A Compact Finite Difference Method for Solving the General Rosenau RLW Equation

Transcription

1 A Compact Fnte Dfference Method for Solvng the General Rosenau RLW Equaton Ben Wongsaja, Kanyuta Poochnapan, Thongcha Dsyadej Abstract In ths paper, a compact fnte dfference method to solve the Rosenau RLW equaton s proposed. A numercal tool s appled to the model by usng a three level average mplct fnte dfference technque. The fundamental conservatve property of the equaton s preserved by the presented numercal scheme, the estence unqueness of the numercal soluton are proved. Moreover, the convergence stablty of the numercal soluton are also shown. The new method gves second fourth order accuracy n tme space, respectvely. The algorthm uses fve pont stencl to appromate the dervatves for the space dscretzaton. The numercal eperments show that the proposed method mproves the accuracy of the soluton sgnfcantly. Inde Terms fnte dfference method, Rosenau RLW equaton. I. INTRODUCTION A nonlnear wave phenomenon s the mportant area of scentfc research, whch many scentsts n the past have studed about mathematcal models eplanng the wave behavor. There are mathematcal models whch descrbe the dynamc of wave behavors for eample, the KdV equaton, the RLW equaton, the Rosenau equaton, many others [1 [1. The KdV equaton has been used n very wde applcatons, such as magnetc flud waves, on sound waves, longtudnal astgmatc waves [ [. The RLW equaton, whch was frst proposed by Peregrne [7, [8 provdes an eplanaton on a dfferent stuaton of a nonlnear dspersve wave from the more classcal KdV equaton. The RLW equaton s one of models whch are encountered n many areas, e.g. on acoustc plasma waves, magnetohydrodynamc plasma waves, shallow water waves. Snce the case of wave wave wave wall nteractons cannot be descrbed by the KdV equaton, Rosenau [9, [1 proposed an equaton for descrbng the dynamc of dense dscrete systems; t s known as the Rosenue equaton. The estence unqueness of the soluton for the Rosenau equaton were proved by Park [11, [1. For the further consderaton of the nonlnear wave, a vscous term u t needs to be ncluded: u t u t u t u u p =, 1 where p s an nteger u s a known smooth functon. Ths equaton s usually called the Rosenua RLW equaton. If p =, then Eq. 1 s called the usual Rosenau RLW equaton. Moreover, f p = 3, then Eq. 1 s called the modfed Rosenau RLW equaton. The behavor of the Kanyuta Poochnapan, Correspondng Author, Department of Mathematcs, Faculty of Scence, Chang Ma Unversty, Chang Ma 5, Thal emal: kanyuta@hotmal.com, k.poochnapan@gmal.com Ben Wongsaja, Department of Mathematcs, Faculty of Scence, Chang Ma Unversty, Chang Ma 5, Thal emal: ben.wongsaja@gmal.com Thongcha Dsyadej, Electrcty Generatng Authorty of Thal, Phtsanulok 5, Thal e-mal: thongcha.d@egat.co.th soluton to the Rosenau RLW equaton wth the Cauchy problem has been well studed for the past years [13 [18. It s known that the soltary wave soluton for Eq. 1 s u, t = e ln{p33p1p1/[p 3p p7}/p1 [ sech /p1 p 1 p 8p ct, where p s an nteger c = p p 3 1p p 5/p p 3 1p 1p 1. The Rosenau RLW equaton has been solved numercally by varous methods for eample, see [13 [18. Zuo et al. [13 have proposed the Crank Nocolson fnte dfference scheme for the equaton. The convergence stablty of the proposed method were also dscussed. Obvously, the scheme n [13 requres heavy teratve computatons because the scheme s nonlnear mplct. Pan Zhang [1, [15 developed lnearzed dfference schemes whch are three level conservatve mplct for both the usual Rosenau RLW p = the general Rosenau RLW p equatons. The second order accuracy uncondtonal stablty were also proved. In ths paper, we consder the followng ntal boundary value problem of the general Rosenau RLW equaton wth an ntal condton: boundary condtons u, = u, l r, u l, t = u r, t =, u l, t = u r, t =, t T. 3 The ntal boundary value problem possesses the followng conservatve propertes: Qt = r l u, td = r l u, d = Q, Et = u L u L u L = E. When l r, the ntal boundary value problem 1 3 s consstent, so the boundary condton 3 s reasonable. By observaton, the total accuracy of a specfc method s affected by not only the order of accuracy of the numercal method but also other factors. That s, the conservatve appromaton property of the method s another factor that has the same or possbly even more mpact on results. Better solutons can be epected from numercal schemes whch have effectve conservatve appromaton propertes rather than the ones whch have nonconservatve propertes [19, [. To create the dscretzaton equaton, a fnte dfference

2 method s appled n the present research snce conservatve appromaton analyss by the mathematcal tools has been developng untl now. The content of ths paper s organzed as follows. In the net secton, we descrbe a conservatve mplct fnte dfference scheme for the general Rosenau RLW equaton 1 wth the ntal boundary condtons 3. Some prelmnary lemmas dscrete norms are gven the nvarant property Q n s proved. We dscuss about the solvablty of the fnte dfference scheme, the estence unqueness of the soluton are also proved n the Secton 3. Secton presents complete proofs on the convergence stablty of the proposed method wth convergence rate O h. The results of valdaton for the fnte dfference scheme are presented n Secton 5, where we make a detaled comparson wth avalable data, to confrm llustrate our theoretcal analyss. Fnally, we fnsh our paper by concludng remarks n Secton. II. FINITE DIFFERENCE SCHEME In ths secton, we ntroduce a fnte dfference scheme for the formulaton of Eqs The soluton doman Ω = {, t l r, t T } s covered by a unform grd: Ω h = {, t n = l h, t n = n, M, n N}, wth spacngs h = r l /M = T/N. Denote u n u, t n, Ω h = {, t n = l h, t n = n, 1 M 1, n N}, Z h = {un = u n u = u M =, 1 M 1}. We use the followng notatons for smplcty: u n 1 = un1 u n t = un1 u n u n u n = un 1 un h u n ˆ = un 1 un 1 h, ū n = un1, u n ˆt = un1 u n1 u n1, u n = un un 1, h, u n, v n = h u n v n, =1 u n = u n, u n, u n = ma 1 un. By settng w = u t u u t u p, Eq. 1 can be wrtten as w = u t. By the Taylor epanson, we obtan w n = t u n = un ˆt O, w n = [u n h ˆt 1 [u n h ˆt [[u n p ˆ h t u n [u n ˆ h t u n 3 u p n,, 3 u n O h. 5 From Eq., we have t u n = t u n u 3 n u 3 p n w n. Then, w n = [u n h ˆt 1 [[u n p ˆ h h t u n 3 u p n [u n ˆ h [ u n ˆt 3 u n [ t u n 3 u n 3 u p n w n O h. 7 Ths mples that w n = u n h ˆt 1 t u n un ˆ [u n p ˆ u n ˆt h w n O h. 8 Usng second order accuracy for appromaton, we obtan u n =un O h, w n = wn O h. The followng method s a proposed fnte dfference scheme to solve the problem 1 3: u n ˆt 1 h u n ˆt 1 h u n 1 ˆt u n ˆ [u n p ˆ = ; where 1 M 1, 1 n N 1, 9 u = u, M, 1 u n = u n M =, u n = u n M =, 1 n N. 11 A three step method s used for the tme dscretzaton of the above descrbed scheme. After the new tme dscretzaton of Eq. 9 s performed, three fve pont stencls appromatng the dervatves for the space dscretzaton are used to obtan an algebrac system. The matr system of Eq. 9 s bed wth penta dagonals we use a stard routne of the MATLAB to solve the system The nonlnear term of Eq. 1 s hled by usng the lnear mplct scheme. Therefore, the equatons are solved easly by usng the presented method snce t does not requre etra effort to deal wth the nonlnear term. Lemma 1: Pan Zhang [15 For any two mesh functons u, v Zh, we have uˆ, v = u, vˆ, u, v = u, v, v, u = v, u, u, u = u, u = u. Furthermore, f u n = u n M =, then t mples u, u = u.

3 Theorem : Suppose that u H, then the scheme 9 11 s conservatve n sense: Q n = h =1 u n1 u n = Q n1 =... = Q, 1 under assumptons u 1 = u 1 = u = u M1 =. Proof: By multplyng Eq. 9 by h, summng up for from to M 1, consderng the boundary condtons, assumng u 1 = u 1 = u = u M1 =, we get h =1 u n1 u n1 =. Then, ths gves Eq. 1. Lemma 3: Dscrete Sobolev s nequalty [1 There est two constants C 1 C such that u n C 1 u n C u n. Theorem : Suppose u H [ l, r, then the soluton u n satsfes u n C u n C, whch yelds u n C. Proof: It follows from the ntal condton 1 that u C. The frst level u 1 s computed by the fourth order method. Hence, the followng estmates are gotten about u 1 C u 1 C. Now, we use the nducton argument to prove the estmate. We assume that u k C for k =, 1,,..., n. 13 Takng the nner product of Eq. 9 wth ū n usng Lemma 1, we obtan u n1 u n1 1 h u n1 u n1 1 h u n1 u n1 1 = u n ˆ, ū n [u n p ˆ, ū n. Accordng to the Cauchy Schwarz nequalty drect calculaton, t gves u ṋ u n, u n ˆ, ū n u n 1 u n1 1 u n1. From Eq. 13, the Cauchy Schwarz nequalty, Lemma 1, we get [u n p ˆ, ū n = h u n p u n1 =1 C u n 1 u n1 1 u n1 ˆ u n1. Settng B n = u n u n1 1 h u n u n1 1 h u n 1 u n1, then B n1 B n C B n1 B n. If s suffcently small, whch satsfes k kc k >, then B n1 1 C 1 C Bn 1 kc B n ep kct B. Hence u n1 C, u n1 C, u n1 C, whch yeld u n1 C by Lemma 3. III. SOLVABILITY In ths secton, we prove the estence unqueness of our proposed scheme that mples the unque solvablty. Theorem 5: The fnte dfference scheme 9 11 s unquely solvable. Proof: By usng the mathematcal nducton, we can determne u unquely by an ntal condton then choose a fourth order method to compute u 1. Now, suppose u, u 1, u,..., u n to be solved unquely. By consderng Eq. 9 for u n1, we have 1 un1 1 1 h 1 u n1 1 h 1 u n1 =. 1 By takng an nner product of Eq. 1 wth u n1, we obtan 1 u n1 1 1 h u n1 1 1 h 1 u n1 =. By the Cauchy Schwarz nequalty Lemma 1, we have u n1 = u n1, u n1 1 u n1 1 u n1. Then, 1 u n1 1 h 1 u n1 =. Therefore, Eq. 1 has the only one soluton Eq. 9 u n1 s unquely solvable. Ths completes the proof of Theorem 5. IV. CONVERGENCE AND STABILITY In ths secton, we prove the convergence stablty of the scheme Let e n = v n un, where vn u n are the solutons of , respectvely. Then, we obtan the followng error equatons: r n = e n ˆt 1 h e n ˆt 1 h 1 e n ˆt e n ˆ [v n p ˆ [u n p ˆ, 15 where r n denotes the truncaton error. By usng the Taylor epanson, t s easy to see that r n = O h holds as, h. The followng lemmas are essental for the proof of convergence stablty of our scheme. Lemma : Dscrete Gronwall s nequalty [1 Suppose that ωk ρk are nonnegatve functons ρk s nondecreasng. If C > k1 ωk ρk C ωl, k, l=

4 then Settng ωk ρke Ck, k. Lemma 7: Pan Zhang [15 Suppose that u E n = e n e n1 1 h e n H [ l, r, then the soluton u n e n1 of Eqs. 1 3 satsfes u L C, u L C, 1 h e n 1 e n1, u L C, u L C. then Eq. can be rewrtten as The followng theorem shows that our scheme converges to the soluton wth convergence rate O h. Theorem 8: Suppose u H [ l, r, then the soluton u n converges to the soluton for the problem n the sense of the rate of convergence s O h. E n1 E n r n C E n1 E n, 1 C E n1 E n r n CE n. Proof: By takng an nner product on both sdes of Eq. 15 wth ē n e n1 e n1, we get If s suffcently small, whch satsfes 1 C >, then e n1 e n1 e 1 h n1 e n1 E n1 E n C r n CE n. 3 e 1 h n1 Summng up Eq. 3 from 1 to n, we have e n1 = r n, ē n n 1 E n1 E 1 C n r k C E k. e ṋ, ē n [v n p ˆ [u n p ˆ, ē n. 1 k=1 k=1 Accordng to the Schwarz nequalty, Lemma 1, Theorem, Lemma 7, we obtan [v n p ˆ [u n p ˆ, ē n Thus, we can use a fourth order method to compute u 1 such that E 1 O h, = h =1 = h =1 = h =1 [[v n p ˆ [u n p ˆ ē n [ e n [v n p u n p ē n ˆ p v n pk u n k ē n ˆ k=1 C e n ē ṋ C e n e n1 ˆ e n1 ˆ. 17 By the Cauchy Schwarz nequalty, Lemma 1, a drect calculaton, we obtan e ṋ e n, 18 e n = e n, e n 1 e n e n, 19 e ṋ, ē n e ṋ 1 e n1 e n1, r n, ē n r n 1 e n1 e n1. 1 From Eqs.1 1, they yeld e n1 e n1 1 h e n1 e n1 1 h e n1 e n1 1 r n C e n1 e n e n1 e n1 e n e n1. n r k n k=1 ma l n1 r l T O h. By Lemma, we obtan E n O h, that s e n O h, e n O h. From Eq., we obtan By Lemma 3, e n O h, e n O h, e n O h. e n O h. Ths completes the proof. Theorem 9: Under the condtons of Theorem 8, the soluton u n of Eqs s stable n norm. V. NUMERICAL EXPERIMENTS In ths secton, we present numercal eperments on a test problem to confrm llustrate the accuracy of our proposed method. The accuracy of the method s measured by the comparson of numercal solutons wth eact solutons as well as other numercal solutons from the method n the lterature [15 by usng norm. The ntal condton assocated for the Rosenau RLW equaton takes the form: u = e ln{p33p1p1/[p 3p p7}/p1 [ sech /p1 p 1 p 8p.

5 TABLE I COMPARISON OF ERRORS WITH =.1, h =.5, l =, AND r = 1 AT t =. e 1 e 1 3 p Pan&Zhang Pan&Zhang TABLE II COMPARISON OF ERRORS WITH =.1, h =.5, l =, AND r = 1 AT t = Pan Zhang [15 e 1 e 1 p Pan&Zhang Pan&Zhang Fg.. Absolute error dstrbuton at p = 8, h =.5, = h, t =..1.5 Pan Zhang [ Pan Zhang [ Fg. 1. Absolute error dstrbuton at p =, h =.5, = h, t =. For u 1, we employ a two level method to estmate the soluton by u n t 1 h u n t 1 h u n 1 t u n 1 ˆ [un p ˆ = ; 1 M 1, 1 n N 1. 5 We make a comparson between the scheme 9 11 the scheme proposed n [15. The rate of convergence s computed usng two grds, accordng to the formula: e h Rate = log e h/. The results n term of errors at t =, =.1, dfferent p, by usng l = r = 1, wth h =.5 h =.5 are reported n Tables I II. It s clear that the results obtaned by the scheme 9 11 are more accurate than the ones obtaned by the scheme n [15. Absolute error dstrbutons for the two methods wth =.5, h =.5, t = are drawn at p = 8 n Fgs. 1, respectvely. The results obtaned by the scheme 9 11 are greatly mproved when compared to those by the scheme n [15. It can be easly observed that the mamum Fg. 3. Error e versus t at p =, h =.5, = h Zhang Pan [ Fg.. Error e versus t at p =, h =.5, = h. error s taken place around the peak ampltude of the soltary wave then the scheme 9 11 s appled n ths area. Fgs. 3 show errors at t [, wth =.5, h =.5, p =, 8 by comparng wth the Pan&Zhang method [15. It s observed that both errors ncrease wth tme qute lnearly but the error of the present method s less than that of the Pan&Zhang method [15. As shown n Tables III IV, on one partcular choce of the parameters, the estmated rate s close to the theoretcally

6 Pan Zhang [ Fg. 5. Error e versus t at p = 8, h =.5, = h Pan Zhang [ Fg.. Error e versus t at p = 8, h =.5, = h. predcted fourth order rate of convergence. We can also say that when we use smaller tme space steps, numercal values are almost the same as eact values. The CPU tme for two methods are lsted n Tables III IV. It can be seen that the computatonal effcency of the present method are slghtly better than that of Pan&Zhang method [15, n term of CPU tme. However, the constructon of the novel scheme requres only a regular fve pont stencl at a hgher tme level, whch s smlar to the stard second order Crank Ncolson scheme Pan&Zhang scheme [15. As n Tables V VI, the values of Q n E n at any tme t [,, whch results from the present method, concde wth the theory. The quanttes Q n E n seem to be conserved on the average,.e. they are contaned n a small nterval but there are fluctuatons. Fgs. 7 8 show numercal solutons at t = wth p = 8. The results from the Pan&Zhang method [15 are slghtly oscllate at the left sde of the soltary wave n case of p = 8. However, the results from the present method are almost perfectly sharp n both cases p = 8. From the pont of vew for the long tme behavor of the resoluton, the present method can be seen to be much better than the method n [15. The soltary waves obtaned by the present scheme are plotted n Fgs. 9 1 usng =.5, h =.5, l =, r =, p =, 8. The soltons at t = TABLE III RATE OF CONVERGENCE AND CPU TIME WITH p = AND t =. =.5, h =.5, h, h 1, h e Rate.39.1 e Rate CPU tme s Pan&Zhang e Rate e Rate CPU tme s TABLE IV RATE OF CONVERGENCE AND CPU TIME WITH p = 8 AND t =. =.5, h =.5, h, h 1, h e Rate e Rate CPU tme Pan&Zhang e Rate e Rate CPU tme TABLE V DISCRETE MASS Q n. =.5, h =.5 t p = p = 8 t = t = t = t = Q TABLE VI DISCRETE ENERGY E n. =.5, h =.5 t p = p = 8 t = t = t = t = E agree wth the solton at t = qute well, whch also shows the accuracy of the scheme. VI. CONCLUSIONS The new conservatve fnte dfference scheme for the Rosenau RLW equaton s ntroduced analyzed. The present method gves an mplct lnear system, whch can be easly mplemented. Ths method shows the second fourth order accuracy n tme space, respectvely. In

7 .7. Eact soluton Pan Zhang [ t = t = t = Fg. 7. Numercal solutons at p =, l =, r = 3, h =.5, = h, t =..8 Eact soluton.7 Pan Zhang [ Fg. 8. Numercal solutons at p = 8, l =, r = 3, h =.5, = h, t = t = t = t = Fg. 9. Numercal solutons at p =. addton, the numercal eperments show that the present method supports the analyss of convergence rate. It s obvous from numercal eperments that the present method, the scheme 9 11, gves the well resoluton for the Rosenau RLW equaton. It s possble that the soltary wave obtaned by ths novel method can be smoothed out, at long tme, by type of the hgh order accuracy. ACKNOWLEDGMENT Ths research was supported by Chang Ma Unversty. Fg. 1. Numercal solutons at p = 8. REFERENCES [1 K. Mohammed, New Eact Travellng Wave Solutons of the 31 Dmensonal Kadomtsev Petvashvl KP Equaton, IAENG Internatonal Jounal of Appled Mathematcs, vol. 37, no. 1, pp , 7. [ J.V. Lambers, An Eplct, Stable, Hgh Order spectral Method for the Wave Equaton Based on Block Gaussan Quadrature, IAENG Internatonal Jounal of Appled Mathematcs, vol. 38, no., pp. 33 8, 8. [3 JC Chen W. Chen, Two Dmensonal Nonlnear Wave Dynamcs n Blasus Boundary Layer Flow Usng Combned Compact Dfference Methood, IAENG Internatonal Jounal of Appled Mathematcs, vol. 1, no., pp , 11. [ A.R. Bahadr, Eponental Fnte Dfference Method Appled to Korteweg de Vres Equaton for Small Tmes, Appled Mathematcs Computaton, vol. 1, no. 3, pp. 75 8, 5. [5 S. Ozer S. Kutluay, An Analytcal Numercal Method Appled to Korteweg de Vres Equaton, Appled Mathematcs Computaton, vol. 1, no. 3, pp , 5. [ Y. Cu D.-k. Mao, Numercal Method Satsfyng the Frst Two Conservaton Laws for the Korteweg de Vres Equaton, Journal of Computatonal Physcs, vol. 7, pp , 7. [7 D.H. Peregrne, Calculatons of the Development of an Undular Bore, Journal of Flud Mechancs, vol. 5, pp , 199. [8 D.H. Peregrne, Long Waves on a Beach, Journal of Flud Mechancs, vol. 7, pp , [9 P. Rosenau, A Quas Contnuous Descrpton of a Nonlnear Transmsson Lne, Physca Scrpta, vol. 3, pp , 198. [1 P. Rosenau, Dynamcs of Dense Dscrete Systems, Progress of Theoretcal Physcs, vol. 79, pp. 18 1, [11 M.A. Park, On the Rosenau Equaton, Mathematca Aplcada e Computaconal, vol. 9, no., pp , 199. [1 M.A. Park, Pontwse Decay Estmate of Solutons of the Generalzed Rosenau Equaton, Journal of the Korean Mathematcal Socety, vol. 9, pp. 1 8, 199. [13 J.-M. Zuo, Y.-M. Zhang, T.-D. Zhang, F. Chang, A New Conservatve Dfference Scheme for the General Rosenau RLW Equaton, Boundary Value Problems, vol. 1, Artcle ID 51, 13 pages, 1. [1 X. Pan L. Zhang, On the Convergence of a Conservatve Numercal Scheme for the Usual Rosenau RLW Equaton, Appled Mathematcal Modellng, vol. 3, pp , 1. [15 X. Pan L. Zhang, Numercal Smulaton for General Rosenau RLW Equaton: An Average Lnearzed Conservatve Scheme, Mathematcal Problems n Engneerng, vol. 1, Artcle I517818, 15 pages, 1. [1 X. Pan, K. Zheng, L. Zhang, Fnte Dfference Dscretzaton of the Rosenau RLW Equaton, Applcable Analyss, vol. 9, no. 1, pp , 13. [17 N. Atouan K. Omran, Galerkn Fnte Element Method for the Rosenau RLW Equaton, Computers Mathermatcs wth Applcatons, vol., pp , 13. [18 R.C. Mttal R.K. Jan, Numercal Soluton of General Rosenau RLW Equaton Usng Quntc B Splnes Collocaton Method, Communcaton n Numercal Analyss, vol. 1, Artcle ID cna-19, 1 pages, 1. [19 J. Hu, Y. Xu, B. Hu, Conservatve Lnear Dfference Scheme for Rosenau KdV Equaton, Advances n Mathematcal Physcs, vol. 13, Artcle ID 3718, 7 pages, 13.

8 [ F.E. Ham, F.S. Len, A.B. Strong, A Fully Conservatve Second Order Fnte Dfference Scheme for Incompressble Flow on Nonunform Grds, Journal of Computatonal Physcs, vol. 177, pp ,. [1 Y. Zhou, Applcaton of Dscrete Functonal Analyss to the Fnte Dfference Method, Inter. Acad. Publshers, Bejng, 199.