交天津航道局有限公司 ,天津300450;中交天津航道局有限公司 ,天津300450【正文语种】中文【中图分类】P731.22珊瑚礁作为一种典型的海洋地貌形态,在深海和浅海区域均有分布。

根据其形式一般可以分为环礁、岸礁和堡礁。

珊瑚礁地形广泛分布于我国的台湾岛附近及南海海域,具有重要的港口、航运、生态、旅游、科研和军事价值。

近年来,随着国家对南海岛礁开发力度的提升,珊瑚礁动力环境的研究也逐步加深。

然而,由于珊瑚礁地形地貌特殊而复杂,其动力特点和形成机制尚待系统、深入地探究。

作为珊瑚礁环境的最主要动力,波浪在珊瑚礁地形上的传播和变形得到了国内外学者广泛的关注。

Gourlay[1]较早开展了规则波在珊瑚礁地形上传播变形的物理模型实验,并提出了非线性系数F co=来描述相对波高的变化规律:当非线性系数F co越小时,相对波高也越小。

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Advanced Optical MaterialsIntroductionAdvanced optical materials are a class of materials that possess unique optical properties and are engineered to enhance light-matter interactions. These materials have revolutionized various fields such as photonics, optoelectronics, and nanotechnology. In this article, we will explore the different types of advanced optical materials, their applications, and the future prospects of this exciting field.Types of Advanced Optical MaterialsPhotonic CrystalsPhotonic crystals are periodic structures that can manipulate the propagation of light. They consist of a periodic arrangement ofdielectric or metallic components with alternating refractive indices. These structures can control the flow of light by creating energy bandgaps, which prohibit certain wavelengths from propagating through the material. Photonic crystals find applications in optical communication, sensing, and solar cells.MetamaterialsMetamaterials are artificially engineered materials that exhibit properties not found in nature. They are composed of subwavelength-sized building blocks arranged in a periodic or random manner. Metamaterials can manipulate electromagnetic waves by achieving negative refractive index, perfect absorption, and cloaking effects. These unique properties have led to applications in invisibility cloaks, super lenses, and efficient light harvesting.Plasmonic MaterialsPlasmonic materials exploit the interaction between light and free electrons at metal-dielectric interfaces to confine light at nanoscale dimensions. This confinement results in enhanced electromagnetic fields known as surface plasmon resonances. Plasmonic materials have diverse applications such as biosensing, photothermal therapy, and enhanced solar cells.Quantum DotsQuantum dots are nanoscale semiconductor crystals with unique optical properties due to quantum confinement effects. Their size-tunable bandgap enables them to emit different colors of light depending ontheir size. Quantum dots find applications in display technologies (e.g., QLED TVs), biological imaging, and photovoltaics.Organic Optoelectronic MaterialsOrganic optoelectronic materials are based on organic compounds that exhibit electrical conductivity and optical properties. These materials are lightweight, flexible, and can be processed at low cost. They find applications in organic light-emitting diodes (OLEDs), organic photovoltaics (OPVs), and organic field-effect transistors (OFETs).Applications of Advanced Optical MaterialsInformation TechnologyAdvanced optical materials play a crucial role in information technology. Photonic crystals enable the miniaturization of optical devices, leading to faster and more efficient data transmission. Metamaterials offer possibilities for creating ultra-compact photonic integrated circuits. Plasmonic materials enable the development of high-density data storage devices.Energy HarvestingAdvanced optical materials have revolutionized energy harvesting technologies. Quantum dots and organic optoelectronic materials are used in next-generation solar cells to enhance light absorption and efficiency. Plasmonic nanoparticles can concentrate light in solar cells, increasing their power output. These advancements contribute to the development of sustainable energy sources.Sensing and ImagingThe unique optical properties of advanced optical materials make them ideal for sensing and imaging applications. Quantum dots are used as fluorescent probes in biological imaging due to their bright emissionand excellent photostability. Metamaterial-based sensors offer high sensitivity for detecting minute changes in refractive index ormolecular interactions.Biomedical ApplicationsAdvanced optical materials have significant implications in biomedical research and healthcare. Plasmonic nanomaterials enable targeted drug delivery, photothermal therapy, and bioimaging with high spatial resolution. Organic optoelectronic materials find applications in wearable biosensors, smart bandages, and flexible medical devices.Future ProspectsThe field of advanced optical materials is rapidly evolving with continuous advancements being made in material synthesis, characterization techniques, and device fabrication processes. Thefuture prospects of this field are promising, with potential breakthroughs in areas such as:1.Quantum Optics: Integration of advanced optical materials withquantum technologies could lead to the development of quantumcomputers, secure communication networks, and ultra-precisesensors.2.Flexible and Wearable Electronics: Organic optoelectronicmaterials offer the potential for flexible and wearable electronic devices, such as flexible displays, electronic textiles, andimplantable medical devices.3.Optical Computing: Photonic crystals and metamaterials may pavethe way for all-optical computing, where photons replace electrons for faster and more energy-efficient data processing.4.Enhanced Optoelectronic Devices: Continued research on advancedoptical materials will lead to improved performance and efficiency of optoelectronic devices such as solar cells, LEDs, lasers, andphotodetectors.In conclusion, advanced optical materials have opened up newpossibilities in various fields by enabling unprecedented control over light-matter interactions. The ongoing research and development in this field promise exciting advancements in information technology, energy harvesting, sensing and imaging, as well as biomedical applications. The future looks bright for advanced optical materials as they continue to revolutionize technology and shape our world.。