D ec. 00 T ransactions of N anjing U niversity of A eronautics & A stronautics V o l. 8 Supp l. D EVELOPM ENT AND APPL ICATIONS OF W ENO SCHEM ES IN CONTINUUM PHY SICS Ξ Ch iw ang S hu (D ivision of A pp lied M athem atics, B rovidence, R hode Island 09,U SA ) ABSTRACT T h is paper briefly p resents the general ideas of h igh o rder accu rate w eigh ted essen tially nono scillato ry (W ENO ) schem es, and describes the sim ilarities and dif ferences of the tw o classes of W ENO schem es: finite vo lum e schem es and fin ite difference schem eṡ W e also briefly m ention a recent developm ent of W ENO schem es, nam ely an adap tive app roach w ith in the finite difference fram ew o rk using smoo th tim e dependent curvilinear coo r dinateṡ Key words: w eigh ted essen tially nono scillato ry; fin ite difference m ethod; fin ite vo lum e m ethod; adap tive m ethod CLC num ber: V. 3 INTROD UCT ION H igh o rder accu rate w eigh ted essen tially nono scillato ry (W ENO ) schem es have been de veloped to so lve a hyperbo lic con servation law u t + g g f (u) = 0 () T he first W ENO schem e w as con structed in R eḟ [ ] fo r a th ird o rder fin ite vo lum e version in one space dim en sion. In R ef. [ ], th ird and fifth o r der fin ite difference W ENO schem es in m u ltip le space dim en sion s are con structed, w ith a general fram ew o rk fo r the design of the sm oo thness indi cato rs and non linear w eigh tṡ L ater, second, th ird and fou rth o rder fin ite vo lum e W ENO schem es fo r D genera l t riangua t ion have been developed, e. g. in R eḟ [ 3 ]. V ery h igh o rder fi n ite differece W ENO schem es ( fo r o rders be tw een 7 and ) have been developed in R ef. [4 ]. Cen tral W ENO schem es have been developed in R ef. [ 5 ]. A techn ique to treat negative linear w eigh ts in W ENO schem es has been developed in R ef. [ 6 ]. In th is conference, J iang, Shan and L iu p resen ted their new resu lt s on develop ing com pact W ENO schem eṡ W ENO schem es a re designed ba sed on the successfu l ENO schem es in R efs. [ 6, 7 ]. Bo th ENO and W ENO schem es u se the idea of adap t ive stencils in the recon st ruct ion p rocedu re based on the local sm oo thness of the num erical so lu tion to au tom atically ach ieve h igh o rder accu racy and nono scillato ry p roperty near discon ti nu itieṡ ENO u ses ju st one (op tim al in som e sen se) ou t of m any candidate stencils w hen do ing the recon struction; w h ile W ENO u ses a convex com b ina t ion of a ll the cand ida te stencils, each being assigned a non linear w eigh t w h ich depends on the local sm oo thness of the num berical so lu tion based on that stencil. W ENO im p roves upon ENO in robu stness, bet ter sm oo thness of fluxes, bet ter steady sta te convergence, bet ter p rovab le convergence p rop ert ies, and m o re efficiencey. Fo r a deta iled review of ENO and W ENO schem es, up to the tim e w hen these no tes w ere Ξ Received 5 O cṫ 00 995-005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
Supp l. Ch iw ang Shu. D evelopm ent and A pp lications of W ENO Schem es in Continuum Physics 7 pub lished, w e refer to the lectu re no tes [ 8, 9 ]. W ENO schem es are already w idely u sed. F IN ITE VOL UM E AND F IN ITE D IFFERENCE W ENO SCHEM ES A fin ite vo lum e schem e fo r a con servation law such as () app rox im ates an in tegral version of iṫ L et s u se the one dim en sional exam p le u t + f (u) x = 0 () to illu strate the ideaṡ Suppo se { I i = [ x i- x i+, ]}, i=,, N, is a partition of the com pu tational dom ain, and x i= x i+ - x i-. If w e in tegrate the PD E ( ) in the cell I i, w e ob tain du θ i (t) dt + [f (u (x i+ x, t ) ) - f (u (x i- i, t ) ) ]= 0 w here θ u i (t) = f I x i u (x, t) dx is the cell average i of u in cell I i. A sem idiscrete (discrete in the spatial vari ab le on ly) fin ite vo lum e schem e fo r Eq. () is an OD E system fo r the cell averages {u θ i ( t) }, i=,, N. In o rder to ob tain such a schem e, w e need the fo llow ing recon struction p rocedu re: Procedure R econ struction O b tain accu rate po in t values {u (x i+, t ) }, i= 0,, N, from the given cell averages {u θ i (t) }, i=,, N. W ENO p rocedu re. version. is sim p ly a specific recon struction L et u s dem on strate the fifth o rder Fo r th is purpo se, the app rox im ation of {u (x i+, t ) } u ses the info rm ation of five cell av erages, from the stencil {I i-, I i-, I i, I i+, I i+ }. T h is stencil is no t symm etric w ith respect to the po in t x i+ of the recon struction. T here is one m o re cell to the left than to the righ ṫ T hu s th is recon struction is good fo r upw inding. du re con sists of the fo llow ing step s: () B reak the final stencil T he p roce T = {I i-, I i-, I i, I i+, I i+ } (3) in to the fo llw o ing th ree sm aller stencils S = {I i-, I i-, I i}, S = {I i-, I i, I i+ } S 3 = {I i, I i+, I i+ } N o tice that each sm all stencil con tain s the tar get cell I i. ( ) Con struct th ree po lynom ials p j (x ) of degree at m o st tw o, w ith their cell averages a greeing w ith that of the function u in the th ree cells in each stencil S j. W e also con struct a po ly nom ial Q (x ) of degree at m o st fou r, w ith its cell averages agreeing w ith that of the function u in the five cells in the larger stencil T. w eigh ts such that (3) F ind th ree con stan ts, also called linear Χ = 0, Χ = 3 5, Χ3 = 3 0 Q (x i+ g) = Χp (x i+ g) + Χp (x i+ g + Χp 3 (x i+ g) fo r all po ssib le given data u θ j, j = i-, i-, i, i+ (4), i+. T h is is to say, the h igher o rder recon struction Q ( x i+ g ) can be w ritten as a linear com b ination of th ree low er o rder recon struction s p j (x i+ g). T he linear w eigh ts given in Eq. (4) depend on local geom etry and o rder of accu racy, bu t no t on u θ j. If som e of these linear w eigh ts are negative, special techn iques m u st be u sed [ 6 ]. (4) Com pute the sm oo thness indicato r, de no ted by Βj, fo r each stencil S j, w h ich m easu res how sm oo th the function p j (x ) is in the target cell I j. T he sm aller th is sm oo thness indicato r Βj, the sm oo ther the function p j (x ) is in the target cell. In m o st of the cu rren tly u sed W ENO schem es the fo llow ing sm oo thness indicato r is u sed [ ] 995-005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. Βj = x l= l- I i d l dx l p j (x ) fo r j =,, 3, fo r th is fifth o rder case. dx (5) T hese sm oo thness indicato rs are quadratic function s of the cell averages in the stencil. (5) Com pute the non linear w eigh ts based on the sm oo thness indicato rs g Ξj Ξj = lξ g, Ξ g j = l Χj (Ε+ Βj) (6)
8 T ransactions of N anjing U niversity of A eronautics & A stronautics V o l. 8 w here Χj are the linear w eigh ts determ ined in Step 3 above, and Ε is a sm all num ber to avo id the denom inato r to becom e 0. T yp icallly, w e can take Ε= 0-6 in all the com putation ṡ T he final W ENO app rox im ation o r recon struction is then given by R (x i+ g) = Ξp (x i+ g) + Ξp (x i+ g) + Ξ3p 3 (x i+ g) (7) W ith th is W ENO recon struction p rocedu re, a fin ite vo lum e W ENO schem e is now ready. T he details can be found [, 8, 9 ]. N ex t w e describe the setup of a fin ite differ ence schem e fo r so lving Eq. (). A sem idiscrete fin ite difference schem e fo r () is an OD E system fo r the po in t values {u i (t) }, i=,,n, w here u i (t) app rox im ate the po in t values of the so lu tion u (x i, t). W e also in sist on a con servative app rox i m ation to the derivative f (u) x in the fo rm of f (u) x g x = x i x (f^ i+ g - f^ i- g) (8) w here f^ i+ g is the num erical flux, w h ich typ ical ly is a L ip sch itz con tinuou s function of several neighbo ring values of u j around x i. A t a first glance, the firile differece schem e has no th ing in comm on w ith the firite vo lum e schem e described above, a s they app rox im a te differen t values of the so lu tion. How ever, the fo llow ing ob serva t ion, first in t roduced in R eḟ [ 8 ], estab lishes a clo se relation sh ip betw een the tw o. If w e iden tify a function h (x ) by f (u (x ) ) = w here w e have supp ressed the f x x + x h (Ν) dν (9) x - dependency of the function as w e are in terested now on ly at spa tial discretization s, then by ju st tak ing deriva tives on bo th sides of the above equality w e ob tai f ( u ( x ) ) x h (x + x x ) - h (x - x ). T h is m ean s that w e on ly need to take num erical flux as f δ i+ g h (x i+ g). If w e cou ld get an app rox im a w ou ld also be of the sam e h igh o rder of accu ra cy. N o tice that Eq. (9) can be w ritten as f (u i) = h i, i. e. w e are given the cell averages of h (since w e know the po in t values u i, hence also f (u i), in a fin ite difference schem e) and w e w ou ld need to recon struct its po in t values h (x i+ g) fo r the nu m erical flux. But th is is exactly the sam e recon struction p rob lem in P rocedu re above fo r fin ite vo lum e schem es! T hu s fo r one space dim en sion, the fin ite dif ference and fin ite vo lum e schem es share the sam e recon st ruct ion p rocedu re, app lied in d ifferen t con tex ts (on cell averages of u fo r the fin ite vo lum e schem es, and on po in t values of f (u ) fo r the fin ite difference schem es). T hey invo lve the sam e com p lex ity and co sṫ F in ite difference schem es are m o re restrictive in the sitnation s that they can be app lied, as they on ly w o rk fo r un ifo rm o r sm oo th varying m eshes and a flux sp litting (fo r upw inding) m u st be sm oo th. How ever, fo r m u ltip le space dim en sion s, there are essen tial differences betw een these tw o classes of m ethods, w hen the o rder of accu racy is a t lea st th ree. W h ile fin ite d ifference schem es can still be app lied in a dim en sionbydim en sion fash ion (no t dim en sion sp litting! ), i. e. ing f (u ) x com put along a x line w ith fixed y u sing the p rocedu re above, and likew ise fo r g (u ) y, tion to h (x ) to h igh o rder the accu racy, the con of accu racy. In retu rn, the fin ite vo lum e schem es servative app rox im ation to the derivative in (8) do allow m o re flex ib ility in their app lication ṡ 995-005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. then add ing them together to fo rm the residue, fin ite vo lum e schem es of th ird o rder of h igher m u st in vo lve m u ltidim en sional recon struction s from cell averages to po in t values and then num erical in te g ra t ion s to get the num erica l fluxes a long the bounda ries of cellṡ T he deta ils of these recon struction s can be found [ 3, 5, 8, 9 ]. A s such, the op eration coun t and CPU tim e fo r a fin ite vo lum e schem e is around tw o fou r tim es m o re expen sive in tw o dim en sion s and around five to n ine tim es m o re expen sive in th ree dim en sion s, com pared w ith a fin ite difference schem e of the sam e o rder
Supp l. Ch iw ang Shu. D evelopm ent and A pp lications of W ENO Schem es in Continuum Physics 9 T hey can be app lied in arb itrary triangu lation s and do no t requ ire sm oo thness of the m esheṡ O n the o ther hand, fin ite difference schem es can on ly be app lied to un ifo rm rectangu lar o r sm oo th cu rvilinear coo rdinateṡ A RECENT D EVELOPM ENT OF W ENO SCHEM ES In th is sect ion w e b riefly describe a recen t developm en t of W ENO schem es, nam ely and adap t ive app roach w ith in the fin ite d ifference fram ew o rk u sing sm oo th tim e dependen t cu rvi linear coo rdinateṡ A s w e can see from the p reviou s section, fi n ite difference W ENO schem es are easier to code and fa ster in CPU t im e than fin ite vo lum e W ENO schem es of the sam e o rder of accu racy fo r m u ltip le space dim en sion ṡ Its m esh adap tivity is needed to concen trate po in ts near the region s of the so lu tion w ith m o re structu res, w e cou ld u ti lize sm oo th tim e dependen t cu rvilinear coo rdi nates as long as the m esh tran sfo rm ation from a un ifo rm com putational m esh to a nonun ifo rm, tim e dependen t, cu rvilinear physical dom ain m esh has as m any sm oo th derivatives as the o r der of the schem e. T he exam p le dem on strates that th is capab il ity of fin ite difference W ENO schem e is the dou b le M ach reflection p rob lem, o riginally given in R ef. [ ] and later u sed often in the literatu re as a benchm ark. T he com puational dom ain is [0, 4 ] [ 0, ], although typ ically on ly the resu lts in [ 0, 3 ] [ 0, ] are show n in the figu reṡ T he re boundary, the flow values are set to describe the exact m o tion of a M ach 0 shock. T he com puta tion is carried ou t to t= 0.. A t a very refined reso lu t ion, the slip line induced in stab ility and ro llup can be ob served. T he capab ility of a nu m erical m ethod to sim u late these ro llup s is an indication of its sm all num erical visco sity and h igh reso lu tion. In F ig., left, w e give the den sity con tou rs of the sim u lation resu lt w ith the fifth o rder W ENO schem e on a fixed, un ifo rm m esh w ith x = y = g960. In F ig. (righ t), w e give the den sity con tou rs of the sim u lation resu lt w ith the fifth o rderw ENO schem e on a nonun i fo rm and m oving m esh, w h ich is sm oo th and concen trates its po in ts near the shock and the re gion under the doub le M ach stem, w ith on ly half the num ber of po in t s in each d irect ion ( 480 po in ts in y ). T he m esh m ovem en ts w ere deter m ined by a given sm oo th tran sfo rm ation w h ich fo llow s the structu re of the so lu tion. F ig. gives a zoom ed in p ictu re. W e can clearly see that the reso lu tion s are com parab le w h ile the m oving nonun ifo rm m esh version u ses on ly g4 as m any D m esh po in ts as the un ifo rm one, hence saving a lo t of com putational effo rṫ flecting w all lies at the bo ttom, starting from x = g6. In itially a righ tm oving M ach 0 shock is po sitioned at x = g6, y = 0 and m akes a 60 angle w ith the x ax iṡ Fo r the bo ttom boundary, the exact po stshock cond it ion is im po sed fo r the part from x = 0 to x = g6 and a reflective bound a ry cond it ion is u sed fo r the resṫ A t the top 995-005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. F ig. D ensity contours, double m ach reflection, fifth o rder fin ite differencew ENO schem e. L eft: u nifo rm m esh w ith x = y = ; R igh t: non 960 unifo rm moving m esh w ith g4 as m any D m esh po ints (480 po ints in y )
0 T ransactions of N anjing U niversity of A eronautics & A stronautics V o l. 8 F ig. A zoom ed in version of the density REFERENCES con tou rs, doub le M ach reflection, fifth o rder fin ite difference W ENO schem e. L eft: unifo rm m esh w ith x = y = ; R igh t: nonunifo rm mov 960 ing m esh w ith g4 as m any D m esh po ints (480 po ints in y ) L iu X D, Isher S, Chan T. W eigh ted essentially non o scillato ry schem eṡ J Comput Phys, 994, 5: 00 J iang G, Shu C W. Efficient imp lem entation of w eigh t ed ENO schem eṡ J Comput Phys, 996, 50: 0 8 3 H u C, Shu C W. W eigh ted essentially nono scillato ry schem es on triangular m esheṡ J Comput Phys, 999, 50: 97 7 4 Balasar D, Shu C W. M ono tonicity p reserving w eigh t ed essentially nono scillato ry schem es w ith increasing ly h igh o rder of accuracy. J Comput Phys, 000, 60: 405 45 5 L evy D, Puppo G, R usso G. Compact centralw ENO schem es fo r m u ltidim en sional con servation law ṡ S IAM J Sci Comput, 000, : 656 67 6 H arten A, Engquist B, O sher S, et al. U nifo rm ly h igh o rder essen tially nono scillato ry schem es, III J Comput Phys, 987, 7: 3 303 7 Shu C W, O sher S. Efficient imp lem entation of essen tially nono scillato ry shock cap tu ring schem eṡ II J Comput Phys, 989, 83: 3 78 8 Shu C W. E ssentially nono scillato ry and w eigh ted es sen tially nono scillato ry schem es fo r hyperbo lic con servation law ṡ In: A dvanced N um erical A pp roxim a tion of N onlinear H yperbo lic Equations, B Coclburn, C Johnson, C W Shu, et al. (Edito r: A. Q uarteroni), L ecture N o tes in M athem atics, vo lum e 697, 998. 35 43 9 Shu C W. H igh o rder ENO and W ENO schem es fo r com pu tational flu id dynam icṡ In: H igho rder M eth ods fo r Computational Physicṡ T J Barth, H D econ inck, edito rs, L ecture N o tes in Computational Science and Engineering, 999, 9: 439 58 0 W oodard P, Co lella P. T he num erical sim ulation of tw odim en sional flu id flow w ith strong shock ṡ J Comput Phys, 984, 54: 5 73 995-005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.