THE CONSERVATIVE DIFFERENCE SCHEME FOR THE GENERALIZED ROSENAU-KDV EQUATION

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1 Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 S93 THE CONSERVATIVE DIFFERENCE SCHEME FOR THE GENERALIZED ROSENAU-KDV EQUATION by J ZHOU a*, Maobo ZHENG b, ad Re-Xi JIANG c a School of Mathematics ad Statistics, Yagtze Normal Uiversity, Chogqig, Chia b Chegd Techological Uiversity, Chegd, Chia c Eglish Teachig ad Research Grop, Forteeth Middle School of Flig, Chogqig, Chia Itrodctio Origial scietific paper DOI:.98/TSCI6S393Z I this paper, merical soltios for the geeralized Rosea-KdV eqatio are cosidered via the eergy ad mometm coservative o-liear implicit fiite differece scheme. Uiqe eistece of the coservative properties of the soltios for the differece scheme is show. Nmerical reslts demostrate that the scheme is efficiet ad reliable. Key words: geeralized Rosea-KdV eqatio, differece scheme, properties, merical eperimet The well-kow Korteweg-de Vries (KdV) eqatio [-4]: t + + = () has bee sed to describe the wave propagatio ad spread iteractio. I this paper, we cosider the followig geeralized Rosea-KdV eqatio: p t t ( ) = () where p is a iteger. Whe p eq. () is called as sal Rosea-KdV eqatio. I [, ], the solitary soltios for the geeralized Rosea-KdV eqatio with sal solitary asatze method were discssed ad the two ivariats for the geeralized Rosea- KdV eqatio were give. I [], the two types of solito soltio, i. e., solitary wave soltio ad siglar solito, were researched. Frthermore, they also sed pertrbatio theory ad semi-variatio priciple to stdy the pertrbed geeralized Rosea-KdV eqatio aalytically. I [3], asatze method was applied to obtai the topological solito soltio or shock soltio of this eqatio. Moreover, three methods, that is, asatze method, G'/G-epasio method as well as the ep-fctio method were applied to etract a few more soltios to this eqatio i [4]. Bt the merical method to the iitial-bodary vale problem of geeralized Rosea-KdV eqatio has ot bee stdied till ow. I [5, 6], two coservative differece schemes for the geeralized Rosea-KdV eqatio were proposed. Bt their schemes ca oly preserve oe coservative law. * Correspodig athor; flzzklm@6.com

2 S94 Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 I this paper, we propose a coservative o-liear Crak-Nicolso-implicit differece scheme for the eqatio. The stdies show that the scheme does ot eed to select aother scheme to help iitial comptatio sch as the average liear scheme i [5]. It shold be oted that the merical simlatios show that the scheme preserves two coservative ivariats, which is better tha those reslts i [5, 6]. Hece, i this paper, we propose a coservative two-level o-liear implicit fiite differece scheme for the geeralized Rosea-KdV eq. () with the bodary coditios: ad iitial coditio: [, ]: X (,) t = X (,) t ( X,) t = ( X,) t l r l r ( X,) t = ( X,) t t [, T] (3) l r (,) = ( ) (4) The iitial-bodary vale problem presets the followig coservative properties Xr Xr (5) X X Mt ( ) = d = d = M() l l Xr (6) L L X Et ( ) = ( + )d = + = E() l Whe Xl, Xr, the iitial-bodary vale problem, eqs. ()-(4), ad the Cachy problem, eq. (), are cosistet. Coservative implicit differece scheme I this sectio, we first give some otatio sed i this paper, ad propose the coservative differece scheme for the problem of eqs. ()-(4). As sal, deote = X l + h, t = τ, J, N, where h = (X r X l )/J ad τ are the iform the spatial ad temporal step size, respectively. Let ( h, τ ), h J J+ Z = { = ( ) = = = J + }. Throghot this paper, we deote C as a geeral costat idepedet of h ad τ. Defie the differece operators, ier prodct ad orms are: t ( ) ( ) h + t ( )ˆ t ( ) h + t, J = v = h v, p p i p i ( ) ( )ˆ =,, + h + + h = ma J I view of ( ) = /( + p) Σ ( ) [7], we ca costrct the followig coservative implicit fiite differece scheme for the problems, eqs. ()-(4): i ( ) ( ) p + / + / + / + / t ˆ ˆ t p ( ) ( ) ( p i ) ( ) = (7) + ˆ

3 Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 S95 = ( ) J (8) = J ( ) ˆ = ( J) ˆ ( ) = ( J ) = (9), h Lemma.. [8] For ay two mesh fctios v Z, oe has: v, = v,, ˆ, v= v, ˆ, v, =, v () The we have: J,, =, = () Frthermore, if ( ) = ( ) the: = () To prove the eistece of soltio for scheme, eqs. (7)-(9), the followig Browder fied poit theorem shold be itrodced. For the proof, see [9]. Lemma.. (Browder fied poit theorem) Let H be a fiite dimesioal ier prodct space. Sppose that g: H H is cotios ad there eists a α sch that * * * g ( ), >, H, = α. The there eists H sch that g ( ) = ad α. Theorem.3. There eists Z h satisfyig the differece scheme (7)-(9). Proof: For N, we assme that,,, Z h satisfy the differece scheme (.)-(.3). Net we prove that there eists + satisfyig eqs. (7)-(9). Defie a operator g o Z as follows: h p τ i p i τ ˆ τ + + p (3) + ˆ gv ( ) = v + v + v + v + ( v ) ( v ) = By comptig the ier prodct of eq, (3) with v, we get: Therefore: vˆ, v vˆ, v p i p i v ( v ), ˆ v = + p gv ( ), v = v, v + v, v v v + v v v v v v v + + v v + It is obvios that gv ( ), v, for all v Z h with v = + +. It follows * from Lemma. that there eists v Z h sch that g(v * ) =. Let + = v *, ad it ca be proved that + is the soltio of the scheme (7)-(9). (4)

4 S96 Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 H [ Xl, Xr ], the the soltio of the iitial- Lemma.4. [6] Sppose that bodary vale problem ()-(4) satisfies: L C, L C, C (5) Theorem.5. Sppose that H [ Xl, Xr ], the the schemes (7)-(9) are coservative for discrete mometm ad eergy, that is: = J = = = = M h M M (6) = + = = = E E E (7) Proof: Mltiplyig eq. (7) with h ad smmig p for from to J, from the bodary coditio i eq. (9), ad lemma., we get: where J + ( ) = (8) = h Therefore, eq. (6) is easily gotte from eq. (8). By comptig the ier prodct of eq. (7) with + (i. e. + + ), we have: h J ( + ) = τ ˆ ˆ + + ( + ) ( ) + + P = τ i p i p + + P = p + ˆ By the defiitio of ( ) t, it follows from the first term of eq. (9) that: J + ( + ) = = τ ( + ) () τ From eq. (), it follows from the secod ad the third term of eq. (9) that: J + + J = + = = = ˆ J + + J + + = + = () = = (9)

5 Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 S97 Similarly: = () J + + = ˆ With the help of the bodary coditio i eqs. (9) ad (), it follows from the forth term of eq. (9) that: J [( ) ( ) ] = = τ τ (3) I view of eq. (), it follows from the last term of eq. (9) that: P, i p i + J p h = p + = ˆ h = p + h = p + i+ p i J p + + = p i i+ J p = ˆ Let i = p ( i+ ). Obviosly, if i =, the i = p. If i = p, the i =. It follows from eq. (4) that: i p i + J h = = p + = i = p ˆ ˆ (4) P, P, (5) Therefore: P, + = (6) By the previos reslts of eqs. ()-(3) ad eq. (6), we have: = The, by the defiitio of E, eq. (7) holds, which implies that the differece scheme is coservative for eergy. I order to prove the bodedess ad the coservative law of the merical soltios, we lead ito the followig lemma [8]. Lemma.6 (Discrete Sobolev s ieqality) There eist two costat C ad C sch that: (7) C C + (8)

6 S98 Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 which yields: Theorem.7. Sppose H [ Xl, Xr ], the the soltios C Proof: It follows from eq. (7) that:, C C (,, N) C, By Lemma. ad Schwartz ieqality, we get: + C From Lemma.6, we have C (,, N). Nmerical eperimets of eqs. (7)-(9) satisfy: C (9) I this sectio, we preset some merical eperimets to verify theoretical reslts obtaied i previos sectios. We ow cosider two cases: p = 3 ad p = 5, respectively. Whe p = 3, the solito soltio is: t (, ) = sech (5 4) t ad the iitial coditio is: (3) (3) (,) = sech (3) 4 4 Whe p = 5, the solito soltio is: t (, ) 4 4 ( 5 34)sech 5 34 = + + (5 34) t ad the iitial coditio is: (,) = ( ) 5 sech (34) First, we simlate the wave graph of the merical soltio of the implicit o-liear scheme eqs. (7)-(9). The compariso of merical soltio (, t ) betwee differet time step ad space step at varios times are give i fig. whe p = 3. Similarly, we ca get the almost same figres for differet time step ad space step at differet times whe p = 5, respectively. (33) Figre. Wave graph of (, t) at varios times whe p = 3 ad τ = h =.5

7 Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 S99 Meawhile, we also list the coservatio ivariats M ad E at differet time i tabs. ad whe p = 3. Similarly, we ca get the coservatio ivariats M ad E at differet time whe p = 5. Table. The mometm of differet time i differet time step ad space step whe p = 3 (h, τ) T = s T = s T = 3 s T = 4 s (/4, /4) (/8, /8) (/6, /6) (/3, /3) Whe p = 5. M = ad E = at differet time. These reslts also verify that the proposed scheme is coservative for two qatities M ad E. Table. The eergy of differet time i differet time step ad space step whe p = 3 (h, τ) T = s T = s T = 3 s T = 4 s (/4, /4) (/8, /8) (/6, /6) (/3, /3) Coclsio I this paper, we costrcted o-liear-implicit fiite differece scheme for the geeral Rosea-KdV eqatio ad ivestigated some properties of its merical soltio. We proved that the o-liear scheme preserved the discrete mass ad eergy coservatio, respectively. The proposed scheme is the coditioally stable ad secod-order covergece by the discrete eergy method. The reslts show that the scheme is reliable ad efficiet. Refereces [] Ami, E., Solitary Wave Soltios for Geeralized Rosea-KdV Eqatio, Commicatios i Theoretical Physics, 55 (), 3, pp [] Polia, R., et al., Pertrbatio of Dispersive Shallow Water Waves, Ocea Egieerig, 63 (3), May, pp. -7 [3] Saha, A., Topological -Solito Soltios for the Geeralized Rosea-Kdv Eqatio, Fdametal Joral of Mathematical Physics, (),, pp. 9-3 [4] Ghodrat, E., et al., Topological Solitos ad Other Soltios of the Rosea-KdV Eqatio with Power Law No-Liearity, Romaia Joral of Physics, 58 (3), -, pp. - [5] Zheg, M. B., et al., A Average Liear Differece Scheme for the Geeralized Rosea-KdV Eqatio, Joral of Applied Mathematics, 4 (4), ID 793 [6] Lo, Y., et al., Coservative Differece Scheme for Geeralized Rosea-KdV Eqatio, Advaces i Mathematical Physics, 4 (4), ID [7] Zo, J. M., et al., A New Coservative Differece Scheme for the Geeral Rosea-RLW Eqatio, Bodary Vale Problems, (), ID 566

8 S9 Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 [8] Zho, Y., Applicatios of Discrete Fctioal Aalysis to the Fiite Differece Method, Iteratioal Academic Pblishers, Beiig, Chia, 99 [9] Browder, F. E., Eistece ad Uiqeess Theorems for Soltios of No-Liear Bodary Vale Problems, Proceedigs, Symposia i Applied Mathematics, New York, USA, 7 (965), 6, pp Paper sbmitted: December 5, 5 Paper revised: Febrary 5, 6 Paper accepted: March 5, 6