A Singly Diagonally Implicit Runge-Kutta- Nyström Method with Reduced Phase-lag

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Sgly Dagoally Implct Ruge-Kutta- Nytröm Method wth Reduced Phae-lag N. Seu, M. Sulema, F. Imal, ad M. Othma btract I th paper a gly dagoally mplct Ruge- Kutta-Nytröm (RKN) method developed for ecod-order ordary dfferetal equato wth perodcal oluto. he method ha algebrac order four ad phae-lag order eght at a cot of four fucto evaluato per tep. h ew method more accurate whe compared wth curret method of mlar type for the umercal tegrato of ecod-order dfferetal equato wth perodc oluto, ug cotat tep ze. Keyword Ruge-Kutta-Nytröm method; Dagoally mplct; Phae-lag; Ocllatory oluto. I. INRODUCION HIS paper deal wth umercal method for ecod-order ODE, whch the dervatve doe ot appear explctly, y f( x y) y( x ) y y( x ) y () 0 0 0 0 for whch t kow advace that ther oluto ocllatg. Such problem ofte are dfferet area of egeerg ad appled cece uch a celetal mechac, quatum mechac, elatodyamc, theoretcal phyc ad chemtry, ad electroc. -tage Ruge-Kutta-Nytröm (RKN) method for the umercal tegrato of the IVP gve by y yh b k y y h h bk y Maucrpt receved October 9, 00; reved Jauary 9, 0. h work wa upported part by the UPM Reearch Uverty Grat Scheme (RUGS) uder Grat 05-0-0-0900RU ad Fudametal Reearch Grat Scheme (FRGS) uder Grat 0-07-0-97FR N. Seu wth the Departmet of Mathematc ad Ittute for Mathematcal Reearch, Uvert Putra Malaya, 4400 UPM Serdag, Selagor, MLYSI. (correpodg author phoe: 60-8946-6848; fax: 60-8947958; e-mal: razak@math.upm.edu.my ). M. Sulema ad F. Imal are wth Departmet of Mathematc ad Ittute for Mathematcal Reearch, Uvert Putra Malaya, 4400 UPM Serdag, Selagor, MLYSI. (e-mal: mohamed@math.upm.edu.my, fudzah@math.upm.edu.my ). M. Othma wth the Ittute for Mathematcal Reearch, Uvert Putra Malaya, 4400 UPM Serdag, Selagor, MLYSI. (e-mal: mothma@fktm.upm.edu.my). () where y j j j k f x chy ch h a k he RKN parameter ajbj bj ad cj are aumed to be real ad the umber of tage of the method. Itroduce the - dmeoal vector cb ad b ad matrx, where c [ ccc ] b [ bb b ] b[ b b b ] [ a j ] repectvely. RKN method ca be dvded to two broad clae: explct ( a jk 0, k j ) ad mplct ( a jk 0, k > j). he latter cota the cla of dagoally mplct RKN (DIRKN) method for whch all the etre the dagoal of are equal. he RKN method above ca be expreed Butcher otato by the table of coeffcet c b b Geerally problem of the form () whch have perodc oluto ca be dvded to two clae. he frt cla cot of problem for whch the oluto perod kow a pror. he ecod cla cot of problem for whch the oluto perod tally ukow. Several umercal method of varou type have bee propoed for the tegrato of both clae of problem. See Stefel ad Bett [], va der Houwe ad Sommejer [], Gautch [6] ad other. Whe olvg () umercally, atteto ha to be gve to the algebrac order of the method ued, ce th the ma crtero for achevg hgh accuracy. herefore, t derable to have a lower tage RKN method wth maxmal order. h wll lee the computatoal cot. If t tally kow that the oluto of () of perodc ature the t eetal to coder phae-lag (or dpero) ad amplfcato (or dpato). hee are actually two type of trucato error. he frt the agle betwee the true ad the approxmated oluto, whle the ecod the ace from a tadard cyclc oluto. I th paper we wll derve a ew dagoally mplct RKN method wth three-tage fourth-order wth dpero of hgh order. umber of umercal method for th cla of problem of the explct ad mplct type have bee extevely developed. For example, va der Houwe ad Sommejer [], Smo, Dma ad Sderd, [5], ad Seu, Sulema

ad Imal [8] have developed explct RKN method of algebrac order up to fve wth dpero of hgh order for olvg ocllatory problem. For mplct RKN method, ee for example va der Houwe ad Sommejer [], Sharp, Fe ad Burrage [4] ad Imo, Otuta ad Ramamoha [7]. I th paper a dpero relato mpoed ad together wth algebrac coo to be olved explctly. I the followg ecto the cotructo of the ew four-tage fourth-order dagoally mplct RKN method decrbed. It coeffcet are gve ug the Butcher tableau otato. Fally, umercal tet o ecod order dfferetal equato problem poeg a ocllatory oluto are performed. II. NLYSIS OF PHSE-LG I th ecto we hall dcu the aaly of phae-lag for RKN method. he frt aaly of phae-lag wa carred out by Bura ad Ngro [0]. he followed by Gladwell ad homa [5] for the lear multtep method, ad for explct ad mplct Ruge-Kutta(-Nytrom) method by va der Houwe ad Sommejer [], []. he phae aaly ca be dvded two part; homogeeou ad homogeeou compoet. Followg va der Houwe ad Sommejer [], homogeeou phae error cotat tme, meawhle the homogeeou phae error are accumulated a creae. hu, from that pot of vew we wll cofe our aaly to the phae-lag of homogeeou compoet oly. he phae-lag aaly of the method () vetgated ug the homogeeou tet equato y ( ) y( t) () lteratvely the method () ca be wrtte a where y h ( ) f t ch Y y y h h b f( t ch Y) y y b y j j j Y y ch h a f( t chy ) By applyg the geeral method () to the tet equato () we obta the followg recurve relato a how by Papageorgou, Famel ad toura [4] y y D z h, hy hy Hb ( I H) e Hb ( I H) c DH ( ) Hb ( I H) e Hb ( I H) c where H z e () c ( c c m ). Here D(H) the tablty matx of the RKN method ad t charactertc polyomal tr( Dz ( )) det( Dz ( )) 0, (4) (5) the tablty polyomal of the RKN method. Solvg dfferece ytem (5), the computed oluto gve by y c co( ) (6) he exact oluto of () gve by yt ( ) co( z) (7) Eq. (6) ad (7) led u to the followg defto. Defto. (Phae-lag). pply the RKN method () to (). q he we defe the phae-lag ( z) z. If ( z) O( z ), the the RKN method ad to have phae-lag order q. doally, the quatty ( z) called r amplfcato error. If ( z) O( z ), the the RKN method ad to have dpato order r. Let u deote R( z ) trace( D) ad S( z ) det( D) From Defto, t follow that Let u deote Rz ( ) ( z) zco S( z ) S( z ) R( z ) ad S( z ) the followg form z z Rz ( ) ˆ ( + z ) z z Sz ( ), ˆ ( + z ) where ˆ dagoal elemet the Butcher tableau. Here the eceary coo for the fourth-order accurate dagoally mplct RKN method () to have hae-lag order eght term of ad gve by (8) (9) 6 4 4 (0) 60 8 4 4 4 4 6 0 () 45 40 Notce that the fourth-order method already dperve order four ad dpatve order fve. Furthermore dperve order eve ad dpatve order odd. III. CONSRUCION OF HE MEHOD I the followg we hall derve a four-tage fourth-order accurate dagoally mplct RKN method wth dperve order eght, by takg to accout the dpero relato

ecto II. he RKN parameter mut atfy the followg algebrac coo to fd fourth-order accuracy a gve Harer ad Waer []. order order order b () b b c () bc b c 6 6 (4) order 4 bc b c b ajcj 4 6 4 4. (5) For mot method the c eed to atfy c aj ( ) (6) j four-tage method of algebrac order four ( p 4 ) wth dperve order eght ( q 8 ) ad dpatve order fve ( r 5 ) ow codered. he coo ()-(6) ad dpero relato (0)-() formed thrtee olear equato wth etee varable to be olved. Now, from algebrac coo ()-(6) ad phae-lag relato of order x (0) ad lettg be a free parameter, the we olve t multaeouly. herefore the followg oluto of oeparameter famly obta c c c c a 6 6 6 6 a a4 a a4 a4 0 6 6 a a a a b b b 0b 4 44 4 (80 ) b 0( 4 4 88 7 ) 60 55 60 0 b4 5( 4 4 88 7 ) From the above oluto, we are gog to derve a method wth dpero of order eght. he eght order dpero relato () eed to be atfed ad th ca be wrtte term of RKN parameter whch correpod to the above famly of oluto yeld the followg equato 7 6 6 5 5 (5806080 4550 4550 490 967680 4 4 60480 8440 80640 4768 44856 976 94 75585 49 ) [0960( )] 00 ad olvg for yeld -0.755795, -0.08545609, 0.0479776, 0.684065,0.49098846, 0.684677664, ad -0.056647. Numercal reult how that choog -0.08545609 wll gve u more accurate cheme compared to the other ad we metoed here oe fourth-order (p=4) wth dperve order eght (q=8) method. For -0.08545609, the followg method wll be produced. h method wll be deoted by DIRKN4(4,8)NEW (ee able I) c able : : he DIRKN(4,6) method 6 6 0 6 6 0 0 6 6 0 b b4 4 0 0 where c =-0.7049006, b =0.957499, b 4 =0.604875, ad = =0.0454747 h method ha PLE BLE I HE DIRKN4(4,8)NEW MEHOD (5) (5) 66970 ad 670. able II compare the properte of our method wth the method derved by va der Houwe ad Sommejer [0], Sharp, Fe ad Burrage [4] ad Imo, Otuta ad Ramamoha [7]. IV. PROBLEM ESED I th ecto we ue our method to olve homogeeou ad homogeeou problem whoe exact oluto cot of a rapdly or/ad a lowly ocllatg fucto. For purpoe of llutrato, we wll compare our reult wth thoe derved by ug three method; DIRKN three-tage fourth-order derved by va der Houwe ad Sommejer [0] ad Imo, Otuta, Ramamoha [7], three-tage fourth-order dperve order x derved by Sharp, Fe ad Burrage [4] ad four-tage fourth-order derved by l-khaaweh, Imal, Sulema [].

BLE II SUMMRY OF HE CHRCERISIC OF HE FOURH-ORDER DIRKN MEHODS Problem (Homogeou) d y t () 00 yt ( ) y(0) y(0) Exact oluto y() t (0) t co(0) t Problem Method q d d y t 5 () yt ( ) t y(0) y(0) Exact oluto yt ( ) ( t) co( t) t Source : lle ad Wg [9] Problem (Ihomogeeou ytem) d y () t vy ( x) vft ( ) f( t) y(0) a f(0) y (0) f(0) d y () t vy () tvft () f() t y (0) f(0) y (0) va f(0) ( p ) Exact oluto y() t aco( vt) f() t y() t a( vt) f() t f () t 005t choe to be e ad parameter v ad a are 0 ad 0. repectvely. Source : Lambert ad Wato [7] DIRKN4(4,8)NEW 8 5 4.84 0 67 0 DIRKN(4,4)IMONI 4-75 0 DIRKN(4,4)HS 4 4 4 0 65 0 4 DIRKN(4,6)SHRP 6 0 0 85 0 DIRKNRaed 4 80 0. 0 Notato : q Dpero order, d Dpato cotat ( p ) Error coeffcet for y ( p) Error coeffcet for y ( p) 6 0 0 59 0 4 66 0 4 7 0 Problem 4 ( almot Perodc Orbt problem) d y t d y () t () y ( t) 000co( t) y (0) y (0) 0 y( t) 000( t) y(0) 0 y (0) 09995 Exact oluto y ( t ) co( t ) 0 0005 t ( t ), y ( t ) ( t ) 0 0005 t co( t ) Source : Stefel ad Bett [] he followg otato are ued able III-VI: DIRKN4(4,8)NEW : four-tage fourth-order dperve order eght method wth mall dpato cotat ad prcpal local trucato error derved th paper. DIRKN(4,4)IMONI : three-tage fourth-order derved by Imo, Otuta ad Ramamoha [7]. DIRKN(4,4)HS : three-tage fourth-order dperve order four derved by va der Houwe ad Sommejer [0]. DIRKN(4,6)SHRP : three-tage fourth-order dperve order x a Sharp, Fe ad Burrage [4]. DIRKN4(4,4)Raed : four-tage fourth-order drved by l-khaaweh, Imal, Sulema []. V. NUMERICL RESULS he reult for the four problem above are tabulated able III-VI. Oe meaure of the accuracy of a method to exame the Emax( ), the maxmum error whch defed by Emax( ) max y( t) y t0 where t t0 h? h able III-VI how the abolute maxmum error for DIRKN4(4,8)NEW, DIRKN(4,4)IMONI, DIRKN(4,4)HS, DIRKN(,6)SHRP ad DIRKN4(4,4)Raed method whe olvg Problem -4 wth three dfferet tep value. From umercal reult able III-VI, we oberved that the ew method more accurate compared wth DIRKN(4,4)IMONI, DIRKN(4,4)HS ad DIRKN4(4,4)Raed method whch do ot relate to the dpero order of the method. lo the ew method more accurate compared wth DIRKN(4,6)SHRP method becaue the ew method ha dpero order eght whch the hghet ad alo the dpato cotat for our method maller tha the DIRKN(4,6)SHRP method (ee able II).

BLE III COMPRING OUR RESULS WIH HE MEHODS IN HE LIERURE FOR PROBLEM h Method =00 =000 =4000 0.005 DIRKN4(4,8)NEW.494(-9).0464(-7) 7.766(-7) DIRKN(4,4)IMONI.5646(-).46(-) 4.7069(-) DIRKN(4,4)HS.56(-7).689(-6) 5.84(-6) DIRKN(4,6)SHRP.050(-7).09(-6).0(-5) DIRKN4(4,4)Raed 9.774(-6) 9.904(-5).79(-4) 0.005 DIRKN4(4,8)NEW.54(-9).56(-8) 5.047(-7) DIRKN(4,4)IMONI.0(-).8480(-) 5.6(-) DIRKN(4,4)HS 6.6977(-7) 6.6966(-6).78(-5) DIRKN(4,6)SHRP.5569(-6).564(-5).055(-4) DIRKN4(4,4)Raed.48(-4).4849(-) 5.99(-) 0.0 DIRKN4(4,8)NEW 4.5984(-8) 4.05(-7).875664(-6) DIRKN(4,4)IMONI 5.9680(-) 4.6(-) 9.8605(-) DIRKN(4,4)HS.05(-5).6(-4).95597(-) DIRKN(4,6) SHRP.4(-4).448(-).6707(-) DIRKN4(4,4)Raed.699(-).786(-) 9.56865(-) BLE IV COMPRING OUR RESULS WIH HE MEHODS IN HE LIERURE FOR PROBLEM h Method =00 =000 =4000 0.065 DIRKN4(4,8)NEW.468(-8).764(-8).8789(-8) DIRKN(4,4)IMONI 5.96(-) 5.(-).004(-) DIRKN(4,4)HS 6.80(-7) 6.86(-6).794(-5) DIRKN(4,6) SHRP 4.007(-6) 4.06(-5).649(-4) DIRKN4(4,4)Raed 5.8594(-5) 5.8706(-4).509(-) 0.5 DIRKN4(4,8)NEW.475(-7) 5.87(-7).0986(-6) DIRKN(4,4)IMONI.04(-).00(-).69(-) DIRKN(4,4)HS.087(-5).090(-4) 4.85(-4) DIRKN(4,6)SHRP.006(-4).98(-) 5.657(-) DIRKN4(4,4)Raed 8.070(-4) 8.09(-).9(-) 0.5 DIRKN4(4,8)NEW 5.8948(-6).77(-5).5775(-5) DIRKN(4,4)IMONI.94(-).868(-) 6.958(-) DIRKN(4,4)HS.7(-4).7444(-) 7.0007(-) DIRKN(4,6) SHRP 4.480(-) 4.644(-).950(-) DIRKN4(4,4)Raed.897(-).969(-) 5.6(-) BLE V COMPRING OUR RESULS WIH HE MEHODS IN HE LIERURE FOR PROBLEM h Method =00 =000 =4000 0.005 DIRKN4(4,8)NEW 8.6798(-0).090(-8).509(-7) DIRKN(4,4)IMONI 5.9756(-) 4.600(-) 9.50(-) DIRKN(4,4)HS.9675(-7).9897(-6).608(-5) DIRKN(4,6)SHRP.8995(-6).9004(-5) 7.6048(-5) DIRKN4(4,4)Raed.9(-5).9(-4).650(-) 0.005 DIRKN4(4,8)NEW.64(-8) 9.49(-8) 4.04(-7) DIRKN(4,4)IMONI.7(-) 7.005(-) 9.90(-) DIRKN(4,4)HS 6.468(-6) 6.496(-5).544(-4) DIRKN(4,6)SHRP 6.59(-5) 6.776(-4).498(-) DIRKN4(4,4)Raed 4.66(-4) 4.6689(-).8754(-) 0.0 DIRKN4(4,8)NEW 5.54(-7).456(-6).84(-5) DIRKN(4,4)IMONI.9988(-) 9.040(-).000(-) DIRKN(4,4)HS.04(-4).056(-) 4.0589(-) DIRKN(4,6) SHRP.089(-).85(-).66(-) DIRKN4(4,4)Raed 7.506(-) 7.7409(-).57(-) BLE VI COMPRING OUR RESULS WIH HE MEHODS IN HE LIERURE FOR PROBLEM 4 h Method =00 =000 =4000 0.065 DIRKN4(4,8)NEW.495(-8).748(-8) 4.6060(-8) DIRKN(4,4)IMONI.998(-) 4.09(-).08(-) DIRKN(4,4)HS 5.605(-7) 5.808(-6).06(-5) DIRKN(4,6) SHRP.498(-6).68(-5).9990(-4) DIRKN4(4,4)Raed 4.8(-5) 4.777(-4).486(-) 0.5 DIRKN4(4,8)NEW.578(-7) 4.899(-7).0(-6) DIRKN(4,4)IMONI 7.90(-) 7.67(-).76(-) DIRKN(4,4)HS 7.6595(-6) 7.9664(-5) 4.794(-4) DIRKN(4,6)SHRP 9.794(-5) 9.749(-4) 5.67(-) DIRKN4(4,4)Raed 5.69(-4) 5.8856(-).0(-) 0.5 DIRKN4(4,8)NEW 5.84(-6).0(-5).604(-5) DIRKN(4,4)IMONI.995(-).655(-) 6.7067(-) DIRKN(4,4)HS.0(-4).75(-) 7.0099(-) DIRKN(4,6) SHRP.09(-).895(-).996(-) DIRKN4(4,4)Raed 9.0479(-) 9.459(-) 5.0(-) Notato :.45(-4) mea 45 0 4 VI. CONCLUSION I th paper we have derved dagoally mplct four-tage fourth-order ad dperve order eght wth mall dpato cotat ad prcpal local trucato error. We have alo performed varou umercal tet. From the reult tabulated able III-VI, we coclude that the ew method more accurate for tegratg ecod-order equato poeg a ocllatory oluto whe compared to the curret DIRKN method derved by va der Houwe ad Sommejer [0], Sharp, Fe ad Burrage [4], Imo, Otuta ad Ramamoha [7] ad l-khaaweh, Imal, Sulema []. REFERENCES [] D. I. Okubor ad R. D. Skeel, Caocal Ruge-Kutta-Nytröm method of order fve ad x, J. Comput. ppl. Math., vol. 5, pp. 75-8, 994. [] E. Harer ad G. Waer, heory for Nytrom Method, Numer. Math., vol. 5, pp. 8-400, 975. [] E. Stefel ad D.G. Bett, Stablzato of Cowell method, Numer. Math, vol., pp. 54-75, 969. [4] G. Papageorgou, I. h. Famel ad Ch. toura, P-table gle dagoally mplct Ruge-Kutta-Nytröm method, Numercal lgorthm, vol. 7, pp. 45-5, 998. [5] I. Gladwell ad R. homa, Dampg ad phae aaly of ome method for olvg ecod order ordary dfferetal equato, Iterat. J. Numer. Method Egrg., vol. 9, pp. 495-50, 98. [6] J.C. Butcher, Numercal Method for Ordary Dfferetal Equato, Wley & So Ltd., Eglad, 00. [7] J. D. Lambert ad I.. Wato, Symmetrc multtep method for perodc tal-value problem, J. It. Math pplc, vol. 8, pp. 89-0, 976. [8] J. D. Lambert, Numercal Method for ordary Dfferetal Sytem-he Ital Value Problem, Wley & So Ltd., Eglad, 99. [9] J. R. Dormad, Numercal Method for Dfferetal Equato, CRC Pre, Ic, Florda, 996. [0] L. Bura ad L. Ngro, oe-tep method for drect tegrato of tructural dyamc equato, Iterat. J. Numer. Method Egrg, vol. 5, pp. 685-699, 980. [] M. M. Chawla ad S. R Sharma, Iterval of perodcty ad abolute tablty of explct Nytrom method for y f( x y), BI, vol., pp. 455-469, 98. [] P. J. va der Houwe ad B. P. Sommejer, Explct Ruge-Kutta(-

Nytröm) method wth reduced phae error for computg ocllatg oluto, SIM J. Numer. al., vol. 4, o., pp. 595-67, 987. [] P. J. va der Houwe ad B. P. Sommejer, Dagoally mplct Ruge- Kutta(-Nytröm) method for ocllatory problem, SIM J. Numer. al., vol. 6, o., pp. 44-49, 989. [4] P. W. Sharp, J. M. Fe ad K. Burrage, wo-tage ad hree-tage Dagoally Implct Ruge-Kutta-Nytröm Method of Order hree ad Four, J. Of Numercal aly, vol. 0, pp. 489-504, 990. [5]. E. Smo, E. Dma,.B. Sderd, Ruge-Kutta-Nytröm method for the tegrato of pecal ecod-order perodc tal-value problem, J. Comput. ppl. Mat., vol 5, pp. 7-6, 994. [6] W. Gautch, Numercal tegrato of ordary dfferetal equato baed o trgoometrc polomal, Numer. Math. vol., pp. 8-97, 96. [7] S. O. Imo, F. O. Otuta ad. R. Ramamoha, Embedded mplct Ruge-Kutta-Nytröm method for olvg ecod-order dfferetal equato, Iteratoal Joural of Computer Mathematc, vol. 8, o., pp. 777-784, 006 [8] N. Seu, M. Sulema ad F. Imal. (009, Jue). embedded explct Ruge Kutta Nytröm method for olvg ocllatory problem. 80(). valable: tack.op.org/physcr/80/05005 [9] R. C. lle, Jr. ad G. M. Wg, varat mbeddg algorthm for the oluto of homogeeou lear two-pot boudary value problem, J. Computer Phyc, vol. 4, pp. 40-58, 974. [0] B.P. Sommejer, ote o a dagoally mplct Ruge-Kutta- Nytr om method, J. Comp. ppl. Math., vol 9, pp. 95-99, 987. [] N. Seu, M. Sulema, F. Imal, ad M. Othma, New Dagoally Implct Ruge-Kutta-Nytröm Method for Perodc IVP, WSES raacto o Mathematc, vol. 9, pp. 679-688, 00. [] l-khaaweh, R.., Imal, F., Sulema, M., Embedded dagoally mplct Ruge-Kutta-Nytröm 4() par for olvg pecal ecod-order IVP. ppl. Math. Comp., vol. 90, 007, 80-84.

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