An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws

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1 Avalable onlne at Journal of Computatonal Physcs 7 (8) 9 An mproved weghted essentally non-oscllatory scheme for hyperbolc conservaton laws Rafael Borges a, Monque Carmona a, Bruno Costa a, *, Wa Sun Don b a Departamento de Matemátca Aplcada, IM-UFRJ, Caa Postal 685, Ro de Janero, RJ, CEP , Brazl b Dvson of Appled Mathematcs, Brown Unversty, Provdence, Rhode Island 9, Unted States Receved September 7; receved n revsed form 5 November 7; accepted November 7 Avalable onlne 8 December 7 Abstract In ths artcle we develop an mproved verson of the classcal ffth-order weghted essentally non-oscllatory fnte dfference scheme of [G.S. Jang, C.W. Shu, Effcent mplementaton of weghted ENO schemes, J. Comput. Phys. 6 (996) 8] (WENO-JS) for hyperbolc conservaton laws. Through the novel use of a lnear combnaton of the low order smoothness ndcators already present n the framework of WENO-JS, a new smoothness ndcator of hgher order s devsed and new non-oscllatory weghts are bult, provdng a new WENO scheme () wth less dsspaton and hgher resoluton than the classcal WENO. Ths new scheme generates solutons that are sharp as the ones of the mapped WENO scheme () of Henrck et al. [A.K. Henrck, T.D. Aslam, J.M. Powers, Mapped weghted essentally non-oscllatory schemes: achevng optmal order near crtcal ponts, J. Comput. Phys. 7 (5) ], however wth a 5% reducton n CPU costs, snce no mappng s necessary. We also provde a detaled analyss of the convergence of the scheme at crtcal ponts of smooth solutons and show that the soluton enhancements of and at problems wth shocks comes from ther ablty to assgn substantally larger weghts to dscontnuous stencls than the WENO-JS scheme, not from ther superor order of convergence at crtcal ponts. Numercal solutons of the lnear advecton of dscontnuous functons and nonlnear hyperbolc conservaton laws as the one dmensonal Euler equatons wth Remann ntal value problems, the Mach shock densty wave nteracton and the blastwave problems are compared wth the ones generated by the WENO-JS and schemes. The good performance of the scheme s also demonstrated n the smulaton of two dmensonal problems as the shock vorte nteracton and a Mach 4.46 Rchtmyer Meshkov Instablty (RMI) modeled va the two dmensonal Euler equatons. Ó 7 Elsever Inc. All rghts reserved. MSC: 65P; 77A Keywords: Weghted essentally non-oscllatory; Hyperbolc conservaton laws; Smoothness ndcators; WENO weghts * Correspondng author. Tel.: E-mal addresses: rborges@ufrj.br (R. Borges), carmona@ufrj.br (M. Carmona), bcosta@ufrj.br (B. Costa), wsdon@dam.brown.edu (W.S. Don). -999/$ - see front matter Ó 7 Elsever Inc. All rghts reserved. do:.6/j.jcp.7..8

2 9 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9. Introducton In the numercal smulaton of compressble flows modeled by means of hyperbolc conservaton laws n the form ou þ $ FðuÞ ¼; ot ðþ the development of fnte tme dscontnutes generates O() oscllatons, known as the Gbbs phenomenon, causng loss of accuracy and numercal nstablty. Among many choces of shock capturng schemes such as the pecewse parabolc method (PPM) [7], the essentally non-oscllatory scheme (ENO) [6], hgh-order weghted essentally non-oscllatory schemes (WENO) [,] have been etensvely used for the smulaton of the fne scale and delcate structures of the physcal phenomena related to shock turbulence nteractons. WENO schemes owe ther success to the use of a dynamc set of stencls, where a nonlnear conve combnaton of lower order polynomals adapts ether to a hgher order appromaton at smooth parts of the soluton, or to an upwnd spatal dscretzaton that avods nterpolaton across dscontnutes and provdes the necessary dsspaton for shock capturng. The nonlnear coeffcents of the conve combnaton, hereafter referred to as weghts, are based on the local smoothness ndcators, whch measure the sum of the normalzed squares of the scaled L norms of all dervatves of the lower order polynomals []. An essentally zero weght s assgned to those lower order polynomals whose underlnng stencls contan hgh gradents and/or shocks, yeldng an essentally non-oscllatory soluton at dscontnutes. At smooth parts of the soluton, hgher order s acheved through the mmckng of the central upwndng scheme of mamum order, when all smoothness ndcators are about the same sze. Hence, an effcent desgn of these smoothness ndcators s a very mportant ssue for WENO schemes. The classcal choce of smoothness ndcators n [] generated weghts that faled to recover the mamum order of the scheme at ponts of the soluton where the frst or hgher dervatves of the flu functon vansh. Ths fact was clearly ponted out at Henrck et al. []. In ther study, necessary and suffcent condtons on the weghts, for optmalty of the order, were derved and a correctng mappng to be appled to the classcal weghts was devsed. The resultng mapped WENO scheme of [] recovered the optmal order of convergence at crtcal ponts of a smooth functon and presented sharper results close to dscontnutes. In ths artcle, we follow a dfferent approach, whch s to mprove on the classcal smoothness ndcators to obtan weghts that satsfes the suffcent condtons for optmal order. Taylor seres analyss of the classcal smoothness ndcators reveals that a smple combnaton of them would gve hgher order nformaton about the regularty of the numercal soluton. The ncorporaton of ths new hgher order nformaton nto the weghts defnton mproves the convergence order at the crtcal ponts of smooth parts of the soluton, as well as decreases the dsspaton close to dscontnutes, whle mantanng stablty and the essentally non-oscllatory behavor. The enhancements of the new scheme come from the larger weghts that t assgns to dscontnuous stencls. Contrary to common belef, the strategy should be to augment the nfluence of the stencl contanng the dscontnuty as much as possble, wthout destroyng the essentally non-oscllatory behavor. A comparson of the weghts of the classcal, the mapped and the new WENO scheme close to a dscontnuty shows that the rato between the weght of a dscontnuous stencl and a contnuous one ncreases slghtly from the classcal weghts to the mapped weghts, and ncreases substantally wth the new weghts proposed n ths artcle. The computatonal cost of the new scheme s the same as the WENO-JS and around 75% of. Ths paper s organzed as follows: n Secton, the classcal WENO scheme of Jang and Shu [] and the mapped weghts verson of Henrck et al. [] are descrbed and some of the relevant analytcal results are revewed. The new WENO scheme s ntroduced n Secton where a detaled dscusson about the new smoothness ndcator s gven. In Secton 4, we analyze the order of convergence of at crtcal ponts of a smooth soluton. In Secton 5, all three schemes are compared wth the numercal smulaton of the lnear advecton of dscontnuous functons, the one dmensonal Euler equatons wth Remann ntal values problems (SOD, LAX and ), the Mach shock densty wave nteracton and the nteractve blastwaves problems. We end the artcle showng results on the two dmensonal Mach shock vorte nteracton and the Mach 4.46 Rchtmyer Meshkov Instablty (RMI) along wth the CPU tmng of the three WENO schemes. Concludng remarks are gven n Secton 6.

3 . Weghted essentally non-oscllatory schemes In ths secton, we brefly descrbe the ffth-order weghted essentally non-oscllatory conservatve fnte dfference scheme when appled to hyperbolc conservaton laws as n (). Wthout loss of generalty, we wll restrct our dscusson to the one dmensonal scalar case. Etensons to system of equatons and hgher spatal dmensons present no etra complety wth regards to our man pont whch s the desgn of new weghts for the WENO scheme. Consder an unform grd defned by the ponts ¼ D; ¼ ;...; N, whch are also called cell centers, wth cell boundares gven by þ ¼ þ D, where D s the unform grd spacng. The sem-dscretzed form of (), by the method of lnes, yelds a system of ordnary dfferental equatons du ðtþ ¼ of dt o ; ¼ ;...; N; ðþ ¼ where u ðtþ s a numercal appromaton to the pont value uð ; tþ. A conservatve fnte dfference formulaton for hyperbolc conservaton laws requres hgh-order consstent numercal flues at the cell boundares n order to form the flu dfferences across the unformly spaced cells. The conservatve property of the spatal dscretzaton s obtaned by mplctly defnng the numercal flu functon hðþ as Z þ D R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 9 f ðþ ¼ hðnþdn; D D such that the spatal dervatve n () s eactly appromated by a conservatve fnte dfference formula at the cell boundares, du ðtþ ¼ dt D h þ h ; ðþ where h ¼ hð Þ. Hgh-order polynomal nterpolatons to h are computed usng known grd values of f, f ¼ f ð Þ. The classcal ffth-order WENO scheme uses a 5-ponts stencl, hereafter named S 5, whch s subdvded nto three -ponts stencls fs ; S ; S g, as shown n Fg.. The ffth-order polynomal appromaton ^f ¼ h þ OðD 5 Þ s bult through the conve combnaton of the nterpolated values ^f k ð Þ, n whch f k ðþ s the thrd degree polynomal below, defned n each one of the stencls S k : ^f ¼ X k^f k ð Þ; ð4þ - - +/ + + S 5 τ 5 S β S β S β Fg.. The computatonal unform grd and the 5-ponts stencl S 5, composed of three -ponts stencls S ; S ; S, used for the ffth-order WENO reconstructon step.

4 94 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 where ^f k ð þ Þ¼^f k þ ¼ X j¼ c kj f kþj ; ¼ ;...; N: ð5þ The c kj are Lagrangan nterpolaton coeffcents (see []), whch depend on the left-shft parameter k ¼ ; ;, but not on the values f. It can be shown by Taylor seres epanson of (5) that ^f k ¼ h þ A k D þ OðD 4 Þ; ð6þ where the values A k are ndependent of D. The weghts k are defned as k ¼ a k d k P l¼ a ; a k ¼ l ðb k þ Þ p : ð7þ The coeffcents d ¼ ; d ¼ ; d 5 ¼ are called the deal weghts snce they generate the central upstream ffth-order scheme for the 5-ponts stencl S 5. We refer to a k as the unnormalzed weghts. The parameter s used to avod the dvson by zero n the denomnator and p ¼ s chosen to ncrease the dfference of scales of dstnct weghts at non-smooth parts of the soluton. The smoothness ndcators b k measure the regularty of the kth polynomal appromaton ^f k ð Þ at the stencl S k and are gven by Z b k ¼ X þ D l l d ^f k ðþ d: ð8þ d l l¼ The epresson of the b k n terms of the cell averaged values of f ðþ; f are gven by b ¼ ð f f þ f Þ þ ð 4 f 4f þ f Þ ; ð9þ b ¼ ð f f þ f þ Þ þ ð 4 f f þ Þ ; ðþ b ¼ f f þ þ f þ þ 4 f 4f þ þ f þ : ðþ and ther Taylor seres epansons at are b ¼ f D þ f f f b ¼ f D þ f þ f f b ¼ f D þ f f f D 4 6 f f f f D 4 þ OðD 6 Þ; D 4 þ 6 f f f f D 5 þ OðD 6 Þ; D 5 þ OðD 6 Þ: The general dea of the weghts defnton (7) s that on smooth parts of the soluton, the smoothness ndcators b k are all small and about the same sze, generatng weghts k that are good appromatons to the deal weghts d k. On the other hand, f the stencl S k contans a dscontnuty, b k s O() and the correspondng weght k s small relatvely to the other weghts. Ths mples that the nfluence of the polynomal appromaton of h taken across the dscontnuty s dmnshed up to the pont where the conve combnaton (4) s essentally non-oscllatory. Fg. shows the case where stencl S s dscontnuous, yeldng b and b to be much smaller than b.by(7), ths results on beng a small number n the conve combnaton (4) (see also Fg. (a) of Secton ). The process syntheszed by (4) and (5) s called the WENO reconstructon step, for t reconstructs the values of hðþ at the cell boundares of the nterval I ¼½ ; þ Š from ts cell averaged values f ðþ n the stencls ðþ ðþ ð4þ

5 fs k ; k ¼ ; ; g.in[], truncaton error analyss of the fnte dfference Eq. () led to necessary and suffcent condtons on the weghts k for the WENO scheme to acheve the formal ffth-order ðoðd 5 ÞÞ convergence at smooth parts of the soluton. It was also found that at frst-order crtcal ponts c, ponts where the frst dervatve of the soluton vanshes ðf ð c Þ¼Þ, convergence degraded to only thrd-order ðoðd ÞÞ, a fact that was hdden by the homogenzaton of the weghts caused by the use of a relatvely large value for n (7). Snce ths analyss s relevant to the descrpton of the new WENO scheme ntroduced at net secton, we recall ts essental steps. Addng and subtractng P d k^f k to (4), gves: ^f ¼ X k d k^f þ X h ð k d kþ^f k ¼ h þ B D 5 þ OðD 6 Þ þ X ð k d kþ^f k ð5þ : (The superscrpts ± corresponds to the ± n the f k Epandng the second term wth the help of (6) we.) obtan: X ð k d kþ^f k ¼ X ¼ h X ð k d kþ h þ A k D þ OðD 4 Þ ð k d kþþd X A k ð k d kþþ X ð k d kþoðd 4 Þ: Substtutng the result above at a fnte dfference formula for the polynomal appromaton ^f : ^f þ ^f ¼ h þ h P þ OðD 5 ðþ k d kþ^f k P þ Þþ ð k d kþ^f k D D D " ¼ f ð ÞþOðD 5 Þþ h P þ ðþ k d P kþ h ð k d # kþ D " # þ D X A k ð þ k k Þþ X ð þ k d kþ X ð k d kþ OðD Þ: The OðD 5 Þ term remans after dvson by D because B þ ¼ B n (5). Thus, necessary and suffcent condtons for ffth-order convergence n () are gven by X X ð k d kþ¼oðd 6 Þ; A k ð þ k k Þ¼OðD Þ; R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 95 k d k ¼ OðD Þ: ðþ Note that due to normalzaton P k ¼ P d k, so the frst constrant s always satsfed and, from (7),we see that a suffcent condton for ffth-order of convergence s smply gven by k d k ¼ OðD Þ: ðþ Let us now check how the classcal WENO weghts k (7) behave wth respect to the restrctons above. It was shown n [] that f the smoothness ndcators b k satsfy b k ¼ Dð þ OðD p ÞÞ, then the weghts k satsfy k ¼ d k þ OðD p Þ, where D s a non-zero constant ndependent of k. Lookng at the Taylor seres epansons of the smoothness ndcators b k n () (4), we see that b k ¼ Dð þ OðD ÞÞ, mplyng that k ¼ d k þ OðD Þ. Ths requres that condton () must be as well satsfed n order for the classcal WENO to have the epected ffth-order convergence. Ths ndeed happens and can be easly confrmed wth any symbolc calculaton. Nevertheless, at crtcal ponts ths stuaton becomes more comple dependng on the number of vanshng dervatves of f. For nstance, f only the frst dervatve vanshes, then b k ¼ Dð þ OðDÞÞ and k ¼ d k þ OðDÞ, degradng the convergence of the scheme to thrd order only. If the second dervatve also vanshes, then convergence decreases even more to second order. ð6þ ð7þ ð8þ ð9þ ðþ ðþ

6 96 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 A f to ths defcency of the classcal weghts k was proposed n [] through the applcaton of a mappng functon that ncreased the appromaton of k to the deal weghts d k at crtcal ponts to the requred thrdorder OðD Þ n (). The mappng functon g k ðþ used n [] s defned as g k ðþ ¼ d k þ d k d k þ d k þ d ð4þ ð kþ and s a non-decreasng monotone functon wth the followng propertes:. 6 g k ðþ 6, g k ðþ ¼ and g k ðþ ¼.. g k ðþ f; g k ðþ f.. g k ðd k Þ¼d k, g k ðd kþ¼g k ðd kþ¼. 4. g k ðþ ¼d k þ OðD 6 Þ,f¼d k þ OðD Þ. Numercal results n [] confrmed the usefulness of the mappng, snce wth the modfed weghts the resultng WENO scheme () recovered the full ffth-order convergence at crtcal ponts of a smooth soluton. Note, however, that f at a crtcal pont the second dervatve also vanshes, then b k ¼ Dð þ OðÞÞ, mplyng k ¼ d k þ OðÞ (see Eqs. () (4)) and the mappng s unable to mprove the weghts appromaton, mantanng the same second order of convergence as the classcal WENO. Remark. For problems wth shocks, the O() truncaton error at the dscontnutes dmnshes the advantages of such order mprovements at crtcal ponts. Nevertheless, the numercal results obtaned by the mapped scheme of [] are clearly superor to the ones of the classcal WENO even for problems wth shocks at the ntal condtons. Dstnctly from [], we do not credt these enhancements of the numercal soluton to the hgher order of appromaton of the mapped weghts at crtcal ponts, but to the smaller dsspaton that results from the assgnment of larger weghts to dscontnuous stencls when usng the mapped scheme. As we shall see n the net secton, the new WENO scheme assgn weghts to stencls wth dscontnutes that are even larger than the mapped WENO ones, generatng even sharper solutons, whle stll mantanng the non-oscllatory property.. The new WENO scheme In ths secton, we devse a new set of WENO weghts k that satsfes the necessary and suffcent condtons () for ffth-order convergence. The novel dea s to use the whole 5-ponts stencl S 5 (see Fg. ) to devse a new smoothness ndcator of hgher order than the classcal smoothness ndcators b k. We denote t by s 5 and t s smply defned as the absolute dfference between b and b at, namely s 5 ¼jb b j: ð5þ It s straghtforward to see from () (4) that the truncaton error of s 5 s jf f jd 5 þ OðD 6 Þ ð6þ and that t s a measure of the hgher dervatves of f, when they est, and s ndeed computed usng the whole 5-ponts stencl S 5. Here, we lst the mportant propertes of s 5 : If S 5 does not contan dscontnutes, then s 5 ¼ OðD 5 Þb k for k ¼ ; ; ; f the soluton s contnuous at some of the S k, but dscontnuous n the whole S 5, then b k s 5, for those k where the soluton s contnuous; s 5 6 ma k b k. We now defne the new smoothness ndcators b k z as b z k ¼ b k þ ; k ¼ ; ; ð7þ b k þ s 5 þ

7 and the new WENO weghts z k as z k ¼ az k P ; a z l¼ az k ¼ d k b z ¼ d k þ s 5 ; k ¼ ; ; ; ð8þ l k b k þ where s a small number (see the remark below) used to avod the dvson by zero n the denomnator of (7). All b z k are smaller than unty and they are all close to at smooth parts of the soluton. They are n fact the normalzaton of the classcal smoothness ndcators b k by the hgher order nformaton contaned n s 5. The followng notaton wll be used n order to dstngush between the three dfferent WENO schemes consdered n ths work. The mapped weghts of [] are denoted as M k and the resultng mapped WENO scheme as. The new WENO scheme, ntroduced below, s referred as and the superscrpt z s added to all quanttes related to t. To keep a coherent notaton throughout the artcle, the classcal WENO weghts and smoothness ndcators carry no superscrpt, although, the classcal WENO scheme of [] s referred as WENO-JS. Remark. The role of the parameter was dscussed n [], where t was shown that the value of 6, commonly suggested n the lterature, would domnate over the smoothness ndcators b k at the denomnators of the classcal WENO weghts, hdng the suboptmal performance of the classcal scheme on crtcal ponts. In ths work, we use much smaller values of for the and schemes, that wll be clearly ndcated along wth the numercal eperments, n order to force ths parameter to play only ts orgnal role of not allowng vanshng denomnators at the weghts defntons. We now nvestgate the order of appromaton of the new weghts z k wth respect to the deal weghts d k. We frst study the case where there are no crtcal ponts. These wll be studed separately n the net secton. We also take ¼ n the analyss below. It s straghtforward to check from () (4) and the propertes of s 5 that, at smooth parts of the soluton, þ s 5 ¼ þ OðD Þ; k ¼ ; ; b k and from (8), z k ¼ d k þ OðD Þ; k ¼ ; ; : ð9þ Thus, the new weghts z k satsfy the suffcent condton (), provdng the formal ffth order of accuracy to the scheme at non-crtcal ponts of a smooth soluton. Let us take a look at a numercal eample through the dscontnuous functon: ( uð; Þ ¼f ðþ ¼ snðpþ < < ; ðþ snðpþ þ 6 6 ; consstng of a pecewse sne functon wth a jump dscontnuty at ¼. Fg. (a) shows the numercal solutons of the wave equaton u t ¼ u at t ¼ 8, for the WENO-JS, and schemes, wth () as ntal condton, along wth the eact soluton. Fg. (b) shows the values of the smoothness ndcators b ; b ; b and of the hgh-order smoothness ndcator s 5 of the scheme wth ¼ 4. Note that s 5 s only comparable to one of the b k at stencls that nclude the dscontnuty. Net, we eamne the new weghts z k on stencls contanng dscontnutes and confrm ther dstncton wth respect to the classcal weghts. Consder the smple case of a shock localzed n stencl S, whle the soluton n stencls S and S are smooth (see Fg. ). The ratos between b ; b and b are larger than the same ratos usng the classcal weghts: b z k b z R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 97 ¼ b k b b þ s 5 b k þ s 5 P b k b ; k ¼ ; ; ðþ where we used the fact that b > b k ; k ¼ ;. Ths has the straghtforward mplcaton that the relatve mportance of the stencl S, the one contanng the dscontnuty, s ncreased n the fnal conve combnaton of the scheme (see Fg. (a) (c)).

8 98 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 a b WENO Z WENO M WENO Soluton β β β τ Fg.. (a) Numercal solutons of the lnear wave equaton wth the dscontnuous ntal condton () at t ¼ 8 for the WENO-JS, and schemes. The eact soluton s shown n a sold lne. (b) The values of the smoothness ndcators b ; b ; b and of the hgh-order smoothness ndcator s 5 of the scheme wth ¼ JS w JS w JS w - - M w M w M w - - Z w Z w Z w -4 d d -4 d d -4 d d -5 d -5 d -5 d (a) WENO-JS (b) (c) Fg.. The dstrbuton of the deal weghts d k and the weghts k ; k ¼ ; ; for (a) WENO-JS ð ¼ 6 Þ, (b) ð ¼ 4 Þ and (c) ð ¼ 4 Þ schemes at the frst step of the numercal soluton of the wave equaton u t ¼ u, wth perodc ntal condton gven by (). The deal weghts d k are shown n lnes and the weghts k are shown n symbols. The vertcal as s shown n log scale. Fg. (a) (c) show the weghts k, for the WENO-JS ð ¼ 6 Þ, ð ¼ 4 Þ and ð ¼ 4 Þ schemes at the frst step of the numercal soluton of the wave equaton u t ¼ u, wth perodc ntal condton gven by (). The deal weghts d k are also plotted as lnes and the vertcal as s n log scale. Far away from the dscontnuty ¼, the weghts k (symbols) for all schemes correctly match the correspondng deal weghts d k (lnes). ¼ : s the frst locaton where the 5-ponts stencl S 5 contans the dscontnuty. At ths grd pont, the rghtmost stencl S has ts weght decreased to a much smaller value than and, whle these two are slghtly ncreased to reflect ther larger relevance at the reconstructon step. At ¼ :, the dscontnuty s present at S and S and a small value s assgned to as well, yeldng OðÞ. At ¼ :, a symmetrc scenaro occurs and assumes the largest value.

9 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 99 Whle ths stuaton s general for all schemes, the man dfference s at the ratos between the weghts for dscontnuous and contnuous stencls, as dscussed above. Whle WENO-JS sets very small values for the dscontnuous stencls, around 8, and the mappng of generates a small ncrease on these values, yelds a more substantal ncrease to 4. In dfferent words, the conve combnatons of WENO- M and are closer to the central scheme than the one of WENO-JS, ncurrng n less dsspaton. Lookng closer at the numercal solutons of Fg. (a), we also see that the appromaton of s slghtly sharper at the dscontnuty. Remark. In the numercal eperments of Secton 5 we wll once agan notce ths slghtly superor sharpness of wth respect to at problems wth dscontnutes. Nevertheless, as we should also see n Secton 5, t s on ther computatonal costs that they dffer more, snce takes about 5% less CPU tme than. The larger values of weghts for the dscontnuous stencls s one of the man dfferences of and over the classcal WENO-JS. Another one s ther superor order of convergence at crtcal ponts of smooth solutons. Ths s the subject of net secton. 4. Convergence at crtcal ponts In [], t was detected that the classcal ffth-order WENO scheme acheves only thrd order at crtcal ponts of smooth solutons. It was also demonstrated that ths defcency was beng hdden by the large value of 6 assgned to the parameter n the defnton of the weghts. The mappng of the weghts proposed n [] recovered the formal ffth order at crtcal ponts and the enhanced numercal results n problems wth shocks were credted to ths f. In ths secton we analytcally demonstrate that s fourth-order accurate at crtcal ponts of a smooth soluton and also present a further modfcaton of the weghts that recovers the ffth order of convergence at the crtcal ponts. Nevertheless, we also show that ths modfcaton ncreases the dsspaton of the scheme. We depart from formula (9) and notce that the order of the WENO scheme s gven by s ¼ mnð5; s þ ; s þ Þ; ðþ where X A k ð þ k k Þ¼OðDs Þ; k d k ¼ OðD s Þ: ðþ ð4þ We wll eamne the values of s and s for a smooth functon wth a crtcal pont of order n cp P, whch s a functon f wth vanshng dervatves for all order less than and equal to n cp : f ð c Þ¼¼f ðncpþ ð c Þ¼; ð5þ where the superscrpt denotes the order of dfferentaton. To smplfy notaton, we drop the superscrpt Z for the weghts Z k n ths secton. Usng the defnton of the weghts (8), t s easly seen that a k d k ¼ d k s 5 b k þ mples k d k ¼ O s 5 b k þ : ð6þ It s also easly seen from the Taylor seres formulae () (4) and (6) for the smoothness ndcators b k and s 5 that n the presence of a crtcal pont of order n cp, we have b k ¼ OðD ðncpþþ Þ and s 5 ¼ OðD5 Þ f n cp 6 ; ð7þ OðD ðncpþþ Þ else:

10 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 Applyng these to (6), we obtan the value of s as below: 8 8 >< OðD Þ f n cp ¼ >< f n cp ¼ ; k d k ¼ OðDÞ f n cp ¼ or s ¼ f n cp ¼ ; >: >: OðÞ f n cp P f n cp P : For s, we epand () as where and X A k ð þ k k Þ¼Xþ X ¼ N þ D N D þ X ¼ X N ¼ s 5 A k k ¼ P F k ða kþ þ P F k ðþ Y b k! X A k d k ; D ¼ Y b k ð8þ D þ D ; ð9þ wth F k ðcþ ¼s 5 c d k b k ; b k! þ s 5 Y b k! X d k b k ð4þ ; ð4þ where we used that P A kd k ¼ and P d k ¼. The leadng order of (9), s, can be found wth the ad of a symbolc algebra program by substtutng the Taylor seres epanson of the smoothness ndcators b k, subjected to a crtcal pont of order n cp (see Table ). The order of the scheme, s ¼ mnð5; s þ ; s þ Þ, s shown at Table. Note that attans order 4 at a frst-order crtcal pont, mprovng over WENO-JS whch attans only order, as shown n []. See also Table that shows the L error and rate of convergence of the WENO-JS and schemes for f ðþ ¼ þ cosðþ, whch has a frst-order crtcal pont at ¼, but whose thrd-order dervatve s not zero. The computatons were done n Matlab usng ts varable precson functons wth 64 dgts and a value of ¼ 4. Ths was necessary because of the roundoff error n the standard double precson due to the very small values of D that had to be used n order to detect the fourth-order convergence of at the crtcal pont. Only a small modfcaton on the formulae of the weghts (8) s necessary to recover the ffth-order accuracy at a frst-order crtcal pont. If nstead we defne the new weghts as z k ¼ az k P ; a z l¼ az k ¼ d k b z ¼ d k l k þ s 5 b k þ q ; k ¼ ; ; ; ð4þ wth q ¼, then ffth order s acheved when n cp ¼. Note that both defntons agree when q ¼. Ths s easy to understand, snce for a smooth functon, ncreasng the value of q n (4) decreases the correcton of the weghts to the deal weghts fd k g, makng the scheme closer to the optmal central scheme. On the other hand, for dscontnuous problems, the rato s 5 b k s much larger for the contnuous substencls than for the one contanng the dscontnuty. Therefore, ncreasng q decreases the relatve mportance of the dscontnuous substencl, makng the scheme more dsspatve. Nevertheless, the results n Fg., along wth the ones of net secton, show that due to the essental O() error of problems wth dscontnutes, fourth order of convergence at crtcal ponts does not keep, as smply defned n (8), wth q ¼, to obtan results n problems wth shocks that are equvalent to the ones of the scheme. Ths confrms that the larger Table The leadng order term of (9), s, as a functon of n cp n cp P N þ D N D þ OðD 8 Þ OðD 6 Þ OðD 6 Þ OðD ðncpþ Þ D þ D OðD Þ OðD 4 Þ OðD 6 Þ OðD ðncpþ Þ s 6

11 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 Table Order of convergence of the scheme n cp P s 6 s s ¼ mnð5; s þ ; s þ Þ 5 4 Table The L error and rate of convergence of the WENO-JS and schemes (for q ¼ and q ¼ ) at the frst-order crtcal pont ¼ of the functon f ðþ ¼ þ cosðþ wth ¼ 4 D WENO-JS, q ¼, q ¼ Error Order Error Order Error Order.e.6e 9.87e.8e 5.e 4.9e e e e 4.66e.97 4.e e e 4.5e.98.6e e e 5 4.e.99.6e e weghts attrbuted to dscontnuous stencls by and, although by dstnct approaches, s the essental mprovement of these schemes wth respect to WENO-JS. 5. Numercal eperments In ths secton, we compare the numercal performance of wth the classcal WENO-JS and ts verson wth mappng,. In all the numercal eperments below, refers to the defnton n (8) and (4) wth q ¼. Ths s to make the pont that the smaller dsspaton of, as ponted out before n Sectons and4, has much more nfluence than ts rate of convergence at crtcal ponts when solvng problems wth shocks. For that matter, the solutons obtaned wth both versons of, q ¼ and q ¼, show the same order of convergence at dscontnuous solutons, not justfyng a separate presentaton. A more detaled analyss of the schemes for several values of q, ncludng hgher orders versons, s left for an upcomng artcle. The numercal presentaton of ths secton starts wth the soluton of a classcal problem of advecton of a functon wth dscontnutes, followed by the soluton of the one dmensonal Euler system of equatons wth Remann ntal value problems such as the Sod, La and ; the Mach shock densty wave nteracton, the nteractve blastwaves problem and fnshes wth D smulatons on the shock vorte nteracton and the Mach 4.46 Rchtmyer Meshkov Instablty (RMI) problem wth a sngle mode perturbaton along a enon and argon gases nterface. In the followng eperments, the ODEs resultng from the sem-dscretzed PDEs are evolved n tme by the thrd-order total varaton dmnshng Runge Kutta scheme (RK-TVD) []: ~U ¼ ~U n þ DtLð~U n Þ; ~U ¼ 4 ð ~U n þ ~U þ DtLð~U ÞÞ; ð4þ ~U nþ ¼ ð ~U n þ ~U þ DtLð~U ÞÞ; where L s the spatal operator. The CFL number s set to be.4 for all WENO schemes and Dt s bounded by D 5. The numercal eperments presented here were run on a AMD Opteron(tm) 5 processor wth GB of memory. In order to ensure farness n the comparson of CPU tmngs, all three schemes shared the same subroutne calls and were compled wth the same complaton optons ncludng optmzaton ones. The only

12 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 dfferences between the mplementaton of the three WENO schemes were on the subroutnes for computng the dfferent related WENO weghts. In all the eperments to follow, we took ¼ 6 for WENO-JS and ¼ 4 for and, n order to compare wth the classcal scheme at ts best. 5.. The lnear advecton problem In ths secton, we apply to the lnear transport of dscontnuous functons n the case of an ntal condton consstng of a Gaussan, a trangle, a square-wave and a sem-ellpse, gven by 8 ½Gð; b; z dþþ4gð; b; zþþgð; b; z þ dþš; ½ :8; :6Š; 6 >< ; ½ :4; :Š; uð; Þ ¼ jð :Þj; ½; :Š; ½F ð; a; a dþþ4fð; a; aþþfð; a; a þ dþš; ½:4; :6Š; 6 >: ð44þ ; otherwse; Gð; b; zþ ¼e bð zþ ; qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff F ð; a; aþ ¼ mað a ð aþ ; Þ; where the constants are z ¼ :7; d ¼ :5; b ¼ log 6d ; a ¼ :5 anda ¼. The advecton equaton u t ¼ u was solved untl the fnal tme t ¼ 8 on the nterval [,] wth perodc boundary condtons and the results of are compared aganst the ones of WENO-JS and, all shown n Table 4. Fgs. 4 and 5 along wth Table 4 show that behaves quanttatvely and qualtatvely equvalent to wth regards to the mprovements over WENO-JS. Note that the lower order of at the crtcal ponts s less relevant than ts smaller dsspaton f one wants to obtan sharper representatons of the dscontnutes. 5.. One dmensonal Euler equatons In ths secton, we present numercal eperments wth the one dmensonal system of the Euler equatons for gas dynamcs n strong conservaton form: Q t þ F ¼ ; ð45þ where Q ¼ðq; qu; EÞ T ; F ¼ðqu; qu þ P; ðe þ PÞuÞ T ; ð46þ the equaton of state s gven by P ¼ðc Þ E qu ; c ¼ :4 ð47þ and q; u; P and E are the densty, velocty, pressure and total energy respectvely. Table 4 The L error and ts rate of convergence for the WENO-JS, and schemes, when solvng the lnear transport equaton wth the dscontnuous ntal condton (44) at the fnal tme t ¼ 8 N WENO-JS Error Order Error Order Error Order e.4e.9e.e..65e.9.69e.9 9.e e.8 7.4e e..e..e. 8.94e.5.5e..59e e e e.98

13 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 (a) Soluton (b) Error.8 WENO-JS Eact WENO-JS - - u.6.4 Error Fg. 4. Numercal soluton and absolute pontwse error of the advecton equaton wth the dscontnuous ntal condton (44) as computed by the WENO-JS, and wth N ¼ at t ¼ 8. (a) Soluton WENO-JS Eact (b) Error WENO-JS u.6 Error Fg. 5. Numercal soluton and absolute pontwse error of the advecton equaton wth the dscontnuous ntal condton (44) as computed by the WENO-JS, and wth N ¼ 4 at t ¼ 8. Followng [], the hyperbolcty of the Euler equatons admts a complete set of rght and left egenvectors for the Jacoban of the system. The egenvalues and egenvectors are obtaned va the lnearzed Remann solver of Roe [4] and the frst-order La-Fredrchs flu s used as the low order buldng block for the hgh-order reconstructon step of the WENO schemes (see Eq. (.5) n []). After projectng the flues on the characterstc felds va the left egenvectors, the hgh-order WENO reconstructon step s appled to obtan the hgh-order appromaton at the cell boundares, whch are projected back nto the physcal space va the rght egenvectors. See [] for further detals of the algorthm Remann ntal value problems: Sod, La and In ths secton, we show that the scheme passes the test of the Remann ntal value problems, also known as the shock-tube problems: the Sod problem, the La problem and the problem. The numercal eperments were conducted usng grd ponts.

14 4 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 For the Sod problem, the densty q, velocty U and pressure P n the left and the rght stages of the shock are: ð:5; ; :Þ 5 6 < ; ðq; U; PÞ ¼ ð48þ ð; ; Þ 6 < 5 and the fnal tme s t ¼. For the La problem, the densty q, velocty U and pressure P n the left and the rght stages of the shock are: ð:445; :698; :58Þ 5 6 < ; ðq; U; PÞ ¼ ð49þ ð:5; :; :57Þ and the fnal tme s t ¼ :. For the problem, the densty q, velocty U and pressure P n the left and the rght stages of the shock are: ð; ; :4Þ 5 6 < ; ðq; U; PÞ ¼ ð5þ ð; ; :4Þ and the fnal tme s t ¼. Numercal results from all schemes follow the same pattern as before, wth and beng more accurate than the classcal scheme due to ther lesser dsspatvty. The smulated densty q of the Sod, La and problems are shown n Fg. 6. The numercal results (symbols) are shown along wth the eact solutons (sold black lnes) Shock densty wave nteracton In ths secton, we consder the one dmensonal Mach shock-entropy wave nteracton [5], specfed by the followng ntal condtons: ðq; u; PÞ ¼ ð:8574; :6969; Þ 5 6 < 4; ð5þ ð þ : snðkþ; ; Þ ; where ½ 5; 5Š and k ¼ 5. The soluton of ths problem conssts of a number of shocklets and fne scales structures whch are located behnd a rght-gong man shock. Fg. 7(a) and (b) provde a comparson for all schemes at t ¼ wth an ncreasng number of ponts. We shall refer to the soluton computed by the scheme wth N ¼ ponts as the eact soluton. At a low resoluton, N ¼, as shown n Fg. 7(a), and capture much more fne scale structures of the soluton than WENO-JS, partcularly at the hgh-frequency waves behnd the shock..8 "Eact" WENO-JS.4. "Eact" WENO-JS.8 Rho.6 Rho Rho "Eact" WENO-JS Fg. 6. The densty profles of the Sod, La and problems wth N ¼ at tmes t ¼, t ¼ : and t ¼, respectvely. The sold black lnes are the eact solutons.

15 R. Borges et al. / Journal of Computatonal Physcs 7 (8) WENO-JS "Eact" (a) N= (b) N= WENO-JS "Eact".5.5 Rho.5 Rho Fg. 7. Soluton of the Mach shock densty wave nteracton wth k ¼ 5 as computed by WENO-JS, and schemes, at tme t ¼ wth (a) N ¼, (b) N ¼ ponts. The eact soluton s computed by the scheme wth N ¼ ponts. Increasng the resoluton to N ¼, as shown n Fg. 7(b), we see that both and converge faster than WENO-JS. In Fg. 8, the wave number k s ncreased to, yeldng rougher numercal appromatons at the sne wave densty feld perturbaton. As before, and show much more accurate results than the classcal WENO-JS Interactng blastwaves The one dmensonal blastwaves nteracton problem of Woodward and Collela [7] has the followng ntal condton, wth reflectve boundary condtons: 4.5 WENO-JS "Eact" 4.5 Rho Fg. 8. Soluton of the Mach shock densty wave nteracton wth k ¼ computed by the WENO-JS, and wth N ¼ 5 ponts. For clarty, only symbols at every fourth data pont are plotted. The eact soluton s computed by the scheme wth N ¼ ponts.

16 6 R. Borges et al. / Journal of Computatonal Physcs 7 (8) WENO-JS "Eact" 5 4 Rho Fg. 9. Soluton of the nteractve blastwaves problem computed by the WENO-JS, and wth N ¼ 4 ponts. For clarty, only symbols at every other data pont are plotted. The eact soluton s computed by the scheme wth N ¼ 4 ponts. 8 ð; ; Þ 6 < :; >< ðq; U; PÞ ¼ ð; ; :Þ : 6 < :9; >: ð; ; Þ :9 6 6 :: ð5þ The ntal pressure gradents generate two densty shock waves that collde and nteract later n tme, formng a profle as shown n Fg. 9 at t ¼ :8. All three schemes converge, as the number of ponts ncrease, to the reference soluton computed by the wth N ¼ 4 ponts. As before, and show an mproved convergence wth respect to WENO-JS, due to ther smaller dsspaton. Fg. presents a separate, and more detaled, comparson between and, at two dfferent portons of the doman. Careful eamnaton of Fg. (a) shows that obtans a sharper peak near ¼ :78. In Fg. (b), the hgh-gradent structure at ¼ :86 s better resolved wth, as well as the contact dscontnuty near ¼ : Two dmensonal Euler equatons In ths secton, we apply the WENO schemes to the Shock vorte nteracton (SV); and the Rchtmyer Meshkov nstablty (RMI) n a rectangular doman. The governng equatons are the two-dmensonal Euler equatons n Cartesan coordnates gven by Q t þ F þ G y ¼ ; ð5þ

17 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 7 (a) [.65,.8] (b) [.79,.875] 7 "Eact" "Eact" 6 Rho 5 Rho Fg.. Zoom n regons of the soluton of the nteractve blastwaves problem computed by the and scheme wth N ¼ 8 ponts. For clarty, only symbols at every other data pont are plotted. The eact soluton s computed by the scheme wth N ¼ 4 ponts. where Q ¼ðq; qu; qv; EÞ T ; F ¼ðqu; qu þ P; quv; ðe þ PÞuÞ T ; G ¼ðqv; quv; qv þ P; ðe þ PÞvÞ T and the equaton of state s ð54þ P ¼ðc ÞðE qðu þ v ÞÞ; c ¼ :4: ð55þ Free stream nflow and outflow boundary condtons are mposed n the nflow and outflow boundares, respectvely, n the -drecton. A perodcal boundary condton s mposed n the y-drecton Shock vorte nteracton The tangental velocty profle of the counter-clockwse rotatng vorte [8] centered at ð c ; y c Þ s gven n polar coordnates by 8 >< Crðr r Þ; 6 r 6 r < r ; UðrÞ ¼ Crðr r Þ; r 6 r 6 r ; ð56þ >: ; r > r ; where r ¼ :; r ¼ :, the vorte strength C ¼ :5 and the Mach number M s ¼. In the Shock Vorte nteracton, an acoustc wavefront s generated and fne scale structures are formed behnd the man shock. These are well captured by all three WENO schemes, wth the same pattern as before, where less dsspaton s noted at the and solutons. The densty q and velocty v at tme t ¼ computed by the scheme wth grd resoluton N ¼ N y ¼ 4 are shown n Fg.. In Table 5, the CPU tmng (n seconds) per Runge Kutta step of the three WENO schemes for ths problem shows that s the most effcent scheme and about 5% faster than the scheme, snce does not make use of any mappng to compute the weghts.

18 8 R. Borges et al. / Journal of Computatonal Physcs 7 (8) y y (a) Densty - 4 (b) Velocty Fg.. (a) The densty q and (b) the velocty v of the Mach shock vorte nteracton at tme t ¼ and grd resoluton N ¼ N y ¼ 4 as computed by the scheme. Table 5 CPU tmng (n seconds) per Runge Kutta step of the shock vorte nteracton as computed by the WENO-JS, and schemes Grd sze WENO-JS Rchtmyer Meshkov nstablty We use a rectangular doman ½; 5Š½ :8; :8Š wth a shock Mach number M s ¼ 4:46 nteractng wth a sngle mode snusodal perturbed nterface along a enon (Xe) and argon (Ar) gases nterface. The ntal condton of the sngle mode Rchtmyer Meshkov nstablty s specfed as follows: Rankne Hugonot condtons for the shock, Pre-shock temperature T ¼ 96 K, Pre-shock pressure P ¼ :5 atm, Xenon and Argon densty are q Xe ¼ :9 normal atmospherc pressure, Specfc heat rato c ¼ 5, Atwood number At ¼ :54, Mach number M ¼ 4:46, Wave length k ¼ :6 cm, Ampltude a ¼ : cm. The dffusve nterface s modeled wth an eponental functon,.e. 8 >< ; d 6 ; Sð; yþ ¼ epð ajdj b Þ; < d < ; >: ; d P ; g and q cm Ar ¼ :89 g respectvely, at half of the cm ð57þ

19 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 9 where d¼ ðs þ a cosðpy=kþ þ dþ ; d ð58þ d ¼ : cm s the nterface thckness, b ¼ 8 s the nterface order, s ¼ :5 cm s the locaton of the nterface and a ¼ ln, where s the machne zero. The conservatve or prmtve varables are scaled accordng to Sð; yþ between the enon and argon gases. Shock Computatonal Doman Mach=4.46 y Xenon Argon Xenon-Argon Interface Fg.. The ntal snusodal perturbaton of the densty q separatng the enon and argon gases at tme t ¼ for the Rchtmyer Meshkov Instablty y y (a) Densty, y y (b) Velocty v, (c) Densty ρ, (d) Velocty v, Fg.. Densty, q and velocty, v of the Rchtmyer Meshkov Instablty wth Mach number M s ¼ 4:46 at tme t ¼ 5 ls as computed by the (a b) scheme and (c d) scheme wth grd resoluton N ¼ N y ¼ 8.

20 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 Table 6 CPU tmng (n seconds) per Runge Kutta step of the Rchtmyer Meshkov nstablty problem as computed by the WENO-JS, and schemes Grd sze WENO-JS The ntal condton of the smulaton s shown n Fg.. As the shock wave colldes wth the nterface separatng the two gases wth dfferent denstes, the snusodal perturbed nterface s accelerated, compressed and amplfed followng the transmsson and refracton of the shock. Baroclnc vortcty generated along the gases nterface amplfes the perturbaton of the nterface. The heaver enon gas (Xe) wll penetrate nto the lghter argon gas (Ar) formng fnger-lke structures bubbles and spkes. A bubble (spke) s a porton of the lght (heavy) gas penetratng nto the heavy (lght) gas. Fg. shows the densty q and velocty v at tme t ¼ 5 ls usng the and schemes wth grd resoluton N ¼ N y ¼ 8. The results ndcate that the scheme works very well for ths class of problems, generatng equvalent solutons as the scheme. Table 6 shows the CPU tmng per Runge Kutta step for all three WENO schemes at the grd resoluton of N ¼ N y ¼ 8 wth smlar results as for the shock vorte nteracton. 6. Conclusons We have devsed an mproved verson of the ffth-order WENO fnte dfferences scheme for conservaton laws of [] that makes use of hgher order nformaton already contaned n the orgnal framework of the classcal WENO scheme. The new smoothness ndcators proposed take nto account the novel etra nformaton on the regularty of the soluton and provde a conve combnaton of stencls wth enhanced order of convergence at crtcal ponts and less dsspaton at shocks, but stll non-oscllatory. The new WENO scheme generates numercal solutons wth the same accuracy as the mapped WENO of [] wth smaller computatonal cost. Our analyss also ndcated that the mprovements obtaned by the mapped WENO over the classcal scheme when solvng problems wth shocks are not due to ts superor accuracy at crtcal ponts of the solutons, but to the larger weghts t assgns to stencls wth dscontnutes. Smlarly, the new WENO scheme assgns even larger weghts to dscontnuous stencls, obtanng solutons that are sometmes sharper, although ts rate of convergence at crtcal ponts s ntermedary between those schemes. Research s currently underway to etend the new WENO dea to hgher order WENO reconstructon schemes and wll be reported at an upcomng paper Acknowledgements The frst, second and thrd authors have been supported by CNPq, Grant 5/98-8. The fourth author gratefully acknowledges the support of ths work by the DOE under Contract Number DE-FG-98ER546 and AFOSR under Contract Number FA , and would also lke to thanks the Departamento de Matemátca Aplcada, IM-UFRJ, for hostng hs vst durng the course of the research. References [] D. Balsara, C.W. Shu, Monotoncty preservng weghted essentally non-oscllatory schemes wth ncreasngly hgh order of accuracy, J. Comput. Phys. 6 () [] G.S. Jang, C.W. Shu, Effcent mplementaton of weghted ENO schemes, J. Comput. Phys. 6 (996) 8. [] A.K. Henrck, T.D. Aslam, J.M. Powers, Mapped weghted essentally non-oscllatory schemes: achevng optmal order near crtcal ponts, J. Comput. Phys. 7 (5) [4] P.L. Roe, Appromaton Remann solvers, parameter vectors, and dfference schemes, J. Comput. Phys. 4 (98) 57. [5] C.W. Shu, S. Osher, Effcent mplementaton of essentally non-oscllatory shock-capturng schemes, J. Comput. Phys. 77 (988)

21 R. Borges et al. / Journal of Computatonal Physcs 7 (8) 9 [6] C.W. Shu, S. Osher, Effcent mplementaton of essentally non-oscllatory shock-capturng schemes, II, J. Comput. Phys. 8 () (989) 78. [7] P. Woodward, P. Collela, The numercal smulaton of two dmensonal flud flow wth strong shocks, J. Comput. Phys. 54 (984) 5 7. [8] D.A. Koprva, Spectral collocaton computaton of the sound generated by a shock vorte nteracton, n: D. Lee, M.H. Schultz (Eds.), Computatonal Acoustcs: Algorthms and Applcatons, vol., 988.