Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave Equation

Transcription

1 Appled Mathematcs do:.46/am..698 Publshed Onlne June ( Septc B-Splne Collocaton Method for the umercal Soluton of the Modfed Equal Wdth Wave Equaton Abstract Turab Geykl Seyd Battal Gaz Karakoc Department of Mathematcs Faculty of Educaton Inonu Unversty Malatya Turkey E-mal: Receved March 5 ; revsed Aprl ; accepted Aprl umercal solutons of the modfed equal wdth wave equaton are obtaned by usng collocaton method wth septc B-splne fnte elements wth three dfferent lnearzaton technques. The moton of a sngle soltary wave nteracton of two soltary waves and brth of soltons are studed usng the proposed method. Accuracy of the method s dscussed by computng the numercal conserved laws error norms L and L. The numercal results show that the present method s a remarkably successful numercal technque for solvng the MEW equaton. A lnear stablty analyss shows that ths numercal scheme based on a Crank colson appromaton n tme s uncondtonally stable. Keywords: Fnte Element Septc B-Splne Collocaton Solton Soltary Waves. Introducton Ths study s concerned wth the numercal soluton usng septc B-splne functons n collocaton method of the modfed equal wdth wave (MEW) equaton whch was ntroduced by Morrson et al. [] as a model equaton to descrbe the nonlnear dspersve waves. Few analytcal solutons of the MEW equaton are known. Thus numercal solutons of the MEW equaton can be mportant and comparson between analytc soluton can be made. Many methods have been proposed to solve the EW and MEW equaton. Gardner and Gardner solved the EW equaton wth the Galerkn s method usng cubc B-splnes as a tral and test functon [] and a Petrov-Galerkn method usng quadratc B-splne element [4]. Zak consdered the soltary wave nteractons for the MEW equaton by Petrov-Galerkn method usng quntc B-splne fnte elements [5] and obtaned the numercal soluton of the EW equaton by usng least-squares method [6]. Wazwaz nvestgated the MEW equaton and two of ts varants by the tanh and the sne-cosne methods [7]. Esen appled a lumped Galerkn method based on quadratc B-splne fnte element has been used for solvng the EW and MEW equaton [89]. Saka proposed algorthms for the numercal soluton of the MEW equaton usng quntc B-splne collocaton method []. A soluton based on a collocaton method ncorporated cubc B-splnes s nvestgated by Dağ and Saka []. Varatonal teraton method s ntroduced to solve the MEW equaton by Junfeng Lu []. Hamd et al. [] derved eact soltary wave solutons of the generalzed EW equaton usng Maple software. D. J. Evans and K. R. Raslan [4] studed the generalzed EW equaton by usng collocaton method based on quadratc B-splnes to obtan the numercal solutons of a sngle soltary waves and the brth of soltons. The modfed equal wdth wave equaton whch s as a model for non-lnear dspersve waves consdered here has the normalzed form [] U U U U () t t wth the physcal boundary condtons U as where t s tme and s the space coordnate s a postve parameter. For ths study boundary condtons are chosen Ua t Ub t Ua t Ub t () U a t U b t and the ntal condton as U f a b where f s a localzed dsturbance nsde the consdered nterval. Copyrght ScRes.

2 74 T. GEYIKLI ET AL.. Septc B-Splne Collocaton Method The nterval ab s parttoned nto unformly szed fnte elements by the knots such that a b and h. The septc B-splnes (= () + ) at the knots are defned over the nterval ab as [5] ( 4 ) 8( ) 8( ) 56( ) h 4 4 otherwse. () The set of splnes U t U t forms a bass for functons defned over ab. The numercal soluton to takes the form where U t t (4) are unknown tme dependent quanttes to be determned from the boundary and collocaton condtons and are septc B-splne. Each septc B-splne covers 8 elements thus each element s covered by 8 splnes. A typcal fnte nterval s mapped to the nterval [ ] by a local coordnate transformaton defned by h. Therefore septc B-splnes () n terms of over [ ] can be gven as (5) Snce all splnes apart from are zero over the element [ ]. For the problem the fnte elements are dentfed wth the nterval. Usng the nodal values U U U and U are gven n terms of the parameter by: U hu hu hu (6) and the varaton of gven by U over the element s U We now dentfy the collocaton ponts wth the knots and use (6) to evaluate U and ts space dervatves n (). Ths leads to a set of ordnary dfferental equatons of the form Copyrght ScRes.

3 T. GEYIKLI ET AL Z h h where If tme parameters and ts tme dervatves fnte dfference appromaton respectvely: Z n (7) are dscretzed by the Crank-colson formula and usual n n n n t We obtan a recurrence relatonshp between two tme levels n and n + relatng two unknown parameters where n n n n n n n n n n n n n n EZ M 56EZ 4 M 9 45EZ 5 M M EZ M EZ M EZ M E t M. h h n (7) n (8) For the frst lnearzaton we suppose that the quantty U n the non-lnear term UU to be locally constant. Ths s equvalent to assumng that n (7) all Z are equal to a local constant. Furthermore we can wrte the nonlnear term U U UUU (9) and apply the Rubn and Graves [6] lnearzaton technque n n n n n n UU U U UU UU () to the UU term n (8) and we can also apply the Caldwell and Smth [7] lnearzaton technque n n n n UU U U U U () to the UU term n (8). The system (8) conssts of lnear equaton n 7 unknowns T. To obtan a unque soluton to ths system we need 6 addtonal constrants. These are obtaned from the boundary condtons and can be used to elmnate and from the set () whch then becomes a matr equaton for the unknowns δ T of the form n n A B. The matrces A and B are septa-dagonal matrces and so are easly solved by septa-dagonal algorthm.. Intal State n To start evoluton of the vector of parameters can be determned from the boundary condtons and the ntal condton U. So we can rewrte appromaton (4) for the ntal condton U where parameters wll be determned. To determne the parameters we requre the ntal numercal appromaton U to satsfy the followng condtons: ) t must agree wth the eact ntal condton U at the knots. ) the frst second and thrd dervatves of the appromate ntal condton agree wth those of the eact ntal condtons at both ends of the range. These two condtons can be epressed as: U U U a U b U a U b U a U b. () The above condtons lead to K b matr equaton whch s solved by usng a varant of Thomas algorthm. Copyrght ScRes.

4 74 T. GEYIKLI ET AL. K and δ T b U U U U 4. Stablty Analyss The stablty analyss wll be based on the von eumann theory n whch the growth factor of a typcal Fourer mode n ˆn e kh () where k s the mode number and h the element sze s determned for a lnearsaton of the numercal scheme. Substtutng the Fourer mode () nto the lnearsed recurrence relatonshp (8) shows that the growth factor for mod k s a b g a b where a 8 4M 97 5M cos hk M hk M hk 4 5 cos cos b 45EZ sn hk 56EZ sn hk EZ sn hk. The modulus of g s therefore the lnearsed scheme s uncondtonally stable. 5. umercal Eamples and Results All computatons were eecuted on a pentum 4PC n the Fortran code usng double precson arthmetc. The conservaton propertes of () wll be eamned by calculatng the lowest three nvarants gven as b n C Ud h U a J b n a J b 4 n a J C U d h U n d 4 C U U h U U T whch correspond to mass momentum and energy respectvely [5]. The accuracy of the method s measured by both the error norm eact eact L U U h U U J and the error norm eact L U U eact ma U U. To mplement the method three test problems: moton of a sngle soltary wave nteracton of two soltary waves and the mawellan ntal condton wll be consdered. 6. Moton of Sngle Soltary Wave For ths problem we consder Equaton () wth the boundary condtons U as and the ntal condton U Asec h k. Ths problem has an eact soluton of the form Asechk vt U t whch represents the moton of a sngle soltary wave wth ampltude A here the wave velocty v A and k. For ths problem the analytcal values of the nvarants are [5] 4 Aπ A ka 4A C C C. k k k The analytcal values of nvarants are obtaned from () C C C.58. For the numercal smulaton of the moton of a sngle soltary wave we have used the parameters h. t.5 A.5 through the nter- Copyrght ScRes.

5 T. GEYIKLI ET AL. 74 val 8. The computatons are done untl tme t and n ths lenght we fnd error norms L L and numercal nvarants C C C at varous tmes. In Table we compare the values of the nvarants and error norms obtaned usng the present method wth the three dfferent appromaton and the results of [54 89] at dfferent tmes. We can easly see from the ths table that the error norms L and L are obtaned suffce ntly small and the quanttes n the varants reman almost constant durng the computer run for the frst and second lnearzaton techn ques but we can not say the same for the thrd lnearzaton technque. For the frst and second lnearzaton the numercal values of nvarants are C C C.58 and for the thrd lnearzaton numercal values of nvarants are C C C.544 at the t. Fgure shows that the proposed method perform the moton of propagaton of a soltary wave satsfactorly whch moved to the rght at a constant speed and preserved ts ampltude and shape wth n- creasng tme as epected. Ampltude s at t whch s located at whle t s.499 at t whch s located at.6. The absolute dfference n ampltudes at tmes t and t s so that there s a lttle change between ampltudes. 7. Interacton of Two Soltary Waves For ths problem we consder () wth boundary condtons U as nteracton of two postve soltary waves s studed by usng the ntal condton U A sec h k. where k. We frst used the parameters h. t.5 A A.5 5 through the nterval 8 whch s used by Zak [5]. These parameters provde soltary waves of magntudes and.5 Table. Invarants and error norms for sngle soltary waves. t Lneerzaton C C C L L Frst Second Thrd [5] [4] [8] [9] Copyrght ScRes.

6 744 T. GEYIKLI ET AL. Fgure. The moton of a sngle soltary wave. and peak postons of them are located at 5 and. The analytcal nvarants are [l4] C πa A C A A C A A Calculaton s carred out wth the tme step t.5 and space step h. over the regon 8. The eperment was ru n from t to t 55 to allow the nteracton to take place. Fgure shows the nteracton of two postve soltary waves. It can be seen that at t 5 the wave wth larger ampltude s on the left of the second wave wth smaller ampltude. The larger wave catches up wth the smaller one as tme ncreases. Interacton started at about tme t 5 over- occurred between tmes t 5 and 4 lappng processes and waves started to resume ther orgnal shapes after tme t 4. For the frst and second lnearzaton technques at t 55 the ampltude of larger waves s.49 at the pont 44.4 whereas the ampltude of the smaller one s.577 at the pont 4.6. It s found that the absolute dfference n ampltude s 7. for the smaller wave and.49 for the larger wave for ths algorthm. For the thrd lnearzaton technque at t 55 the ampltude of larger waves s.9959 at the pont 44.7 whereas the ampltude of the smaller one s at the pont 4.6. It s found that the absolute dfference n ampltude s 7.4 for the smaller wave and 4 for the larger wave for ths algorthm. In Table we compares values nvarants of the two soltary waves wth results from frst second and thrd lnearzaton. We see from the Table that for the frst and second lnearzaton technques all nvarants are conserved by less than durng the eperment. Thus we have found that the conservaton quanttes are satsfactorly constant Table. Invarants for nteracton of two soltary wave. A A.5 t Lneerzaton C C C Frst Second Thrd Copyrght ScRes.

7 T. GEYIKLI ET AL. 745 Fgure. Interacton of two soltary waves at dfferent tmes. wth the proposed algorthm. We have also studed the nteracton of two soltary waves wth the followng parameters: 5 A A together wth tme step t.5 and space step h =. n the range 5. The e perment was run from t to t 55 to allow the nteracton to take place. Fgure shows the development of the soltary wave nteracton. As s seen from the Fgure at t a wave wth the negatve ampltude s on the left of another wave wth the postve ampltude. The larger wave wth the negatve ampltude catches up wth the smaller one wth Copyrght ScRes.

8 746 T. GEYIKLI ET AL. Fgure. Interacton of two soltary waves at dfferent tmes. the postve ampltude as the tme ncreases. For the frst lnearzaton technque at t 55 the ampltude of the smaller wave s at the pont 5.5 whereas the ampltude of the larger one s.9896 at the pont.8. For the second lnearzaton technque at t 55 the ampltude of the smaller wave s at the pont 5.5 whereas the ampl- on e s.9867 at the pont tude of the larger.8. It s found th at the absolute dfference n am- pltudes s.5.7 for the smaller wave and.. 9 for the larger wave respectvely. Copyrght ScRes.

9 T. GEYIKLI ET AL The Mawellan Intal Condton As a last study we consder here s the numercal soluton of the Equaton () wth the Mawellan ntal condton wth the boundary condtons U e (4) U t U t U t U t. As t s known Mawellan ntal condton (4) the behavor of the soluton depends on the values of. So we have consdered varous values for. For the frst lnearzaton technque the computatons are carred out for the cases and.5 whch are used n the earler papers [54]. When.5 s used as shown Fgures 4(a) and (b) at tme t the Mawellan ntal condton does not cause development nto a clean soltary wave. However wth smaller values of..5. and.5 Ma- Table. Invarants for nteracton of two soltary wave. A A t Lneerzaton C C C Frst Second The analytcal nvarants by usng Equaton () can be found as C.4597 C. C Table lsts the values of the nvar- ants of the two soltary waves wth ampltudes A A. It can be seen that the values obtaned for the nvarants are satsfactorly constant durng the computer run. wellan ntal condton breaks up nto more soltary waves whch were drawn n Fgures 4(c)-(f) at tme t. The numercal conserved quanttes wth and.5 are gven n Table 4. It s observed that the obtaned values of the nvarants reman almost constant durng the computer run. Table 4. Invarants for Mawellan ntal condton dfferent µ. t C C C Copyrght ScRes.

10 748 T. GEYIKLI ET AL. (a) ( b) (c) (d) (e) (f) Fgure 4. Mawellan ntal condton state at tme t = (a) µ = (b) µ =.5 (c) µ =. (d) µ =.5 (e) µ =. (f) µ =.5. Copyrght ScRes.

11 T. GEYIKLI ET AL Conclusons In ths study a numercal soluton of the MEW equaton based on the septc B-splne fnte element has been presented wth three dfferent lnearzaton technques. Three test problems are worked out to eamne the performance of the algorthms. The performance and accuracy of the method were demonstrated by calculatng the error norms L and L on the moton of a sngle soltary wave. For the frst and second lnearzaton technques the error norms are suffcently small and the nvarants are satsfactorly constant n all computer run. The obtaned results from the frst and the second lnearzaton technques are almost the same and the computed results show that the present method s a remarkably successful numercal technque for solvng the MEW equaton and can also be effcently appled to other types of non-lnear problem.. References [] P. J. Morrson J. D. Mess and J. R. Carey Scatterng of Regularzed-Long-Wave Soltary Waves Physca D: onlnear Phenomena Vol. o. 984 pp do:.6/67-789(84)94-9 [] L. R. T. Gardner and G. A. Gardner Soltary Waves of the Regularzed Long Wave Equaton Journal of Computatonal Physcs Vol. 9 o. 99 pp do:.6/-999(9)947-5 [] L. R. T. Gardner and G. A. Gardner Soltary Waves of the Equal Wdth Wave Equaton Journal of Computatonal Physcs Vol. o. 99 pp. 8-. do:.6/-999(9)954- [4] L. R. T. Gardner G. A. Gardner F. A. Ayoup and. K. Amen Smulatons of the EW Undular Bore Communcatons n umercal Methods n Engneerng Vol. o pp do:./(sici)99-887(9977):7<58::aid-c M9>..CO;-E [5] S. I. Zak Soltary Wave Interactons for the Modfed Eual Wdth Equaton Computer Physcs Communcatons Vol. 6 o. pp. 9-. do:.6/s-4655(99)47- [6] S. I. Zak A Least-Squares Fnte Element Scheme for the Ew Equaton Computer Methods n Appled Mechancs and Engneerng Vol. 89 o. pp do:.6/s45-785(99)-6 [7] A. M. Wazwaz The tanh and sne-cosne Methods for a Relable Treatment of the Modfed Equal Wdth Equaton and Its Varants Communcatons n onlnear Scence and umercal Smulaton Vol. o. 6 pp do:.6/.cnsns.4.7. [8] A. Esen A umercal Soluton of the Equal Wdth Wave Equaton by a Lumped Galerkn Method Appled Mathematcs and Computatonal Vol. 68 o. 5 pp do:.6/.amc.4.8. [9] A. Esen A Lumped Galerkn Method for the umercal Soluton of the Modfed Equal Wdth Wave Equaton Usng Quadratc B Splnes Internatonal Journal of Computer Mathematcs Vol. 8 o pp do:.8/ [] B. Saka Algorthms for umercal Soluton of the Modfed Equal Wdth Wave Equaton Usng Collocaton Method Mathematcal and Computer Modellng Vol. 45 o. 9-7 pp do:.6/.mcm.6.9. [] B. Saka and İ. Dag Quartc B-Splne Collocaton Method to the umercal Solutons of the Burgers Equato n Chaos Soltons and Fractals Vol. o. May 7 pp do:.6/.c haos.5..7 [] J. F. Lu He s Varatonal Method for the Modfed Equal Wdth Wave Equaton Chaos Soltons and Fractals Vol. 9 o. 5 9 pp. -9. do:.6/.chaos [] S. Hamd W. E. Enrght W. E. Schesser J. J. Gottleb and A. Alaal Eact Solutons of the Generalzed Equal Wdth Wave Equaton Proceedngs of the Internatonal Conference on Computatonal Scence and Its Applcaton LCS Sprnger-Verlog Berln Vol. 668 pp [4] D. J. Evans K. R. Raslan and A. Alaal Soltary Waves for the Generalzed Equal Wdth (Gew) Equaton Internatonal Journal of Computer Mathematcs Vol. 8 o. 4 5 pp [5] T. Geykl Modellng Soltary Waves of a Ffth-Order on-lnear Wave Equaton Internatonal Journal of Computer Mathematcs Vol. 84 o. 7 July 7 pp do:.8/ [6] S. G. Rubn and R. A. Graves A Cubc Splne Appromaton for Problems n Flud Mechancs ASA TR R-46 Washngton DC 975. [7] J. Caldwell and P. Smth Soluton of Burgers Equaton wth a Large Reynolds umber Appled Mathematcal Modellng Vol. 6 o. 5 October 98 pp do:.6/s 7-94X(8)8-9 [8] A. Esen and S. Kutluay Soltary Wave Solutons of the Modfed Equal Wdth Wave Equaton Communcatons n onlnear Scence and umercal Smulaton Vol. o. 8 8 pp do:.6/.cnsns [9] K. R. Raslan Collocaton Method Usng Cubc B-Splne for the Generalzed Equal Wdth Equaton Internatonal Journal of Smulaton and Process Modellng Vol. o. - 6 pp Copyrght ScRes.