Conservative and Easily Implemented Finite Volume Semi-Lagrangian WENO Methods for 1D and 2D Hyperbolic Conservation Laws

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1 Joural of Appled Mahemacs ad Physcs, 07, 5, 59-8 hp:// ISSN Ole: ISSN Pr: Coservave ad Easly Implemeed Fe Volume Sem-Lagraga WENO Mehods for D ad D Hyperbolc Coservao Laws Fuxg Hu Deparme of Mahemacs, Huzhou Uversy, Huzhou, Cha How o ce hs paper: Hu, F.X. (07) Coservave ad Easly Implemeed Fe Volume Sem-Lagraga WENO Mehods for D ad D Hyperbolc Coservao Laws. Joural of Appled Mahemacs ad Physcs, 5, hp://dx.do.org/0.436/amp Receved: December, 06 Acceped: Jauary 6, 07 Publshed: Jauary 9, 07 Copyrgh 07 by auhor ad Scefc Research Publshg Ic. Ths work s lcesed uder he Creave Commos Arbuo Ieraoal Lcese (CC BY 4.0). hp://creavecommos.org/lceses/by/4.0/ Ope Access Absrac The paper s devsed o propose fe volume sem-lagrage scheme for approxmag lear ad olear hyperbolc coservao laws. Based o he dea of sem-lagraga scheme, we rasform he egrao of flux me o he egrao space. Compared wh he radoal sem-lagrage scheme, he scheme devsed here res o drecly evaluae he average fluxes alog cell edges. I s hs dfferece ha makes he scheme hs paper smple o mpleme ad easly exed o olear cases. The procedure of evaluao of he average fluxes oly depeds o he hgh-order spaal erpolao. Hece he scheme ca be mplemeed as log as he spaal erpolao s avalable, ad o addoal emporal dscrezao s eeded. I hs paper, he hgh-order spaal dscrezao s chose o be he classcal 5horder weghed esseally o-oscllaory spaal erpolao. I he ed, D ad D umercal resuls show ha hs mehod s raher robus. I addo, o exhb he umercal resoluo ad effcecy of he proposed scheme, he umercal soluos of he classcal 5h-order WENO scheme combed wh he 3rd-order Ruge-Kua emporal dscrezao (WENOJS) are chose as he referece. We fd ha he scheme proposed he paper geeraes comparable soluos wh ha of WENOJS, bu wh less CPU me. Keywords Sem-Lagraga Mehod, Average Flux, WENO Scheme, Hgh-Order Scheme, Hyperbolc Coservao Laws. Iroduco The characersc of hyperbolc coservao laws s ha hey ca geerae dscouous soluos eve f he al codo s smooh. I s hs propery DOI: 0.436/amp Jauary 9, 07

2 ha leads o he challege developg he hgh-order umercal mehods. The ma dffculy of desgg hgh-order mehods s how o maa he hghorder accuracy aroud he smooh regos ad meawhle suppress he spurous oscllaos aroud he regos wh large grades. I he pas few decades, may hgh-order umercal mehods were developed o evolve he hyperbolc problem, e.g. he esseally o-oscllaory schemes (ENO) [] [] [3] [4], he weghed esseally o-oscllaory schemes (WENO) [5] [6] ad he dscouous Galerk fe eleme mehod [7] [8], ad so o. The dscrezao procedure of he classcal WENO schemes [5] ca be dvded o wo sages. Frs, hrough spaal dscrezao mplemeed by WENO recosruco, leavg he equaos couous me; hs leads o a sysem of ordary dffereal equaos me. The, ay umercal mehods for ordary dffereal equaos, e.g. Ruge-Kua mehods, ca be used o evolve hs sysem. I order o make he soluo sable oal varao orm, Shu ad Osher [3] ad Shu [9] devsed a se of TVD Ruge-Kua mehods as ODE solvers. Takg hree-sage TVD Ruge-Kua mehods for example, hree WENO recosruco processes are eeded o evolve oe me sep. Ad hey also suffer from a CFL me sep sably resrco. To fx he problem above, he sem-lagraga mehods [0] [] [] [3] avod he me dscrezao by characersc racg echque ad have more allevave resrco for CFL umber. Recely, he papers [4] [5] [6] [7], he auhors combed he dea of sem-lagraga wh WENO recosruco for solvg adveco problems. The sem-lagraga mehods above deped o he characersc racg. However, for geerally olear cases, s mpossble o fd he race pos exacly (eve fdg he race pos wh hgh accuracy s very hard). I s hs resrco ha makes he sem-lagraga mehods very hard o apply o olear problems. The auhors Huag ad Arbogas [8] devsed a sem-lagraga mehod for olear coservao laws. However, a cosly flux correco s eeded for keepg hgh accuracy ad umercal sably. I hs paper, we ry o drecly evaluae he average flux ( ( )) F = f u x, d, () where s he me sep ad x s he mesh boudary. The ma dea of evaluao of he average fluxes s he rasformao of he egrao of flux fuco me o he egrao space. A smlar rasformao had appeared [6]. The compuao for egrao space oly depeds o he spaal erpolao polyomal. Hece, he scheme here ca be mplemeed easly as log as he procedure of spaal erpolao s avalable. Several hgh-order polyomal erpolaos ca be chose, e.g. he PPM [9], ENO [] [] or WENO [5] [6], ec. I hs paper, we choose he classcal 5h-order WENO erpolao o recosruc he polyomal space ad we call he scheme here as WENOEL. I s oed ha he scheme s obvously coservave sce we drecly evaluae he average flux a each cell edge. As he sem-lagraga mehods preseed leraure, we also eed he rackg back pos alog cha- 60

3 racersc les o evaluae he average flux. Sce he characersc pos ca be racked back exacly (or wh hgh-order accuracy) oly for some smple lear equaos, we do o ry o fd he characersc pos wh hgh-order accuracy. For lear problems, he scheme ca acheve opmal accuracy because he average flux ca be evaluaed wh hgh order of accuracy. For olear problems, esg he order of accuracy for smooh soluo, o opmal accuracy ca be guaraeed. For esg he resoluo olear problems of he scheme, we choose he classcal WENOJS (he 5h-order WENO recosruco wh 3rd-order Ruge-Kua me dscrezao) as bechmark ad compare hem for several D ad D examples. We fd ha he scheme proposed he paper geeraes comparable soluos wh ha of WENOJS bu eeds less ha half compug me. The free CFL resrco s aoher advaage of sem-lagraga mehods. For adveco problems, he CFL umber ca be chose geerally larger ha. However, for olear hyperbolc coservao laws, he CFL > wll lead o umercal sably geerally f here exs shocks. I hs paper, he CFL = 0.6 s chose for he WENOEL scheme all he umercal examples. I he paper ha follows, we wll prese he evaluao procedure of he average flux Seco deal. I Seco 3, o show he hgh resoluo ad effcecy of our scheme, we compare he schemes WENOEL proposed here ad WENOJS for several D ad D examples. Ad from he comparsos of resoluo ad effcecy for hese D ad D examples, we ca fd ha he proposed mehod hs paper ca prese comparable resuls wh he classcal WENOJS scheme, bu wh less CPU me.. The Fe Volume WENOEL Scheme Cosder he coservao laws subec o al codo x( ) u x, f u x, = 0, a x b, 0 () (,0) = u x u x wh proper boudary codos. Iegrag he Equao () o corol volume x, x [, ] gves x = x u x, d x u x, dx x x Rearragg hs equao ad dvdg by x x u( x, ) d (, ) d x x= u x x x x x x Deoe he h cell average by 0 ( ( )) ( ( )) f u x, d f u x, d. x yelds ( f ( u( x, ) ) d f ( u( x, ) ) d ) (3) 6

4 ad average flux a cell edge x by he (3) ca be wre as The value x u = u( x, ) d, x x x ( ( )) f = f u x, d, (4) u = u f f x ( ). U wll approxmae he average value U (5) u. Gve he approx- maed cell average U, he formula (5) ells us ha a ex me sep ca be updaed f he me egrals o he rgh of (4) ca be evaluaed effcely. Deog he approxmaed flux fuco f by F he we oba.. The Lear Equao ( ( )) F f u x, d, U = U F F x ( ). Here, we maly cosder he 5h-order fe volume verso because of s populary ad oher cases ca be obaed smlarly. Cosder he lear adveco equao u x, cu x, = 0, a x b, 0. (7) =. For hs equao, he - u x smply propagaes rgh f c > 0 (or lef f c < 0 ) wh I hs case, he flux fuco f( u( x, )) cu( x, ) al codo 0 uchaged shape. x For evaluag he average flux f (4), we ca frsly apply a 5h-order N o oba - recosruco based o pecewse cosa average fluxes { f k } k= erpolao polyomal P ( x ) o cell I 5 ( ) = ( ) (6) f u x, P x x. (8) The 5h-order polyomal P ( x ) s recosruced over he secl S = { I, I, I, I, I }. Here, we assume he adveco velocy 0 o he rgh wh velocy c, whch gves for ay me [, ] f( u( x, )) a cell edge x s c >, he u x smply spreads, ( (, )) ( (, ) ) f u x = f u x c (9). Combg he Formulas (8) ad (9), he flow rae 5 ( ) = ( ( ) ) ( ) f u x, P x c x. (0) I hs case, he average flux f ca be expressed approxmaely as 6

5 ( ( )) f = f u x, d = P x c x x 5 = P ( x) d x ( x ). c x c 5 Omg he hgh-order erm ( x ) 5 ( ) d (), we oba he umercal flux F x P ( x ) d. x = c () x c The las equaly () s obaed by egrao of subsuo x = x c( ). From he Equao (), he egrao me [, ] s rasformed o egrao space x c x,. Due o he egrad P ( x ) s recosruco polyomal, he las egrao () ca be compued exacly. Remark. Ths rasformao s he ma dea of hs paper. Ad s smlar o he equao (3.9) [6] bu wh a slghly dffere way. The ma dfferece s ha he operao [6] rasforms he flux egrao me a cell ceer o he egrao of u( x, ) space x ( ( )) = f u x, d u x, d, x where x s he backward characersc po of cell ceer x. Ad here we rasform he flux egrao me a cell edge o he egrao of erpolao polyomal of flux fuco space. Ths dfferece makes he exeso of he scheme o olear cases much easer. I he followg of hs subseco, we wll prese he 5h-order WENO recosruco process o approxmae he egral () x x P x d. x x c The 5h-order WENO recosruco procedure s represeed as he covex combao of hree 3rd-order recosrucos. Frs, we ed o recosruc he 3rd-order coservave polyomals o cell I = x, x based o he pecewse cosa average fluxes. { f k }, k =,,. I s oed ha here are 3 hree-po secls coaed he cell I ca be used o recosruc he polyomals. We deoe hese secls respecvely by { } S = I, I, I, 0 { } S = I, I, I, { } S = I, I, I. The correspodg polyomal of each secl s P x = b0 bx bx, = 0,,, (3) where he subscrp deoes he polyomal o cell I ad superscrp 63

6 deoes he recosruco based o secl gve by (for smplcy, we om superscrp b 0 f 7 f f = 6 ( ) f f f x x f 3f f S. The coeffces (3) are f ) ( 3 ) ( ) = b x f f f = b x, f f f x, x f f f x where = 0,, ad x = x x s he legh of cell I. So far, we have P x o each secl x obaed he coservave erpolao polyomals S. I he ed, he egral () ca be expressed as where he lear opmal weghs x x = x c x c = 0 P x dx d P x d, x d are 0 3 d = λ λ, d = λ λ, d = λ λ, c where λ =. For allevag he effec of he o-smooh secls, he olear weghs ca be cosruced as x follows ad The w a a = a a a = 0 d. ε β β s he dcaor of smoohess based o he secl 0 β = S, 3 ( f f f ) ( 3f 4 f f ), 4 3 ( f f f ) ( f f ), 4 β = 3 ( f f f ) ( f 4f 3 f ). 4 β = Fally, he umercal flux should be expressed as F w P x x c x c = 0 Subsug he Formulas (3) o (6) gves, (4) (5) = x d. (6) 64

7 F w P x dx x = x c c = 0 0 ( λ ( ) λ( 3 3 ) ( 5 ) ) ( λ ( ) λ( 3 3 ) ( 5 ) ) ( λ ( ) λ( ) ( )) = w f f f f f f f f 6 w f f f f f f f f w f f f 3 f 9 f 6 f f 7 f f. I coras, whe he adveco velocy 0 wh velocy c, whch gves ad he flow rae c <,, f ( u( x, ) ) f ( u( x c( ), ) ) f( u x, ) a cell edge x s 5 ( ( )) = ( ( ) ) ( ) (7) u x spreads o he lef = (8) f u x, P x c x. (9) Smlarly, he average flux F ca be expressed as F w x c P ( x ) d. x c x = (0) = 0 The olear wegh w ca be cosruced smlarly as (4) ad (5). The lear opmal weghs are 0 3 d = λ λ, d = λ λ, d = λ λ Subsug he formula (3) o (0) gves ( λ ( ) λ( ) ( 7 )) ( λ ( ) λ( 3 3 ) ( 5 )) ( λ ( ) λ( 3 3 ) ( 5 )). F = w f f f f f f f f f 6 w f f f f f f f f w f f f f f f f f The scheme proposed he paper s coservave for lear ad olear cases. The propery of coservao s obvous sce he scheme (6) s represeed coservao form. Remark. I [6], he auhors Qu ad Shu preseed a sem-lagraga fe dfferece WENO mehod for adveco compressble flow. Acually, here are wo recosruco procedures eeded o oba he umercal flux a cell edge x. I s oed ha, for lear equao wh cosa coeffce, he formulas of average flux (7) ad () are oally same as oes [6]. However, hese wo mehods are devsed by oally dffere procedure. They maly have he followg dffereces: he mehod prese here s fe volume; our mehod ca be easly exeded o olear cases by freezg locally ad dealed descrpo s gve Subseco.3; here are always exs lear opmal weghs o cosruc he WENO schemes for our scheme whch s o sasfed he schemes [6]. () 65

8 .. Algorhm Here, we coclude he algorhm for compug umercal flux F a x = x durg a me sep =. ) Dsgush he propagao dreco of soluo a x = x used Rake-Hugoo ump codo or c = f ( u ) f u u u ) The propagao velocy ca be chose o be f u c = f c < 0. u= u u. f u c = u= u u f c > 0, 3) Iser he propagao velocy o (7) or () o ge umercal flux F..3. The Nolear Equaos I hs subseco, we wll exe he mehod above o olear case. Frsly, s oed ha for olear case he mehod here cao approach he opmal accuracy sce he soluo o loger smply raslaes uformly. Isead, deforms as evolves ad he characersc curves are o loger parallel sragh les, whch lead o he evaluao of average fluxes s hard o oba. Ad geerally he rackg back pos cao be foud exacly (eve cao fd he pos wh hgh accuracy). Hece for olear case, raher ha ryg o fd he rackg back pos, we freeze he equao o lear formao locally ad apply he procedure Subseco. o. For solvg he olear case, he propagao dreco s frsly dsgushed by Rake-Hugoo ump f ( u) codos ad propagao velocy s chose o be c = u= u u f ( u) c f propagao dreco < 0. ( f propagao dreco > 0) or = u= u u Remark 3. The sem-lagraga fe volume mehod proposed here s largely depede o he recosruco from he cell flux average x f = f( u( x, ) ) d. x x x For fe volume mehod, oly he cell average x u = u( x, ) dx x x s avalable. For lear problem, f( u( x, )) au( x, ) be apparely chose o be f = au =, he cell flux average ca However, for olear problem, he cell flux average mus be compued by some umercal quadraure. I hs paper, we choose Gauss quadraure o evaluae he cell flux average 66

9 x = = x k k x x k= k= 3 g g ( ) f f u x, d x A f u x,, g g where A k ad x k are he Gauss weghs ad Gauss pos, respecvely. For g smple, he value of u( x, ) a Gauss po x k ca be obaed by he 5horder Lagraga erpolao. I addo, for solvg Euler equaos, he Roe speed s used o defy he upwd dreco whch may lead o some umercal sables, especally for D cases. I Subseco 3.3, compug he umercal ess: double Mach refleco ad Mach 3 wd uel wh a sep, here deed exs umercal sables. I addo, s oed ha hs umercal sably o oly appears he WENOEL scheme, bu also appears he WENOJS scheme. I s sad ha he umercal sably s maly due o he choce of Roe speed. The dealed roducos for he umercal sables of upwd schemes are referred o [0] [] []. I order o elmae such kd of sables, he H-correco procedure [] s ake o solve hese flaws. For hese wo D problems, he umercal flux F, a he cell edge ( x, y) s compued as follows. [] ) If m ( λ,, λ, ) η,, he he umercal flux F, ca be chose o be (7) or () ad he average fluxes f k hese wo equaos are correspodgly subsued by fk, ( k =,, 3). The λ, s he egevalue ad he value η, s deermed by x y y y y η = max η, η, η, η, η, where,,,,,, x η, = u, u, a, a,, = y η,, v, v, a, a, ad a, s he speed of soud. ) Oherwse, a more dsspave Lax-Fredrch flux splg mehod s used o spl he flux f k, o posve ad egave fluxes f k, = ( fk, αuk, ), f k, = ( fk, αuk, ). For he posve ad egave fluxes, he propagao veloces are chose o be c = ( λk, α), c = ( λk, α). Iserg he f k,, c ad f k,, c o (7) ad (), respecvely, we ca ge he posve ad egave umercal fluxes F, ad F,. Fally, he umercal flux a cell edge ( x, y) s F = F F,,,. 67

10 3. Numercal Resuls I hs seco, we wll apply hs mehod o D ad D hyperbolc problems. A prese, he Srag dmesoal splg mehod [3] [4] s appled o D problems, whch s used o spl he D equao o wo D equao. I each coordae dreco, he equao s evolved by he proposed D WENOEL mehod. The dealed comparsos of resoluo ad effcecy are show bewee our mehod ad WENOJS mehod (5h-order spaal ad 3rd-order TVD RK emporal dscrezao). I he ed, s oed ha, he WENOJS mehod, he upwdg (o flux vecor splg) s used o cosruc he umercal flux. So he WENOJS mehod hs paper ca also be called by WENOJS-Roe. The dealed descrpo of WENOJS-Roe s referred o he procedure.4 o page 33 [5]. For he choce of CFL umber, I s foud ha whe 0 < CFL < he WENOEL scheme s umercally sable all he examples we have esed. Especally, we had chose CFL = 0.9 o smulae all he examples appeared hs paper ad o usable pheomeo had bee foud. For comparg he resoluo ad compug me bewee he WENOEL ad WENOJS schemes, we choose CFL = 0.6 for hese wo schemes for all he ess coaed he dscoues. Ths choce of CFL umber for he WENOJS scheme s also adoped may classcal leraures, e.g. [5]. For he problem () (3), whch s used o es he 53 order of accuracy of schemes, we choose = x for WENOJS order ha spaal error s doma bu sll le CFL = 0.6 for WENOEL. 3.. D Scalar Examples I hs subseco, we wll cosder lear adveco, Burger s ad Buckley-Levere equaos. For he lear adveco equao, wo al codos are used o es he schemes. We use a smooh al codo o es he order of accuracy of he scheme ad a codo coaed dscoues ad hgh-frequecy waves o es he performace of he scheme smulag dscouous ad large-grade soluo. For he Burger equao, we also cosder wo al codos whch commoly used o valdae he proposed scheme. For he ocovex Buckley-Levere equao, we sudy a smple model for wo-phase flud flow a porous medum. Excep for he comparso of resoluo bewee he WENOJS ad WENOEL schemes, he compug me s also show hs subseco Lear Adveco Equao Cosder he lear adveco equao u x, u x, = 0, x, 0, () x subec o wo al codos ad perodc boudary codos. Oe al codo s u x,0 = s π x, (3) 68

11 ad he oher s ( ( β δ) ( β δ) ( β )) G x,, z G x,, z 4 G x,, z x 0.6, 0.4 x 0., u( x,0) = 0x 0.0 x 0., ( F( x, α, a δ) F( x, α, a δ) 4 F( x, α, a) ) x 0.6, 0 oherwse, where G( x, β, z) = exp β( x z), F( x α a) ( α( x a) ) ad he parameers are gve by (4),, = max,0, l a = 0.5, z = 0.7, δ = 0.005, α = 0, δ =. 36δ The problem () (3) here s used o es he accuracy of our mehod lear case. The Table ad Table prese he comparso of errors ad accuracy bewee he WENOEL ad WENOJS mehods. The oupu me s chose o be = ad he me seps are chose o be = 0.6 x ad = ( x) 53 for he WENOEL ad WENOJS, respecvely. As he classcal WENOJS mehod, he WENOEL also arrves 5h-order of accuracy. For he l ad l errors, he WENOEL mehod shows much beer resuls ha he WENOJS mehod. Ths Table. The es of order of accuracy for he l ad l errors for al value problem () (3) wh WENOEL mehod. The me sep s chose o be = 0.6 x ad fal me s =. WENOEL N l r l r e e e e e e e e e e e e Table. The es of order of accuracy for he l ad l errors for al value problem () (3) wh WENOJS mehod. The me sep s chose o be = ( x) 53 ad fal me s =. WENOJS N l r l r e e e e e e e e e e e e

12 resul s expeced sce he rackg po ca be foud exacly ad he umercal flux s obaed much accuraely. The soluo of () (4) coas a dscouous square pulse ad several couous bu hgh-grade profles. Ths problem s coveoally employed o es hgh-resoluo schemes. The oupu me s chose o be = ad he me sep s chose o be = 0.6 x for he WENOEL ad WENOJS schemes. Fgure gves he soluos of he WENOEL ad WENOJS mehods wh cells N = 00. From Fgure, we ca fd ha he WENOEL ad WENOJS mehods ca boh capure he dscouous ad seep soluos. Ad for he frs ad hrd hgh-frequecy waves, he WENOEL performs slghly beer ha WENOJS mehod. Fgure s used o es he log-me behavor of hese wo mehods Fgure. The umercal soluos of () (4) s compued by WENOEL ad WENOJS mehods wh N = 00 ul me =. The blue ad red deoe he soluos of WENOEL ad WENOJS mehods, respecvely. Fgure. The umercal soluos of () (4) s compued by WENOEL ad WENOJS mehods wh N = 00 ul me = 0. The blue ad red deoe he soluos of WENOEL ad WENOJS mehods, respecvely. 70

13 afer 0 cycles,.e. = 0. Obvously, followg log-me evoluo, he WENOEL gves hgher-resoluo soluo aroud four waves ha ha of WENOJS mehod. I addo, we prese he comparso of CPU me ( secods) ad l error for hs problem bewee he WENOEL ad WENOJS mehods Burger Equao I hs subseco, we cosder he Burger s equao u u = 0, x, 0 subec o wo al codos. oe al codo s ad he oher s (,0) u x x x 0, = 0 0 x, ( x) (5) (6) u x,0 =.5 s π. (7) For he olear scalar equao, we frs use Rake-Hugoo ump codo o defy he upwdg dreco, he compue he average flux by (7) or (). I addo, a eropy fx s used o modfy he average flux whe rarefaco wave appears. For he problems (5) (6) ad (5) (7), we boh se 0.6 x f = for he WENOEL ad WENOJS schemes, where λ = max λ N u. For he problem (5) (6), he umercal soluos are compued wh cells N = 00 ul me =. From Fgure 3, he soluo of hs problem coas a rarefaco ad a shock, ad hese wo mehods arrve almos same resoluo. I Fgure 4, he problem (5) (7) has he same se bu he oupu me = 0.37, ad has he same cocluso wh las problem. I a word, for olear Burger s Fgure 3. The umercal soluos of (5) (6) are compued by he WENOEL ad WENOJS mehods wh N = 00 ul me =. The blue ad red deoe he soluos of WENOEL ad WENOJS mehods, respecvely. 7

14 Fgure 4. The umercal soluos of (5) (7) are compued by he WENOEL ad WENOJS mehods wh N = 00 ul me = The blue ad red deoe he soluos of WENOEL ad WENOJS mehods, respecvely. equao, he WENOEL scheme ca also oba comparable umercal resuls ad oly eed half CPU me. The preseao of CPU me s omed for savg space Buckley-Levere Equao As a example where ocovex flux fucos arse, we sudy a smple model for wo-phase flud flow a porous medum. Cosder he scalar coservao law u u = 0, x, 0, (8) u ( u) 4 x wh he al codo 0.5 x 0, u( x,0) = (9) 0 oher. Fgure 5 shows he soluos compued by he WENOEL ad WENOJS mehods wh cells N = 00 ul me = 0.4. Also, we choose he me seps 0.6 x f = for WENOEL ad WENOJS schemes, where λ = max λ N u. From Fgure 5 he WENOEL mehod geeraes he soluo whch s comparable wh ha of he WENOJS mehod. For he ocovex flux problem, he WENOEL scheme ca also have a robus performace as WENOJS scheme. 3.. The D Euler Equao I hs subseco, we cosder D Euler equaos sce oe of he ma applcao areas of hgh-resoluo scheme s compressble gas dyamcs, 7

15 Fgure 5. The umercal soluos of (8) (9) are compued by he WENOEL ad WENOJS mehods wh N = 00 ul me = 0.4. The blue ad red deoe he soluos of WENOEL ad WENOJS mehods, respecvely. ρ ρu E E p u ρu ρu p = ( ) x 0, (30) where ρ, u, p, E are desy, velocy, pressure ad oal eergy, respecvely. The sysem of equaos s closed by he equao of sae for a deal polyropc gas: p E = ρu, γ where he rao of specfc heas γ =.4. The followg hree al codos combed wh Euler Equaos (30) are cosdered, whch are ofe used o exame he mehods for solvg Euler equaos: ( ρ, u, p) ( ρ, v, p) 0.445, 0.698, 3.58, f 5 x < 0, = 0.5, 0, 0.57, f 0 x 5, ,,, f 5 x < 4, = ( 0. s ( 5 x), 0, ), f 4 x 5. ( ρ u p), 0,000, f 5 x < 4,,, =, 0, 0.0, f 4 x< 4,, 0,00, f 4 x 5, For solvg Euler equaos, Roe learzao (.e. Roe average) s used o locally freeze a olear sysem o a lear sysem. The hs sysem s decoupled o hree adveco equaos ad each equao s solved by procedure (3) (3) (33) 73

16 Seco 0. I s oed ha he propagao dreco of soluo of each adveco equao s dsgushed by correspodg egevalue of Roe marx. For he problem (30) (3), called Lax problem [6], we solve wh = 0.6 x λ for he WENOEL ad WENOJS mehods. Fgure 6 s compued wh cells N = 00 ad oupu me =.3. Apparely, he WENOEL mehod performs almos he same as WENOJS mehod ad preses a robus soluo for hs shock-ube problem. The exac soluo s obaed by he solver preseed [4]. The problem (30) (3), called shock eropy wave eraco [4], s usually used o es he hgh-order mehods for capurg he hgh-frequecy waves. The CFL umber s also se o be = 0.6 x λ for he WENOEL ad WENOJS mehods. We evolve he equaos up o =.8 wh cells N = 400. For hs problem, a Mach 3 shock wave moved rgh eracs wh se waves a desy whch lead o a feld obaed smooh ad dscouous srucures. The referece soluo Fgure 7 s compued by WENOJS mehod wh cells N = 000. I s plaly see from Fgure 7 ha, as WENOJS, he WENOEL mehod performs wh he hgh resoluo. Furhermore, from Fgure 8, we ca fd ha he par of hgh-frequecy waves he WENOEL mehod performs less dsspave ha he WENOJS mehod. The problem (30) (33) was orgally proposed as a bechmark for esg several umercal mehods by Woodward ad Colella [9]. The reflecve boudary codos are appled o boh boudares. The CFL umber s also se o be = 0.6 x λ for he WENOEL ad WENOJS mehods. We evolve he equaos up o = 0.38 wh cells N = 400. The referece soluo Fgure 9 s compued by WENOJS mehod wh cells N = 000. As he las wo problems, he WENOEL mehod gves almos he same resoluo as WENOJS mehod. Fgure 6. The shock-ube problem (30) (3) s compued by he WENOEL ad WENOJS mehods wh cells N = 00 ul me =.3. The blue ad red deoe he soluos of WENOEL ad WENOJS mehods, respecvely. 74

17 Fgure 7. The shock eropy wave problem (30) (3) s compued by he WENOEL ad WENOJS mehods wh cells N = 400 ul me =.8. The blue ad red deoe he soluos of WENOEL ad WENOJS mehods, respecvely. Fgure 8. The zoomed fgure of Fgure 7 aroud he hgh-frequecy waves The D Euler Sysem Fally, we cosder a umercal experme for D Euler equaos for gas dyamcs, ρ ρu ρv ρu ρu p ρuv = 0, ρv ρuv ρv p E ( E p) u ( E p) v x y where he equao of sae s p = ( γ ) E ρ( u v ), γ =.4. (34) 75

18 Fgure 9. The blas wave problem (30) (33) s compued by he WENOEL ad WENOJS mehods wh cells N = 400 ul me = The blue ad red deoe he soluos of WENOEL ad WENOJS mehods, respecvely Double Mach Refleco Here we frsly apply he WENOEL ad WENOJS schemes o he D double- Mach shock refleco problem where a srog vercal shock moves horzoally o a wedge whch cled wh some agle wh he rao of specfc heas γ =.4. Ially, hs problem was proposed by Woodward ad Colella [9] ad had bee ake exesvely as a es example for hgh-order schemes. The 0, 4 0, ad he reflecve wall les compuaoal doma s chose o be [ ] [ ] o he boom of he compuaoal doma for x 4. I he begg, a 6 Mach 0 shock, movg rgh, s locaed a x =, y = 0 ad makes a agle 6 60 wh he x -axs. For he boudary codos, he exac posshock codo s mposed for boom boudary from x = 0 o x =, ad he 6 reflecve boudary codo s mposed for he res; he flows are mposed o he op boudary such ha here s o eraco wh he Mach 0 shock; flow ad ouflow boudary codos are se for he lef ad rgh boudares respecvely. The ushocked flud has a desy of.4, a pressure of ad hs problem s ru ul = 0. wh cells I addo, here exss umercal sably compug hs problem by uwd mehods. A H-correco procedure preseed Subseco.3 s used o elmae hs sably. The H-correco procedure ecs much dsspao usable rego by he way of shfg he upwd scheme o flux splg scheme. The coour les of he desy are show Fgure 0. These wo mehods boh oba hgh-resoluo soluos. I rego of complcaed srucure, he WENOJS scheme performs beer ha our scheme. 76

19 Fgure 0. Double Mach problem. The desy ρ s show wh meshes equally spaced coour les are ploed from.73 o The op: WENOEL scheme. The boom: WENOJS scheme Mach 3 Tuel wh a Sep Ths wd problem s also orgally preseed [9]. Ths problem s se up as follows. The wd uel s legh u wde ad 3 legh us log. The sep s 0. legh us hgh ad s locaed 0.6 legh us from he lef-had ed of he uel. The problem s alzed by a uform rgh-gog Mach 3 flow. Reflecve boudary codos are appled alog he wall of he uel. The flow ad ouflow boudary codos are appled a he erace ad ex, respecvely. The corer of s a sgular po ad we use he same echque o correc [9], whch s based o he assumpo of a early seady flow he rego ear he corer. I addo, o elmae he umercal sably orgaed from he upwd scheme, he same H-correco procedure as las problem s used o elmae hs flaw. We evolve he al daa ul me 4 wh a grd of cells by he WENOEL ad WENOJS mehods. The CFL umber s chose o be 0.6 for hese mehods. The coour les of he desy are dsplayed Fgure. We observe he good resoluo ad srog 77

20 Fgure. Mach 3 uel wh a sep problem. The desy ρ s show wh meshes equally spaced coour les are ploed from 0. o 6.4. The op: WENOEL scheme. The boom: WENOJS scheme. reflecve waves hs es for hese mehods. Moreover, a beer resoluo ca be observed for he WENOJS mehod D Rema Problem Ths s a smple D Rema problem o square [ 0,] [ 0,] s used o es our mehod whch s orgally proposed [7]. The square s dvded o four quadras by sragh les 7x = 0.8 ad y = 0.8. Ially, four dffere cosa saes are defed o each of quadras ( ρ, uvp,, ) 0.533,.06, 0, 0.3, f 0 x< 0.8, 0.8 y,.5, 0, 0,.5, f 0.8 x, 0.8 y, = 0.38,.06,.06, 0.09, f 0 x< 0.8, 0 y < 0.8, 0.533, 0,.06, 0.3, f 0.8 < x, 0 y < 0.8. (35) We solve he problem ul = 0.8 wh cells The me seps s se o be = 0.6 x λ for he WENOEL ad WENOJS mehods. Fgure gves he desy profles compued by he WENOEL ad WENOJS mehods, respecvely, ad hese wo hgh-resoluo scheme boh perform well Raylegh-Taylor Isably Problem Fally, we cosder Raylegh-Taylor sably problem [8]. Ths problem descrbes he flow moos o he erface bewee fluds wh dffere desy. The heavy flud moves o he rego of he lgh flud wh a fgerg aure, whch lead o he bubbles of lgh flud rsg o he heavy flud ad he spkes of heavy flud fallg o he lgh flud. The compuaoal doma s 78

21 Fgure. The soluos of D Rema problem compued by he WENOEL ad WENOJS mehods. The lef: WENOEL scheme. The rgh: WENOJS scheme. 0, [ 0,] 4, ad he al codos s γ p, 0, 0.05 cos( 8π x), y, f 0 y, ρ < ( ρ, uvp,, ) = (36) γ p, 0, 0.05 cos( 8π x), y.5, f y, ρ 5 where he rao of specfc heas γ =. For he boudary codos, he 3 reflecve boudary codos are mposed o he lef ad rgh boudares; he flows are se as ( ρ, uvp,, ) = (, 0, 0,.5) ad ( ρ, uvp,, ) = (, 0, 0,) for he op ad boom boudares, respecvely. I addo, he source erms ρ ad ρ v are added o he rgh of he hrd ad fourh equaos of Euler equaos (34), respecvely. For hs problem, from Fgure 3, we ca fd ha he WENOEL ad WENOJS mehods boh geerae hgh-resoluo umercal soluos. From he performace of hese four D problems, as he D problems, we ca fd ha he smlar resoluo wh he WENOJS scheme ca be observed. 4. Cocluso I hs paper, we propose a smple ad easly mplemeed mehod o solve hyperbolc coversao laws. The ma dea of hs mehod s he rasformao of egrao of flux fuco me o egrao space. I s hs procedure ha leads o he average flux a cell edge whch ca be drecly evaluaed. I addo, he evaluao of he average flux s easly mplemeed ad ca be combed wh ay o-oscllaory spaal recosruco. Through he performace o los of classcal examples, we ca fd hs mehod s raher robus. We compare our scheme wh he classcal WENOJS scheme ad almos he same performace o resoluo s observed. Ad from he comparsos of resoluo ad effcecy for hese D ad D examples, we ca fd ha he 79

22 Fgure 3. The soluos of Raylegh-Taylor sably problem compued by he WENOEL ad WENOJS mehods wh cells The lef: WENOEL scheme. The rgh: WENOJS scheme. proposed mehod hs paper ca prese comparable resuls wh he classcal WENOJS scheme, bu wh less CPU me. Ackowledgemes Ths work was suppored by he Naoal Naural Scece Foudao of Cha (No. 5038), Naural Scece Foudao of Guagdog Provce (No. 0A030339) ad Suppored by he Maor Proec Foudao of Guagdog Provce Educao Deparme (No. 04KZDXM070). Refereces [] Hare, A., Egqus, B., Osher, S. ad Chakravarhy, S. (987) Uformly Hgh Order Esseally No-Oscllaory Schemes III. Joural of Compuaoal Physcs, 7, hps://do.org/0.06/00-999(87) [] Hare, A. ad Osher, S. (987) Uformly Hgh Order Esseally No-Oscllaory Schemes I. SIAM Joural o Numercal Aalyss, 4, hps://do.org/0.37/0740 [3] Shu, C.W. ad Osher, S. (988) Effce Implemeao of Esseally No-Oscllaory shock-capurg Schemes. Joural of Compuaoal Physcs, 77, hps://do.org/0.06/00-999(88) [4] Shu, C.W. ad Osher, S. (989) Effce Implemeao of Esseally No-Oscllaory Shock-Capurg Schemes II. Joural of Compuaoal Physcs, 83, hps://do.org/0.06/00-999(89)90- [5] Jag, G.S. ad Shu, C.W. (996) Effce Implemeao of Weghed ENO Schemes. Joural of Compuaoal Physcs, 6, 0-8. hps://do.org/0.006/cph

23 [6] Lu, X.D., Osher, S. ad Cha, T. (994) Weghed Esseally No-Oscllaory Schemes. Joural of Compuaoal Physcs, 5, 00-. hps://do.org/0.006/cph [7] Cockbur, B. ad Shu, C.W. (998) The Ruge-Kua Dscouous Galerk Mehod for Coservao Laws V: Muldmesoal Sysems. Joural of Compuaoal Physcs, 4, hps://do.org/0.006/cph [8] Cockbur, B. ad Shu, C.W. (989) TVB Ruge-Kua Local Proeco Dscouous Galerk Fe Eleme Mehod for Coservao Laws II: Geeral Framework. Mahemacs of Compuao, 5, [9] Shu, C.W. (998) Esseally No-Oscllaory ad Weghed Esseally No-Oscllaory Schemes for Hyperbolc Coservao Laws. I: Cockbur, B., Shu, C.-W., Johso, C. ad Tadmor, E., Eds., Advaced Numercal Approxmao of Nolear Hyperbolc Equaos, Lecure Noes Mahemacs, Sprger, Berl, hps://do.org/0.007/bfb [0] Arbogas, T. ad Huag, C. (006) A Fully Mass ad Volume Coservg Implemeao of a Characersc Mehod for Traspor Problems. SIAM Joural o Scefc Compug, 8, hps://do.org/0.37/ [] Arbogas, T. ad Wheeler, M.F. (995) A Characerscs-Mxed Fe Eleme Mehod for Adveco-Domaed Traspor Problems. SIAM Joural o Numercal Aalyss, 3, hps://do.org/0.37/07307 [] Douglas, J. ad Russell, T.F. (98) Numercal Mehods for Coveco-Domaed Dffuso Problems Based o Combg he Mehod of Characerscs wh Fe Eleme or Fe Dfferece Procedures. SIAM Joural o Numercal Aalyss, 9, hps://do.org/0.37/ [3] Wag, H. ad Al-Lawaa, M. (006) A Locally Coservave Eulera-Lagraga Corol-Volume Mehod for Trase Adveco-Dffuso Equaos. Numercal Mehods for Paral Dffereal Equaos,, hps://do.org/0.00/um.006 [4] Huag, C., Arbogas, T. ad Qu, J. (0) A Eulera-Lagraga WENO Fe Volume Scheme for Adveco Problems. Joural of Compuaoal Physcs, 3, hps://do.org/0.06/.cp [5] Qu, J.M. ad Chrsleb, A. (00) A Coservave Hgh Order Sem-Lagraga WENO Mehod for he Vlasov Equao. Joural of Compuaoal Physcs, 9, hps://do.org/0.06/.cp [6] Qu, J.M. ad Shu, C.W. (0) Coservave Hgh Order Sem-Lagraga Fe Dfferece WENO Mehods for Adveco Icompressble Flow. Joural of Compuaoal Physcs, 30, hps://do.org/0.06/.cp [7] Qu, J.M. ad Shu, C.W. (0) Coservave Sem-Lagraga Fe Dfferece WENO Formulaos wh Applcaos o he Vlasov Equao. Commucaos Compuaoal Physcs, 0, hps://do.org/0.408/ccp a [8] Huag, C. ad Arbogas, T. (06) A Eulera-Lagraga WENO Scheme for Nolear Coservao Laws. Numercal Mehods for Paral Dffereal Equaos, I Press. hps://do.org/0.00/um.09 [9] Woodward, P. ad Colella, P. (984) The Numercal Smulao of Two-Dmesoal Flud Flow wh Srog Shocks. Joural of Compuaoal Physcs, 54, hps://do.org/0.06/00-999(84)904-6 [0] Km, S., Km, C., Rho, O. ad Hog, S. (003) Cures for he Shock Isably: Developme of a Shock-Sable Roe Scheme. Joural of Compuaoal Physcs, 85, 8

24 hps://do.org/0.06/s00-999(0) [] Padolf, M. ad D Ambroso, D. (00) Numercal Isables Upwd Mehods: Aalyss ad Cures for he Carbucle Pheomeo. Joural of Compuaoal Physcs, 66, hps://do.org/0.006/cph [] Saders, R., Morao, E. ad Drugue, M. (998) Muldmesoal Dsspao for Upwd Schemes: Sably ad Applcaos o Gas Dyamcs. Joural of Compuaoal Physcs, 45, hps://do.org/0.006/cph [3] Srag, G. (968) O he Cosruco ad Comparso of Dfferece Schemes. SIAM Joural o Numercal Aalyss, 5, hps://do.org/0.37/ [4] Toro, E.F. (009) Rema Solver ad Numercal Mehods for Flud Dyamcs. 3rd Edo, Sprger, New York. hps://do.org/0.007/b7976 [5] Shu, C.W. (988) Toal-Varao-Dmshg Tme Dscrezaos. SIAM Joural o Scefc ad Sascal Compuao, 9, hps://do.org/0.37/ [6] Lax, P.D. (954) Weak Soluos of Nolear Hyperbolc Equaos ad Ther Numercal Compuao. Commucaos o Pure ad Appled Mahemacs, 7, hps://do.org/0.00/cpa [7] Schulz-Re, C.W., Colls, J.P. ad Glaz, H.M. (993) Numercal Soluo of he Rema Problems for Two-Dmeoal Gas Dyamcs. SIAM Joural o Scefc Compug, 4, hps://do.org/0.37/09408 [8] Garder, C.L., Glmm, J., McBrya, O., Mekoff, R., Sharp, D.H. ad Zhag, Q. (988) The Dyamcs of Bubble Growh for Raylegh-Taylor Usable Ierfaces. Physcs of Fluds, 3, hps://do.org/0.063/ Subm or recommed ex mauscrp o SCIRP ad we wll provde bes servce for you: Accepg pre-submsso qures hrough Emal, Facebook, LkedI, Twer, ec. A wde seleco of ourals (clusve of 9 subecs, more ha 00 ourals) Provdg 4-hour hgh-qualy servce User-fredly ole submsso sysem Far ad swf peer-revew sysem Effce ypeseg ad proofreadg procedure Dsplay of he resul of dowloads ad vss, as well as he umber of ced arcles Maxmum dssemao of your research work Subm your mauscrp a: hp://papersubmsso.scrp.org/ Or coac amp@scrp.org 8