THE FORCED KORTEWEG DE VRIES EQUATION

Transcription

1 THE FORCED ORTEWEG DE VRIES EQUATION 4. INTRODUCTION We flid flow is disred y sll p i c geere srfce wve. Te flow of flid over oscle is clssicl d fdel prole i flid ecics. I is well kow rscriicl flow over loclied oscle geeres psre d dowsre olier wveris. Te flow s ee sccessflly odeled i e frework of e forced oreweg-de Vries eqio fdv, were ericl d sypoic lyicl eods re e oly eods for solvig i. Te fdv eqio s ee derived for wer wves y Akyls [], Cole [7], Mei [69], W [94], Lee e l. [66], d for ierl wves y Grisw d Sy [46]. Noe for wer wves, e forcig provided y e oo oscle, or y pplied srfce pressre field, or y sleder sip, re eqivle i e wekly olier pproiio. Grisw d is coworkers [47] sowed e wve-ris vig e srcre of sedy dlr ore lyiclly d lso cofired y ericl solios sig eplici fiie differece eod. We cosider oe-diesiol fdv eqio of wer srfce elevio, sisfyig 6 f 4.. s e flow is disred y p f for sll iervl derived y Cole [7]. Here is e correcio fcor of sll order o e Frode er F, s defied o e disred wer dep, is F. Te eqio is solved ericlly y 5

2 qiic B-splie collocio eod s i is pplied o e dvb eqio y Zki [00] d e sed o oserve wve geerio for vrios disrces f. Te proposed eod deosres e olier wve ris of e wer srfce geered y e forcig er. 4. QUINTIC B-SPLINE COLLOCATION METHOD Cosider e fdv eqio were is e preer d e sscrips d idice o e differeiio wi respec o d. were O pplyig Glerki pproc, we oied wek for of Eq. 4.. s W f d W is weig fcio. If ow e weig fcios re ideified wi Dirc del fcios W e, e reslig se of eqios kes e for f 0, 4.. were, 0,,,..., N. Tese eqios y lso e ideified s poi collocio codiios. Te odry codiios will e cose fro e pysicl odry codiio ssocied wi e fdv eqio,, d e collocio odry codiios reqired for isrig iqe qiic B-splie solio,, ,,

3 Cosider, / N,,,, N, e : 0 N is eql disce priio of e iervl, y e kos. Te se of fcios,,..., N fors sis for fcios defied over e fiie regio,. A glol pproiio, is give y N, N, 4..6 were e re ie-depede preers o e deeried. Te B-splie d is priciple derivives vis oside e regio. Te iervls, re, ideified wi fiie elees, wic re ec covered y si B-splies. Over e elee e vriio of e fcio, is ke i e fro,, j j 4..7 j I ers of locl coordie syse give y, were d 0, epressios for e elee splies re [7] Te odl vles ,, d e ko re give i ers of y

4 were e dses deoe differeiio wi respec o. To oi recrrece reliosip for e ericl solio, ie ceer o /, were is ie sep, d se Crk-Nicolso pproc wi / 4.. / 4..4 / / 4..5 were e sscrip is e ode lel wile e sperscrips d re ie lels. Eq is qsi-lieriio oied rog rieic e i ie. Tese resls re ll secod order ccre pproiios o e vles ie /. Use eqios o evle d is spce derivives d we ve for ec ko eqio relig preers djce ie levels, d i e for: f e d c e d c 4..6 were c e d c

5 d e Te syse of eqios 4..6 cosiss of N qsi-lier eqios i N 5 kows,,,,,,. To oi iqe solio o is syse for 0 N N N ddiiol cosris o e derivives e ed pois re oied fro e collocio odry codiios 4..4 d Terefore e se of eqios 4..6 ogeer wi e odry codiios 4..4 d 4..5 c e wrie i e ri eqio of e for Ad Bd F 4..7 were A d B re defecive pe-digol N N rices d F is col vecor wose eries re f. of vecor iiil vecor Te ie evolio of e pproie solio N, is deeried fro d wic is fod y repeedly solvig e ri eqio 4..7 oce e 0 d s ee coped fro e iiil codiio. As e iiil codiio, vrios fcios c e sed sc s,0 0 represeig o wve or soliry wve i for of sec yperolic fcio. I or cse, e iiil vecor 0 d is se o ero. 4. NUMERICAL RESULTS Te ericl procedre descrie i e previos secio is sed o oserved wves geered y flow disred y p. Mos of or clclios sed 0.5, 0. d er of ie sep 000 p is plced i e iddle of e doi. N for oservio doi,00 0. Te 57

6 Te scee is firs esed for dv eqio y givig e force f 0 i e rig d side of Eq. 4.. d iiil codiio is,0 sec wi e cres loced 0. Alyiclly e iiil codiio 4.. prodces soliry wve rvelig wi cos speed d wio cgig e spe we for e dv Eq. 4.. wi f 0. Tese crcers re deosred i 4 Fig. 4. s solio of Fig. 4.: Plo of, for soe vles of clcled for fdv eqio wi f 0, sig. 5. dv eqio for. 5 d e iiil codiio give i Eq. 4.. for 0., Plo of, for soe vle of is sifed pwrd proporiol o e vle of. For sller vles of e wve rvels o e rig slower d i is coied decresig e wve rvels i differe direcio. A e wve reces sedy. Tis grees wi e lyicl solio, i.e., sec

7 Now e scee is sed o solve e eqio 4.. ivolvig e forcig er. Te preer d e fcio f re ip cosidered effecig o e solio. Firs we perfored e solio for vrios vles of coied wi p 0.0sec f , for 84 96, oerwise 4.. We sow plo of, i Fig. 4. correspodig o.0,0.5,0. 0 d. 0. Two pois re idiced s e plce were e wve ppers i.e. e lef d rig eds of e p, 84 d 96. Te srfce ove e lef ed is psed p d e oer ed is opposie direcio, s flow is disred d e rs dow freely fer pssig e p. Tis coiosly grows p e elevio d followed y pperig oer wves. Differe vles of e elevio of o wves c e see i Fig.. I is idiced e elevio ove 84 d 96 grows p wio c propgig oer wves we eds o e plide of e p. We c see is i Fig. 4.c for e se vle of d e plide. We e oserved e effec of e forcig er y replcig e secyperolic fcio wi oer fcio sc s sisoidl. We oi siilr profile of e elevio. B we e forcig er is egive of Eq. 0 we oi opposie coposiio of e srfce elevio descrie ove. 59

8 c 0. 0 d. 0 Figre 4.: Plo of, priclr ie for vrios vles of c 0. 0 d d CONCLUSION We ve developed qiic B-splie collocio eod o solved e forced dv eqio represeig wve geerio s ifor flow disred y p o e oo of cel. Te eod s ee esed y coprig e ericl solio of e dv eqio o e lyicl solio i e for of soliry wve d we oi greee ewee o solios. We forcig er is iclded we oi e solio descries geerig ri of wves sc s oied y Cole [7]. Te evolio of e wves is oserved for vrios vles of e preer correspodig o Frode er. 60