FourierSpectralMethodsforNumericalSolvingoftheKuramotoSivashinskyEquation

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1 Global Joral of Rsarhs i Egirig: I rial Mhods Vol Iss Vrsio. Yar Typ: Dobl Blid Pr Rviwd Iraioal Rsarh Joral Pblishr: Global Jorals I. USA Oli ISS: & Pri ISS: orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio By Gia Zavalai Polyhi Uivrsiy of Tiraa Albaia Absra- I his papr I prsd a rial hiq for solvig Kraoo-Sivashisy qaio basd o spral orir hods. This qaio dsribs raio diffsio probls ad h dyais of visos-fid fils flowig alog walls. Afr w wro h qaio i ri spa w g a sys. I his as h poial i diffrig hods igra h sys h or araly ha ohr hods si h poial i diffrig hods ass i hir drivaio ha h solio varis slowly i i. Wh valaig h offiis of h poial i diffrig ad h poial i diffrig Rg Ka hods via h Cahy igral. All opaioal wor is do wih Malab paag. Kywords: disr forir rasfor poial i diffrig poial i diffrig rg a hods ahy igral raoo-sivashisy qaio. GJRE-I Classifiaio : OR Cod: 6 6 orirspralmhodsforrialsolvigofhkraoosivashisyeqaio Srily as pr h oplia ad rglaios of:. Gia Zavalai. This is a rsarhrviw papr disribd dr h rs of h Craiv Coos Aribio- oorial. Upord is hp:raivoos.orglissby-. priig all o orial s disribio ad rprodio iay di providd h origial wor is proprly id.

2 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio Gia Zavalai Absra- I his papr I prsd a rial hiq for solvig Kraoo-Sivashisy qaio basd o spral orir hods. This qaio dsribs raio diffsio probls ad h dyais of visos-fid fils flowig alog walls. Afr w wro h qaio i ri spa w g a sys. I his as h poial i diffrig hods igra h sys h or araly ha ohr hods si h poial i diffrig hods ass i hir drivaio ha h solio varis slowly i i. Wh valaig h offiis of h poial i diffrig ad h poial i diffrig Rg Ka hods via h Cahy igral. All opaioal wor is do wih Malab paag. Kywords: disr forir rasfor poial i diffrig poial i diffrig rg a hods ahy igral raoo-sivashisy qaio. I. Irodio orir aalysis ors i h odlig of idpd phoa ha ar aly or approialy priodi. Eapls of his ild h digial prossig of iforaio sh as sph; h aalysis of aral pho sh as arhqas; i h sdy of vibraios of sphrial irlar or raglar srrs; ad i h prossig of iags. I a ypial as orir spral hods wri h solio o h parial diffrial qaio as is orir sris. orir sris doposs a priodi ralvald fio of ral arg io a s of sipl osillaig rigoori fios ssssssssss ha a b robid o obai h origial fio. Sbsiig his sris io h parial diffrial qaio givs a sys of ordiary diffrial qaios for h i-dpd offiis of h rigoori rs i h sris h w hoos a isppig hod o solv hos ordiary diffrial qaios II. orir Sris Th orir sris of a sooh ad priodi ral-vald fio ff [; ] wih priod is ao f a os b si Ahor: Polyhi Uivrsiy of TiraaAlbaia. -ail: zavalaigia@hoail.o os a Si h basis fios si ad ar orhogoal h offiis ar giv by b f si d... f os d... orir sris a b prssd aly i opl for as follows a f Th o a i i i If w dfi b i i ao a ib a ib 5 f i whr h offiis fro h forlas of a ad b as III. i f 6 a b drid d Disr orir Trasfor I ay appliaios parilarly i aalyzig of ral siaios h fio f o b approiad is ow oly o a disr s of saplig pois" of. H h igral 7 ao b valad i a losd for ad orir aalysis ao b applid dirly. I h bos ssary o rpla oios orir aalysis by a disr vrsio of i. Th liar disr orir rasfor of a priodi disr sq of opl vals wih priod is a sq of priodi opl vals dfid by 7 Yar Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Global Jorals I. US

3 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Yar Th liar ivrs rasforaio is i i 8 9 Th os obvios appliaio of disr orir aalysis osiss i h rial allaio of orir offiis. Sppos w wa o approia a ralvald priodi fio f dfid o h irval [; ] ha is sapld wih a v br of grid pois h h... by i is orir sris. irs w op approia vals of h orir offiis f M i - - Siilarly h ivrs disr orir rasfor has h for M Whr M ad - M Bas h disr orir rasfor ad is ivrs hibi priodiiy wih priod i.. his propry rsls fro h priodi ar of i as o ss o s or ha rs i h sris ad i sffis o alla o fll priod. Th orir sris ford wih h approia offiis is f i Th fio ff o oly approias b aally irpolas ff a h saplig grid pois I ari for h disr orir rasfor 8 a b wri as M... Whr 6 M ad i -... i whr is opl oga of Th T algorih rds h opaioal wor rqird o arry o a disr orir rasfor by rdig h br of lipliaios ad addiios of opaioal i is rdd fro OO o OO log. To apply spral hods o a parial diffrial qaio w d o vala drivaivs of fios. Sppos ha w hav a priodi ral-vald fio f [; ] wih priod ha is disrizd wih a v br of grid pois so ha h grid Global Jorals I. US

4 siz h.th opl for of h orir sris rprsaio of f is A i f i h abov sris givs a r whih alras bw ± a h grid poi h... ad si i ao b diffriad w shold s is drivaiv o b zro a h grid pois. Th rial drivaivs of h fio f a b illsrad as a ari lipliaio.or h firs drivaiv w liply h orir offiis by h orrspodig diffriaio ari for a v br of grid pois. i Λ Diag This ari has o-zro ls oly o h diagoal. or a odd br of grid pois h Th w op a ivrs disr orir rasfor sig o rr o h physial spa ad dd h firs drivaiv of f o h grid. Siilarly aig h sod drivaiv orrspods o i diffriaio ari orrspodig o h firs drivaiv is diagoal wih ls. h lipliaio of h orir offiis by h orrspodig diffriaio ari for a v br of grid pois. i Λ Diag I gral i as of a v br of grid pois approiaig h -h rial drivaivs of a grid fio f orrspods o h lipliaio of h orir offiis by h orrspodig diffriaio ari whih is diagoal i wih ls for wih h pio ha for odd drivaivs w s h drivaiv of h highs od o b zro. IV. orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio Epoial Ti Diffrig Th faily of poial i diffrig shs. This lass of shs is spially sid o si-liar probls whih a b spli io a liar par whih oais h siffs par of h dyais of h probl ad a oliar par whih varis or slowly ha h liar par. Epoial i diffrig shs ar i igraio hods ha a b ffiily obid wih spaial approiaios o provid ara sooh solios for siff or highly osillaory si-liar parial diffrial qaios.i his papr I will prs h drivaio of h plii Epoial i diffrig shs for arbirary ordr followig h approah i [ ] [ ] [ ] ad prss h plii Rg-Ka vrsios of hs shs osrd by Co ad Mahws [ ]. W osidr for sipliiy a sigl odl of a siff ordiary diffrial qaio d whr is h d oliar forig r. To driv h s -sp Epoial i diffrig shs w liply hrogh by h igraig faor ad h igra h qaio ovr a sigl i sp fro o o obai. Yar Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Global Jorals I. US

5 d Th sp is o driv approiaios o h igral i qaio. This prodr dos o irod a wad fas i sal io h solio ad h shs a b gralizd o arbirary ordr. If w apply h wo Baward Diffr orla w a wri a polyoial approiaio o i h for G G G s s......! s o ha! If w sbsi io w g d G s d G s W will idia h igral by d ϑ If w sbsi ad io w g s ϑ 5 Whih rprs h gral graig forla of h poial i diffrig shs of ordr s Th firs-ordr poial i diffrig sh is obaid by sig s Th sod-ordr poial i diffrig sh is obaid by sig s { } By sig s w g h forh-ordr poial i diffrig sh 6 Θ Θ Θ Θ Θ Θ Θ Θ. orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio Global Jorals I. US Global Joral of Rsarhs i Egirig Vol XIV Iss I Vrsio I Yar I

6 V. O h Sabiliy of Edr Mhod Th sabiliy aalysis of h ETDRK hod is as follows s [] or[]. or h oliar ODE d. d wih h oliar par w sppos ha hr iss a fid poi his as ha. iarizig abo his fid poi if is h prrbaio of ad δ ' h whr y d δ. d ad h fid poi is sabl if R δ <. Th appliaio of h ETDRK hod o. lads o a rrr rlaio ivolvig ad. Irodig h prvios oaio δ h ad y h ad sig h Mahaia algbra paag w obai h followig aplifiaio faor r y. y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y 5 5 y y y y y y y y y y y y y y y y y y y y y y y y y y y y 5 5 y y y y y y y y y y y y y y y y y y y 6 6 y y y y A ipora rar: opig by h abov prssios sffrs fro orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio rial isabiliy for y los o zro. Bas of ha for sall y isad of h w will s hir asypoi pasios y y y y y Oy y y y y y Oy y y y y y Oy y y y y y Oy Yar 5 Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Global Jorals I. US

7 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio W a wo obsrvaios: As y y or approiaio bos r 6 whih is h sabiliy fio for all h -sag Rg Ka hods of ordr for. Bas ad δ ay b opl h sabiliy rgio of h ETDRK hod is for-disioal ad hrfor qi diffil o rprs. Uforaly w do o ow ay prssio for ry w will oly b abl o plo i. Th os oo ida is o sdy i for ah parilar as; for apl assig o b fid ad ral i [] or ha boh ad δ ar pr iagiary brs i []. Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Yar 6 igr : Bodary of sabiliy rgios for svral gaivy igr : Eprial bodaris ad llips for y 75 Global Jorals I. US

8 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio or dissipaiv PDEs wih priodi bodary odiios h salars ha aris wih a orir spral hod ar gaiv. s a for apl Brgr s qaio ε [ ] whr < ε. Trasforig i o h orir spa givs iζ εζ ζ.5 whr ζ is h orir wav-br ad h εζ < spa ovr a wid rag of offiis vals wh all h orir ods ar osidrd. or high vals of ζ h solios ar arad o h slow aifold qily bas < ad <<. I igr. w draw h bodary sabiliy rgios i h opl pla for y Wh h liar par is zro y w rogizd h sabiliy rgio of h forh-ordr Rg Ka hods ad as y h rgio grows. Of ors hs rgios oly giv a idiaio of h sabiliy of h hod. I fa for y < y << h bodaris ha ar obsrvd approah o llipss whos parars hav b fid rially wih h followig rsl. R I y.7.6 I igr. w draw h prial bodaris ad h llipss.6 wih y 75. Th spr of h liar opraor irass as ζ whil h igvals of h liarizaio of h oliar par lay o h iagiary ais ad iras as ζ. O h ohr had aordig o.6 wh R h irsio wih h iagiary ais I irass as y i.. asζ.si h bodary of sabiliy grows fasr ha h ETDRK hod shold hav a vry good bhavior o solv Brgr s qaio whih ofirs h rsls of papr [6]. VI. Epoial Ti Diffrig Rg-Ka Mhods Grally for h o-sp i-disrizaio hods ad h Rg-Ka hods all h iforaio rqird o sar h igraio is availabl.howvr for h li-sp i-disrizaio hods his is o r.ths hods rqir h valaios of a rai br of sarig vals of h oliar r a h -h ad prvios i sps o bild h hisory rqird for h allaios.thrfor i is dsirabl o osr poial i diffrig hods ha ar basd o Rg-Ka hods. Basd i [ ] ad [ ] Pig s i qaio ss 5 o g a ss 6 Th r a approias h val of a Th sp is o approia i h irval wih ad sbsi io ss yild Eqaio ss 8 rprs h firs-ordr Rg Ka poial i diffrig sh I a siilar way w a also for h sodordr Rg Ka poial i diffrig sh a ss 9 As w a s qaio ss 9 is ford by aig half a sp of ss 6 Th sp is o approia i h irval wih By sbsiig ss 9 io ss w g a O ss { a } ss By sig s a forh-ordr Rg Ka poial i diffrig sh is obaid as follows b a a a O ss 7 a a ss 8 Yar 7 Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Global Jorals I. US

9 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio Global Jorals I. US Global Joral of Rsarhs i Egirig Vol XIV Iss I Vrsio I Yar 8 I b a { } a b ss I gral h poial i diffrig Rg-Ka hod ss has lassial ordr for b Hohbr ad Osra[] showd ha his hod sffrs fro a ordr rdio. Thy also prsd rial pris whih show ha h ordr rdio prdid by hir hory ay i fa aris i praial apls. I h wors as his lads o a ordr rdio o ordr hr for h Co ad Mahws hod ss []. Howvr for rai probls sh as h rial pris odd by Kassa ad Trfh[] [6] for solvig varios o-disioal diffsio-yp probls ad h rial rsls obaid i for solvig so dissipaiv ad disprsiv PDEs h forh-ordr ovrg of h forh-ordr Rg Ka poial i diffrig hod [] is ofird rially. ially w o ha as i h offiis of h s -ordr poial i diffrig Rg-Ka hods h hods rd o h orrspodig ordr of h Rg-Ka shs. V. Th Kraoo-Sivashisy Eqaio Th Kraoo-Sivashisy qaiois o of h sipls PDEs apabl of dsribig opl bhavior i boh i ad spa. This qaio has b of ahaial irs bas of is rih dyaial propris. I physial rs his qaio dsribs raio diffsio probls ad h dyais of visosfid fils flowig alog walls. Kraoo-Sivashisy qaio i o spa disio a b wri Eqaio a b wri i igral for if w irod ζ h ζ ζ ζ ζ 5 or i for Th Kraoo-Sivashisy qaio wih priodi bodary odiios i orir spa a b wri as follows i 6 i 6. i d d 6. i i 6. i i 6. i i 6.5 If w sbsi io w g i i i i i i i i d d By siplifyig ad o ha

10 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio I fial for will b i i i Whr ϖ d i T[ ] Eqaio a b wri as fllows i ϖ h h i i [ ] T 8 Eqaio has srog dissipaiv dyais whih aris fro h forh ordr dissipaio r ha provids dapig a sall sals. Also i ilds λ igr : Th growh ra λ for prrbaios of h for Kraoo-Sivashisy K-S qaio Th zro solio of h K-S qaio is liarly sabl h growh ra λ > for prrbaios of λ i h for o ods wih wav-brs < for a wavlgh ϑ ad is dapd for ϑ ods wih > lada s igr. hs ods ar opld o ah ohr hrogh h o-liar r lada h haiss of a liar gaiv diffsio r whih is rsposibl for a isabiliy of ods wih larg wavlghi. sall wav-brs. Th oliar adviospig r i h qaio rasfors rgy bw larg ad sall sals. i o h zro solio of h Th siffss i h sys 7 is d o h fa ha h diagoal liar opraor wih h ls has so larg gaiv ral igvals ha rprs day bas of h srog dissipaio o a i sal h shorr ha ha ypial of h oliar r. Th ar of h solios o h h Kraoo-Sivashisy qaio varis wih h sys siz of liar opraor. or larg siz of liar opraor Yar 9 Global Joral of Rsarhs i Egirig Vol XIV Iss I Vrsio I I Global Jorals I. US

11 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Yar ogh sabl orir ods is o a h sys haoi. or sall siz of liar opraor isffi orir ods is asig h sys o approah a sady sa solio. I his as h poial i diffrig hods igra h sys vry h or araly ha ohr hods si h h poial i diffrig hods ass i hir drivaio ha h solio varis slowly i i. VI. rial Rsl or h silaio ss w hoos wo priodi iiial odiios os.7 os.si.6 os.si Wh valaig h offiis of h poial i diffrig ad h poial i diffrig Rg Ka hods via h "Cahy igral" approah [ 5][6] w hoos irlar oors of radis R. Eah oor is rd a o of h ls ha ar o h diagoal ari of h liar par [ ] [ ] of h si-disrizd odl. W igra h sys 7 sig forh-ordr Rg Ka poial i diffrig sh sig 6 wih i-sp siz. igr : Ti volio of h rial solio of h Kraoo-Sivashisy p o os 6 wih h iiial odiio [ ] Global Jorals I. US

12 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio Yar igr : Ti volio of h rial solio of h Kraoo-Sivashisy p o os Th solio i h figr wih h iiial ad i- os odiio [ ] wih 6 sp siz 6 wih h iiial odiio [ ] appars as a sh plo ad shows wavs propagaig ravlig priodially i i ad prsisig wiho hag of shap. igr : Ti volio of h rial solio of h Kraoo-Sivashisy p o 6 wih h iiial odiio.7 os.si.6os.si [ ] I h figr wih h iiial odiio.7 os.si.6os.si [ ] wih ad i-sp siz h 6 solio appars as a sh plo ad shows wavs propagaig ravlig priodially i i ad prsisig wiho hag of shap. Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Global Jorals I. US

13 orir Spral Mhods for rial Solvig of h Kraoo-Sivashisy Eqaio Global Joral of Rsarhs i Egirig I Vol XIV Iss I Vrsio I Yar igr : Ti volio of h rial solio of h Kraoo-Sivashisy p o si 6 wih h iiial odiio [ ] I h figr wih h iiial odiio si [ ] wih 6 ad i-sp siz h solio appars or lar as a sh plo ad shows wavs propagaig ravlig priodially i i ad prsisig wiho hag of shap. VII. Colsios I his papr h ai obiv of his sdy was for fidig h solio of o disioal siliar forh ordr hyprboli Kraoo-Sivashisy qaio dsribig raio diffsio probls ad h dyais of visos-fid fils flowig alog walls. I ordr o ahiv his w applid orir spral approiaio for h spaial disrizaio. I addiio w valad h offis of h poial i diffrig ad h poial i diffrig forh ordr Rg Ka hods via h Cahy igral.so ypial apls hav b dosrad i ordr o illsra h ffiiy ad aray of h poial i diffrig hods hiq i his as. or h silaio ss w hos priodi bodary odiios ad applid orir spral approiaio for h spaial disrizaio. I addiio w valad h offiis of h Epoial Ti Diffrig Rg-Ka hods via h "Cahy igral" approah. Th qaios a b sd rpadly wih ssary adapaios of h iiial odiios. Rfrs Référs Rfrias. G. Byli J. M. Kisr ad. Vozovoi. A w Class of Ti Disrizaio Shs for h Solio of oliar PDEs. J. Cop. Phys. 7: J. Crai. Th Solio of Ordiary Dirial Eqaios wih arg Ti Cosas. I Mahaial Mhods for Digial Coprs A. Ralso ad H. S. Wilf ds.:8- Wily w Yor 96.. ridli. Gralizd Rg-Ka Mhods for h Solio of Siff Dirial Eqaios. I rial Tra of Dirial Eqaios R. BrlirshR. Grigori ad J. Shr dr ds. 6 r os i Mahais:5-5Sprigr Brli S. P. ors. A A-Sabl Modiaio of h Adas- Bashforh Mhods. I Cof. o rial Solio of Dirial Eqaios r os i Mah. 9969:-9 Sprigr-Vrlag Brli C. Kli. orh Ordr Ti-Sppig for ow Disprsio Korwg-d Vris ad oliar Shr digr Eqaios. Elroi Trasaios o ria Aalysis 9: A. K. Kassa ad.. Trfh. orh-ordr Ti Sppig for Siff PDEs. SIAM J. Si. Cop. 6: R.. Brd ad J. D. airs. rial Aalysis. Wadsworh Grop svh diio. 8. M. Hohbr ad A. Osra. Eplii Epoial Rg-Ka Mhods for Si-liar Paraboli Probls. SIAM J. r. Aal. : S. M. Co ad P. C. Mahws. Epoial Ti Diffrig for Siff Syss. J. Cop. Phys. 76:-55.. A. K. Kassa. High Ordr Ti sppig for Siff Si-iar Parial Diffrial Eqaios. PhD hsis Oford Uivrsiy. Global Jorals I. US