An Adaptation of the Scheifele Method to Stiff Systems of Differential Equations

Transcription

1 86 he Ope Appled Mhecs Jourl Ope Access A Adpo of he Schefele Mehod o Sff Syses of Dfferel Equos J.A. Reyes F. Grcí-Aloso d Y. Vllcp* Depre of Appled Mhecs. Hgher Polyechc School (EPS). Uversy of Alce Sp Absrc: My brches of Physcs d Egeerg use perurbed ler ODE s. Oe ehod of resoluo s bsed o he use of Schefele fucos for syses. hs pper syseclly epds hree pplcos of he Schefele ehod dped o sff probles. For hs purpose fly of rces -fucos d he uercl ehod re preseed for he egro of perurbed ler syses wh cos coeffces whch ebles he soluo o be epressed s seres of -fucos. he seres coeffces re obed hrough recurrece relos volvg he perurbo fuco. Oe of he dffcules ers of pleeg he ehod s he eed o deere hese relos for ech cse. Furherore hs pper he ecessry dpos re de order o fcle he clculo of he recurrece relos requred for syse egro. I ech proble he uercl lgorh s desged wh vew o eblg copuol pleeo. hs lgorh preses he se good properes s he egro ehod for hroc oscllors oher words c ccurely egre he o-perurbed proble usg us he frs er he seres. he resuls show cresed ccurcy he pplco of he odel whe copred o oher ow ehods pleeed Mple V. Keywords: Nuercl soluos for perurbed ler ODE syses; Sff probles. INRODUCION I 97 Schefele [] cosruced oe-sep uercl ehod for he egro of perurbed oscllors proble h frequely ppers y brches of Physcs d Egeerg. hs ehod s bsed o he cosruco of fly of fucos G whch ebles he soluo o he oscllor o be epressed ers of he seres () = bg = [] d lso offers he dvge of beg ble o egre ccurely he o-perurbed oscllor usg us he frs wo ers. he geerlzo of hs ehod o perurbed frs-order ler dfferel equo syses c be foud [] where ew fly of fucos s cosruced dped o he egro of hs ype of syse. he soluo ers of -fucos s gve by he seres () = = b where he coeffces b re obed usg recurreces volvg he perurbo fuco of he syse. hs es h he -fucos seres ehod for syses preses dffcules ers of s pplco prculrly whe he perurbo fuco hs coplced lycl epresso. However gve s good perforce relo o sff probles d hghly oscllory probles he -fucos ehod s crucl for he cosruco of ulsep lgorhs h solve hs drwbc such s he SMF [4-6] SVF [7] d EI p PC [8] ehods og *Address correspodece o hs uhor he Depre of Appled Mhecs Hgher Polyechc School (EPS). Uversy of Alce Sp; E-l: vllcp@u.es ohers. Furherore he -fucos seres ehod res he good properes dsplyed by he seres ehod relo o dfferel equos oher words f specfc e he perurbo ers dspper he -fucos seres ehod wll ccurely egre he hoogeous proble usg us he frs er he seres. I ddo o provde deled preseo of fucos d her os relev properes s well s he clculo of resduls hs pper syseclly epds hree pplcos of he ehod o he egro of sff syses. I hese probles ll he ecessry dpos re de order o fcle her egro pplyg Seffese echques [9 ] d defg ech of he he uercl lgorh h llows for copuol pleeo. Flly he relve error for he soluos obed usg he -fucos ehod s copred wh h of soluos provded by oher well-ow codes such s M d pleeed Mple V. GENERAION AND PROPERIES OF - FUNC- IONS Le us cosder he followg l vlue proble IVP: = A+ f( ) = () () where A s regulr -order r s sll perurbo preer d he copoes of vecor re fucos () wh =. he copoes of he vecor perurbo feld f re f ( ) wh = d he feld s couous wh couous dervves ul cer order so h ssfes he codos for esece 874-4/8 8 Beh Ope

2 A Adpo of he Schefele Mehod o Sff Syses he Ope Appled Mhecs Jourl 8 Volue 87 d uqueess of he soluo. hs ype of syse s clled perurbed ler syse. Le us suppose h he perurbo fuco g() = f( ) ds he followg seres epso: f( ) = c + ()! = I hs cse he l vlue proble c be epressed s follows: = A+ =. () c + () =! he soluo o IVP () c be obed by pplyg he superposo prcple oher words by cosrucg he geerl soluo for he hoogeeous syse usg he l codo gve d ddg soluo o he coplee proble wh ull l codo. he soluo o he coplee proble wh ull l codo c be obed by resolvg he followg dvdul IVP s: ( ) + = A + + e (4)! () = =... =... where: M( ;R ) (5) ( ) e = ( ) d effecvely cobg he wh coeffces d c. I order o splfy he oo squred -order rces X re roduced whch he colus re rces oherwords: ( ) X = wh =.... (6) Clerly hese rces re soluos o he followg IVP: X = AX + I X () = =.... (7)! where X s rel fuco wh vlues o rg M ( R) of he squred -order rces whch I d re he uy d eurl elees of hs rg respecvely. he soluos o (7) re he Schefele -fucos [] d re epressed s: () () X = + =.... (8) Alhough he oo does o dce hs eplcly hese fucos deped o A. he -fucos obey: () () = =.... (9) By usg he dffereo rule (9) he frs -fuco s defed () s gve by: () () =. () g o ccou (8) (9) d () he followg recurrece relo s obed: () = A + () + I =.... ()! he porce of hs recurrece relo les s pplco o he clculo of -fucos. Fro () d () c be deduced h he fuco s he soluo o he followg IVP: () = A = I () oher words: () = A () d () = I. () Aoher sgfc propery relo o he clculo of -fucos s h such fucos c be epressed usg seres epsos: + = M M M ( R). (4) () = ( + )! By lyclly epdg he fuco we see h: () = M where M M ( R). (5) =! By g he dervve of he bove epresso we ge: () = M = M+ = ( )! =! by () d () (6) M+ = A M = AM =! =! =! (7) by defyg hese ers we ge: M+ = AM =.... (8) d hece: M = A M =.... (9) O he bss of () d () d hvg evlued s = we c deduce h: M () = I M = A =... () he:

3 88 he Ope Appled Mhecs Jourl 8 Volue Reyes e l. A = A = e =! () () A prculr soluo o he proble: = A+ = () s: c + =! p = () c = (). () he soluo for he IVP: = A+ f( ) = (4) es o he for: = () () = () + () c. (5) hs resul ebles he uercl egro o be defed o he bss of -fucos. FINIE EXPANSIONS IN -FUNCIONS AND HE CALCULAION OF RESIDUALS By perforg ylor seres epso d rucg he soluo () for IVP () pproo of he soluo c be obed s follows: ) () = wh = () R. (6) =! By replcg epresso () (6) d g o ccou h rces () d A coue he followg c be epressed: () = () + ()( + A) + () A. (7) + = By defg ew sequece of coeffces s: b = b = A (8) + + (7) s reduced o: () b + () () = A. (9) = Elg he fl er gves dffere pproo: () X () = b () = whch provdes greer ccurcy h () =. ()! = I fc he coeffces he epresso of () = d X () = () b for IVP () re:! = = = ( ) + = A + f () () b = b = f () = c () ( ) + + so: = + = () () ( ) X () f (). (4) By serg (4) o () we ge he resdul: R () = f() ( X () AX ()) ( = f ) (). (5) =! Furherore by g o ccou h: () A () = ( () A + () )( + A ) = ( () + ()) A A (6) d by usg () he resdul correspodg o () s gve by: ( ) r () = f() () A () ( = f ) () + A. (7) =!! Sgfc coclusos c be drw fro he epressos r () d R ().I R () he perurbo preer s fcor of he resdul herefore R () s sll wh. O he corry he resdul r () spls o he su of wo ques of whch oly oe s proporol o ; so geerl r () s o sll wh.if = he ylor seres ehod produces ruco error d ye he -fucos ehod usg us he frs er c ccurely egre he syse of dfferel equos [ 5]. A NUMERICAL INEGRAION MEHOD BASED ON -FUNCIONS Followg slr procedure o he oe descrbed [ 5] order o egre IVP () usg he -fucos s epded s follows: ehod he fuco f f( ) = c + (8) =! d he soluo ers of -fucos s gve by

4 A Adpo of he Schefele Mehod o Sff Syses he Ope Appled Mhecs Jourl 8 Volue 89 () = () + () c (9) where c = re he dervves of he perurbo fuco f( () ) = g () = g ()... g () (4) Deog he po esh [ b ] s = + h =... d he pproo o d ssug h we hve clculed pproo o he soluo o po whch we wll cll c be verfed h = A+ f( ) =. (4) I order o ob pproo o he soluo ( + ) we swch he depede vrble = herefore (4) becoes: = A+ f( + ) () (4) = whch leds bc o he l suo. he recurrece relo s used o clcule he epso coeffces for: f( + ) = c+ (4) =! wh = d g() d ( ) + d = g c (44) d d hvg clculed he ecessry -fucos s possble o epress he pproo o he soluo po + s follows: p + = ( h+ ) + ( h+ ) = g (45) whch cosues he uercl egro ehod for perurbed ler dfferel equos o he bss of fucos. I cer cses lerve o swchg he depede vrble = s he echque used by Seffese [9] whch volves soe ulry vrbles whch eble he syse s perurbo fuco o be rsfored o fuco h does o deped eplcly o e or whch rsfors he syse o o-perurbed syse chevg egro hs cse wh us oe -fuco. RESOLUION OF SIFF PROBLEMS I hs seco we prese hree eples showg he behvour of he -fucos seres ehod gs he ow codes: s ehods cuses uercl soluo o be foud usg he Lverore Sff ODE solver. I solves sff d osff syses. I uses Ads ehods (predcorcorrecor) he osff cse d Bcwrd Dffereo Forul (BDF) he sff cse. cuses uercl soluo o be foud by wy of Burlrsch-Soer rol erpolo ehod. he ehod hs hgher precso d clculo effcecy especlly solvg sff dfferel equos. M [seppr] s ul-sep ehod suble for sff syses. Usg he ls oes he pleeos of MAPLE V o esure h he resuls re o dsored by defce progro h fvours he ew code. Proble I Le us cosder he followg sff proble whch ppers [-]: = + + s (46) = ( + ) + ( + )( cos + s) wh A = ( + ) + () = () = d soluo depede of : = e + s() = l codos () = e + cos() (47) he egevlues of he syse re d whch ebles s degree of sffess o be reguled. For he cse = he followg sff proble s obed proposed [4]: () s () = () () ( cos s ) (48) () = () =. Cosderg vrbles = s 4 () = cos() s possble o cle he followg o-perurbed proble: () () () () = () () (49) 4() 4() = ( () () () 4() ) = ( ). he roduco of ew vrbles creses he deso of he proble bu ebles s egro usg us he frs of he -fucos. Le () be he soluo o he bove proble whch we ssue o be lycl: ( () () () ()) = 4 4 =! =! =! =! (5)

5 9 he Ope Appled Mhecs Jourl 8 Volue Reyes e l. d whe s replced (49) we ge: + + =! =. (5)! = 4 Idefyg he coeffces gves = (5) wh =. By defg ew sequece of coeffces s: b = = b = (5) wh =. Hece s possble o epress he soluo o he proble ers of -fucos: b b () = () (54) = b b 4 d hvg obed he vlue of he -fucos; () A = e r hs bee clcule by he Rchrdsoype ler elo procedure [56]; deog by he pproo o ( h ) hs pproe soluo s gve by: b b b b = ( h) ( h) =. (55) = b b b b 4 4 I order o clcule (.e. he pproo o ( h) ) he se process s crred ou g s he l vlue. Ech sep s copleed usg he followg lgorh: = = b = for = b = for =... b b b b + = ( h) = ( h). (56) = b b b b 4 4 Fg. () d Fg. () show he resuls obed wh 4 d 6 dgs respecvely whe he logrh of he relve error vs of he -fucos ehod usg us oe fuco sepsze h =. d eros s copred wh he (ol = -6 ) M (errorper = Flo( -6 )) d (errorper = Flo( -6 )) fucos M Fg. (). Lber s proble h =. 4 dgs oe -fuco fucos M Fg. (). Lber s proble h =. 6 dgs oe -fuco.

6 A Adpo of he Schefele Mehod o Sff Syses he Ope Appled Mhecs Jourl 8 Volue 9 he error s e o be he dfferece bewee he soluo of he severl ehods d he ec soluo. Is oed he well behvour of he -fuco seres ehod g he error bouded hroughou he egro. Fg. () d Fg. () show h wh he se codos cresg he uber of dgs he error decreses. Proble II Le us cosder he followg sff es proble proposed geerl ers [7]: ( ) () = + () + () (57) () = () ()( + ()) wh he l codos () = () = d soluo depede of : () = e () = e (58) he egevlues of he syse re ( + ) whch ebles he degree of sffess o be reguled. A prculrso for = s represeed [8 9] ses forh he Kps proble for = obg he followg sff syse: () () = + () () () (59) = () () =. Le () be he soluo o he boveeoed proble whch we ssue o be lycl: ( ) () () =! =! =. (6) By subsug (6) (59) d pplyg he Cuchy s rule for he seres produc we ge: + =!! = + = + = =! = Idefyg coeffces gves + = + = + =. (6) (6) epresso h ebles us o defe he followg successo of coeffces: b = = b = = = = wh =. (6) Hece s ow possble o epress he soluo o (59) ers of -fucos: b () = (). (64) = b () A = e r hs bee clcule by he Rchrdsoype ler elo procedure [56] d () for = re clculed by recurrece relo (). Deog he pproo o ( h ) s sd pproo o he soluo s gve s follows: b = ( h) = b. (65) I order o clcule he se process s crred ou g s he l vlue. Ech sep s copleed usg he followg lgorh: = + = = + + = b = b = = = = for =.... for =.... b + = ( h). (66) = b Fg. () shows he resul obed whe he logrh of he relve error vs of he 6 -fucos ehod sep sze

7 9 he Ope Appled Mhecs Jourl 8 Volue Reyes e l. h =. d eros s copred wh he (ol = - )d(errorper = Flo( - )) ehods wh 4 dgs. Fg. (). Kps proble h =. 4 dgs see -fucos. he error s e o be he dfferece bewee he soluo of he severl ehods d he ec soluo. Is oed he well behvour of he -fuco seres ehod g he error bouded hroughou he egro. Proble III hs eple shows pplco of he -fucos ehod o proble of qusperodc orbs suded by [] whch c lso be foud [ -] og ohers. Le () + () = e (67) wh he l codos: () = () =.9995 (68) for whch he lycl soluo s: 4 () = ( 5 ) e ( cos 5 s ) ( s 4 5 cos ) = + +. (69) he soluo represes oo o perurbo of crculr orb he cople ple. he proble y be solved eher s sgle equo cople rhec or pr of ucoupled equos. Nog () = u () + v () d by subsug (67) we ge he followg secod order syse: u + u = v () + () v = s() () () cos() -fucos (7) wh he l codos: u() = u() = v() = v() = By defg he vrbles: () = u() () = u() 4 (7) () = v() () = v(). (7) (7) becoes he syse of frs order ler equos: () = () () = () + cos() () = 4() 4 () = () + s() wh he l codos: 4 (7) () = () = = () = (74) he ulry vrbles [9 ]: () = cos() () = s() (75) 5 6 eble (7) o be epressed s o-perurbed frs order ler syse: () () () () () () = (76) 4() 4() 5() 5() 6() 6() wh he l codos: = () () () () () () = (.9995 ). (77) Alhough hs procedure creses he deso of he proble ebles egro usg us he frs of he fucos. Le () be he soluo o he prevous proble whch we ssue o be lycl: = =...6. (78) =! By subsug (78) (76) d defyg he coeffces we ge: = (79) epresso h ebles us o defe he followg sequece:

8 A Adpo of he Schefele Mehod o Sff Syses he Ope Appled Mhecs Jourl 8 Volue 9 b = = b = wh =. (8) M Hvg obed he vlue of he -fucos; () A = e r hs bee clcule by he Rchrdsoype ler elo procedure [56]; deog he pproo o ( h ) s hs pproo o he soluo would be gve s follows: = ( h) b b b b 4 b 5 b 6 =. (8) I order o clcule he se process s crred ou g s he l vlue. Ech sep s copleed usg he followg lgorh: = = + for =.... b = b = = ( h) b b b b b b = for =.... (8) Fg. (4) shows he resul obed whe he logrh of he relve error vs of he -fucos odel wh us oe -fuco sep sze h =. d eros s copred wh he (ol = -6 ) M (errorper = Flo( -6 )) d (errorper = Flo( -6 )) ehods pleeed Mple V wh 6 dgs. Fg. (5) shows he se coprso s he prevous fgure bu wh sep sze e es greer oher words h =.9 d eros. -5 -fucos Fg. (4). Sefel d Bes proble h =. 6 dgs oe - fuco fucos M Fg. (5). Sefel d Bes proble h =.9 6 dgs oe - fuco. he error s e o be he dfferece bewee he soluo of he severl ehods d he ec soluo. Is oed he well behvour of he -fuco seres ehod g he error bouded hroughou he egro. Fg. (5) shows he well behvour of he -fuco seres ehod f crese he sep sze. CONCLUSIONS he ehod here cosdered s he drec geerlzo of he lgorh for he egro of perurbed oscllors descrbed [] o perurbed ler syses of dfferel equos; he good properes of hs lgorh for equos re reed. Nuercl eperes cofr he srog perforce of he uercl egro ehod bsed o he Schefele -fucos seres; whe reg sff probles ebles he o-perurbed proble o be ccurely egred usg us he frs er he seres.

9 94 he Ope Appled Mhecs Jourl 8 Volue Reyes e l. Furherore he ehod c be e s he foudo for ulsep uercl ehods lso bsed o -fucos or soe refee such s SMF [4] SVF [7 ] d EI P PC [8]. REFERENCES [] Sefel EL Schefele G. Ler d regulr celesl echcs Sprger New Yor 97. [] Grcí-Aloso F Reyes JA Vllcp Y. Iegro of hghly oscllory probles hrough G-fucos. Ope Appl Mh J 7; : 9-4. [] Schefele G. O uercl egro of perurbed le oscllg syses. ZAMP 97; : 86-. [4] Mrí P Ferrádz JM. Behvour of he SMF ehod for heuercl egro of selle orbs. Celesl Mech Dy Asro 995; 6 : 9-4. [5] Mrí P Ferrádz JM. Mulsep uercl ehods bsed o Schefele G-fucos wh pplco o selle dycs. SIAM J Nuer Al 997; 4 : [6] Mrí P Ferrádz JM. Nuercl egro of perurbed ler syses. Appl Mh 999; : 8-9. [7] Vgo-Agur J Ferrádz JM. Hgher-order vrble-sep lgorhs dped o he ccure uercl egro of perurbed oscllors. Copu Phys 998; (5) : [8] Reyes JA Grcí-Aloso F Ferrádz JM Vgo-Agur J. Nuerc ulsep vrble ehods for perurbed ler syse egro. Appl Mh Copu 7; 9 : [9] Seffese JF. O he dfferel equos of Hll he heory of he oo of he oo. Ac Mh 955; 9 : [] Seffese JF. O he dfferel equos of Hll he heory of he oo of he oo (II). Ac Mh 955; 95: 5-7. [] Iru L Gr Vde Berghe G De Meyer H. Frequecy evluo epoel fg ulsep lgorhs for ODEs. J Copu Appl Mh ; 4 : 4-4. [] Iru L Gr Vde Berghe G De Meyer H. Epoelly fed vrble wo-sep BDF lgorhs for frs order ODEs Copu Phys Cou ;: [] V de Vyver H. wo-sep hybrd ehods dped o he uercl egro of perurbed oscllors rxv:h/667v [h.na] Dec 6. [4] Lber JD. Nuercl ehods for ordry dfferel syses. Joh Wlley d Sos Ld New Yor 99. [5] Wlz G. Approo vo Fuoe durch syposche Ewcluge ud Elosprozedure. Dssero. Uversä Mhe 987. [6] Wlz G. Copug he r epoel d oher r fucos. J Copu Appl Mh 988; : 9-. [7] Kps P Rosebroc-ype ehods. I: Dhlqus G Jelsch R Eds. Nuercl ehods for sff l vlue probles Berch Nr. 9 Is. für Geoere ud Prsche Mhe der RWH Ache 98. [8] v der Houwe PJ Soeer BP. he use of pproe fcorzo sff ODE solvers. Repor MAS- R97 CWI Aserd 997. [9] Fr JE Houwe PJ. Prllel ero of he eeded bcwrd dffereo foruls. IMA J Nuer Al ; : [] Sefel E Bes DG. Sblzo of Cowell s ehod. Nuer Mh 969; : [] Vgo-Agur J Ferrádz JM. A geerl procedure for he dpo of ulsep lgorhs o he egro of oscllory probles. SIAM J Nuer Al 998; 5(4): [] Sos E Vgo-Agur J. Epoelly fed syplec egror. Phys Rev E ; 67: -7. [] Ros JI. Pecewse-lerzed ehods for l-vlue probles wh oscllg soluos. Appl Mh Copu 6; 8: -46. Receved: Februry 6 8 Revsed: Aprl 9 8 Acceped: My 9 8 Reyese l.; Lcesee Beh Ope. hs s ope ccess rcle dsrbued uder he ers of he Creve Coos Arbuo Lcese (hp://crevecoos.org/lceses/by/.5/) whch pers uresrcve use dsrbuo d reproduco y edu provded he orgl wor s properly ced.