New Energy-Preserving Finite Volume Element Scheme for the Korteweg-de Vries Equation

Transcription

1 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_47 3 New Energy-Preserving Finie Volume Elemen Scheme for he Koreweg-de Vries Equion Jin-ling Yn nd Ling-hong Zheng Absrc In his pper, n -preserving finie volume elemen scheme is proposed for he Koreweg-de Vries equion. The scheme is combinion of he discree vriionl derivive mehod in ime nd he finie volume elemen mehod in spce. The scheme cn precisely conserve he globl mss nd he discree level, s well s hs higher ccurcy. For comprison, we lso propose momenum-preserving scheme nd finie volume elemen scheme. The numericl resuls demonsre he remrble ccurcy nd efficiency of our mehod compred wih oher schemes. Inde Terms Mss, Momenum, Energy, Finie volume elemen mehod, KdV equion. I. INTRODCTION THE ubiquious Koreweg de-dries KdV equion ws firs inroduced by Boussinesq in 877 nd rediscovered by Koreweg nd his Ph.D suden de Vries ] in 895. The KdV equion models vriey of nonliner phenomenon, such s shllow wer wves, cousic wves in hrmonic crysl nd ion-cousic wves in plsms. The simples form ] of KdV equion is given by u +εuu +µu, where he funcion u u, represens he wer s free surfce in non-dimensionl vrible. The derivive u chrcerizes he ime evoluion of he wve propging in one direcion, he nonliner erm uu describes he seepening of he wve, nd he liner ermu ccouns for he spreding or dispersion of he wve. In his pper we consider he following form of KdV equion: u +αu +βuu +γu, b, where α, β, γ re consns given by α c gd, β 3c d, γ cd 6, wih c gd, he shllow wer speed. Here g is he grviionl ccelerion nd d is he verge deph of wer. The KdV equion is compleely inegrble 3] nd give rise o muliple solion soluions. The eisence of conservion lws hve been considered s n indicion of he inegrbiliy of he KdV. There is n infinie se of Mnuscrip received December, 6; revised Mrch 4, 7. This wor ws suppored in pr by he PhD Sr-up Fund of Wuyi niversiy Grn No YJ7, he Educion Foundion of Fujin Province for Young Techers Grn No JA439 nd ndergrdue Technology Innovion Projec of Fujin Province Grn No SJ9. Jin-Ling Yn is wih he deprmen of Mhemics nd Compuer, Wuyi niversiy, Wuyi Shn, Fujin, 3543, Chin, e-mil: ynjinling3333@63.com. Ling-hong Zheng is wih he deprmen of Informion nd Compuer Technology, No middle school of Nnping, Nnping, Fujin, 353, Chin. e-mil: @qq.com. independen conservion lws for he KdV equion. The firs hree conservion lws of his se re: M J u, K α u + β 6 u3 γ u, u, which correspond o mss, momenum nd conservion lw, respecively. In his pper, he proposed -preserving scheme nd momenum-preserving scheme re consruced using he discree vriionl derivive mehod DVDM 4], which is mehod of designing specil numericl schemes h rein he conservion/dissipion properies of he originl pril differenil equions PDEs. As o DVDM, reserchers hve done lo of wor, for emple, Furih nd Mori 5] proposed sble finie difference scheme for he Chn- Hillird equion. Koide nd Furih 6] designed four conservive schemes for he regulrized long wve equion. Furher, Msuo nd Furih 7] eended he generl sudies o comple-vlued PDEs, lie he nonliner Schrödinger equion. Recenly, he mehod hs been eended in vrious wys, for insnce, Yguchi, Msuo nd Sugihr 8] eended he mehod o nonuniform grids. Msuo nd Kurme 9] proposed n lerning DVDM, nd so on. Finie volume elemen mehod FVEM, s ype of imporn numericl ool for solving he differenil equions, hs long hisory. This mehod is lso nown s bo mehod in some erly references ], or nown s generlized difference mehod ] in Chin. The mehod hs been widely used in severl engineering fields, such s fluid mechnics, he nd mss rnsfer nd peroleum engineering. Perhps he mos imporn propery of FVEM is h i cn preserve he conservion lws mss, momenum nd he flu on ech compuionl cell. This imporn propery, combined wih deque ccurcy nd ese of implemenion, hs rced more people o do reserch in his field ] 5]. In his pper, we will propose n -preserving scheme for he KdV equion. The conservion lw is n imporn propery of he KdV equion. Thus, in he numericl simulion of he KdV equion, we hope rein his propery. Moreover, o our nowledge, he preserving scheme ofen hs beer sbiliy, s well s smller errors. Li nd Vu-Quoc 6] once sid h in some res, he biliy o preserve some invrin properies of he originl differenil equion is crierion o judge he success of numericl simulion. Zhng 7] poined ou h he nonconservive schemes my esily show nonliner blow-up. Thus, in view of his poin, we hope design n -preserving scheme for he KdV equion. Abou he KdV equion, reserchers hve done lo of wor, for Advnce online publicion: 4 My 7

2 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_47 3 insnce, Th nd Ablowiz 8] proposed new scheme using he inverse scering rnsform noion. Zhng nd Wng 9] proposed n improved homogeneous blnce mehod for Muli-Solion Soluions of Grdner Equion. Ascher nd McLchln ] developed nd compred some symplecic nd muli-symplecic finie difference schemes for he KdV equion. Bh nd Bhi ] presened new lgorihm for pproiming he numericl soluion of he KdV equion in modified B-polynomil bsis. Drvishi, Kheybri nd Khni ] proposed pseudospecrl mehod for he KdV equion. Dğ nd Dereli 3] developed meshless mehod bsed on he rdil bsis funcions. In his pper, we develop n -preserving scheme, nd sudy heir conservive properies, ccurcy nd long ime behvior, nd so on. The orgnizion of he pper is s follows. In Secion, we presen some noions nd preliminries bou FVEM. In Secion 3, we derive he proposed schemes, nd nlyze heir conservive properies. In Secion 4, we nlyze he liner sbiliy of he -preserving scheme. In Secion 5, we presen he numericl emples o illusre he effeciveness of he new scheme. A ls, we give some concise conclusions. II. NOTATION AND PRELIMINARIES In his secion, we define some noions nd he frmewor of he FVEM. Firs, we use uniform grid T h o discreize he soluion domin, < < < < n < n b wih grid spcing h i i b /n. Then we plce dul grid Th, < / < 3/ < < n / < n b wih i / i + i /, i,,...,n, nd I, / ], Ii i /, i+/ ] i,,...,n nd IN N /, N ] denoe he dul elemens. The ril funcion spce h is en s he liner elemen spce wih respec o T h. The bsis funcion wih respec o i is given by { h i, i i+, φ i, elsewhere. Thus, he funcions {φ i : i,,...,n} form bsis of h nd ny u h h hs he following epression u h u i φ i, i where u i u h i,. Furher, on he elemen I i, we hve u h u i ξ+u i ξ, u h u i u i /h, I i, i,,...,n, where ξ i /h. Accordingly, he es funcion spce V h is chosen s he piecewise consn funcion sep funcion spce. The bsis funcions of V h re {, j / j+/, ψ j, elsewhere, where j,,...,n. Any v h V h hs he form v h v i ψ i, i where v i v h i,. In he sequel, if no specilly illusre, we will use m o denoe he numericl soluion nd m, where denoes he ime sep size. On he oher hnd, in his pper, we will dop he following periodic boundry condiions, j u j j u b j j,,. 3 III. NMERICAL SCHEMES In his secion, we derive he proposed schemes nd nlyze heir conservive properies. A. Concree form of he proposed scheme For convenience, we define free or locl of he KdV equion s Gu,u α u + β 6 u3 γ u, nd is spil inegrion Ju Gu,u s he globl. Then Eq. cn be rewrien s u δg, 4 δu where δg/δu is he vriionl derivive of Gu,u defined by δg δu G u G. u In he following, we sr o derive he proposed preserving scheme. To his end, we firs give scheme of he locl G d, m α m + β 6 m 3 γ δ + m +δ m 5, nd he ssocied globl is defined by J d m G d, m, 6 where g g +g + +g N + g N. By resoring o Eq. 5, discree scheme of he vriionl derivive corresponding o -preserving scheme is given by α δ m+, m m+ m+ + m+ m + β 6 + γ δ m+ + m, + m + m 7 Advnce online publicion: 4 My 7

3 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_47 3 whereδ denoes he cenrl difference quoien of /. The bove scheme is obined by he following difference: G d, m+ G d, m m+ δ m+, m + boundry erm. m The bove mehod is lso nown s he discree vriionl derivive mehod, more deils bou i plese refer o 4]. Afer h we cn obin he fully discree -preserving finie volume elemen scheme by subsiuing Eq. 7 ino he following we form δ m + m,ψ i δ m+, m,ψ i, 8 where δ m + m m+ m /, is he ime sep, m h nd ψ i V h i,,...,n. On he oher hnd, in order o reflec he superioriy of he -preserving mehod, momenum-preserving finie volume elemen scheme is lso derived. Le m+/ + m+ + + m + /, m+/ m+ + m /, nd subsiue hem ino Eq. 7 nd respecively in plce of m+ nd m, hen discree scheme of he vriionl derivive corresponding o he momenum-preserving scheme is obined s follows α m+/ δ m+/ +, m+/ + + m+/ + β m+/ + 6 +m+/ + m+/ + m+/ γ + δ m+/ + + m+/. Subsiuing i ino he following we form δ m + m,ψ j,ψ δ m+/ +, m+/ j, 9 where m h nd ψ j V h j,,...,n, we obin he fully discree momenum-preserving scheme. A ls, for comprison, we lso derived he following implici midpoin finie volume elemen scheme δ m + m,ψ β m+/ m+/,ψ α m+/,ψ +γ m+/,ψ, B. Conservion properies of he proposed schemes In he following, we sr o sudy he conservive properies of he KdV equion. Proposiion III.. Le u be he nlyicl soluion of 4, nd ssume he following boundry condiion δg δu is sisfied, hen he coninuous mss M is consn, h is d d u d d δg δu u. u δg δu. Proposiion III.. Le u be he nlyicl soluion of 4, nd ssume he following boundry condiions u δg, δu Gu,u re sisfied, hen he coninuous momenum K is consn, h is d b u. d d u d u δg δu u δg + δu uu uδg δu Gu,u Gu,u. where m+/ m+ + m /, m h, nd ψ V h,,...,n. In ddiion, in order o illusre he conservive properies of he schemes 9 nd, we lso consider he following discree quniies, i.e., he globl mss nd he globl momenum M d m K d m m, m. Proposiion III.3. Le u be he nlyicl soluion of 4, nd ssume he following boundry condiions G u, u ] δg b δu re sisfied, hen he coninuous J is consn, h is d b Gu,u. d Advnce online publicion: 4 My 7

4 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_47 3 d G Gu,u d G u u + G u u G u b u u G u δg u b δu δg δg δu δu δg ] δg b δu δu. Gd m+ G d m m+ m δ m+, m δ m+, m δ m+, m δ m+, m. δ m+, m Similrly we hve he following conservive properies. Theorem III.. Discree mss conservion lw Le m be he soluion of 8, nd ssume he following boundry condiion δ m+, m is sisfied, hen he discree mss M d is consn, nmely m cons. Theorem III.3. Discree mss conservion lw Le m be he soluion of 9, nd ssume he following boundry condiion δ m +,m is sisfied, hen he discree mss M d is consn, nmely m cons. m+ m m+ m δ m+, m δ m+, m. Theorem III.. Discree conservion lw Le m be he soluion of 8, nd ssume he following boundry condiion δ m+, m is sisfied, hen he discree J d is consn, nmely G d m cons. m+ m m+ m δ m +,m ] b δ m +,m. Theorem III.4. Discree momenum conservion lw Le m be he soluion of 9, nd ssume he following boundry condiions m+ + m δ m Gd m, m+ +,m re sisfied, hen he discree momenum K d is consn, nmely m cons. Advnce online publicion: 4 My 7

5 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_47 3 m+ m ] m+ + m m+ m m+ + m δ m + m m+ + m m+ + m G d m, m+ G d m, m+. δ m +, m δ m +, m In he bove equions, inegrion by prs formul nd periodic boundry condiions re used. A ls, for he sndrd finie volume elemen scheme, we lso hve he following conservive propery, nmely Theorem III.5. Discree mss conservion lw. Le Z be he soluion of, hen he discree mss M d is consn, nmely m cons. The proof of Theorem III.5 is similr o he one of Theorem III.. IV. STABILITY ANALYSIS In his secion, we sudy he sbiliy of he preserving scheme 8 for solving Eq.. Here we only consider equions wihou nonliner erms, which llow us o sudy he liner sbiliy of he proposed schemes using he Fourier mehod. Firsly, from Eq. 8, we ge he following fully discree -preserving scheme: α m+ +α m+ +α 3 m+ α m+ + α m+ + α m α m +α 3 m +α m + +α m + m+ +α 4 + m+ m +m m+ + + m+ + m + +m + ], where j,,...,n, α γ,α 6γ 3αh, α 3 h 3,α 4 βh. For he liner Fourier nlysis, we only consider he liner version of Eq., which is given by α m+ +α m+ +α 3 m+ α m+ + α m+ + α m α m +α 3 m +α m + +α m +. 3 Then ssume h m j is periodic in -direcion, nd grid node j, le m j V m e iωjh, 4 where V m is mpliude ime level m, nd ω is phse ngle in -direcion. Subsiuing 4 ino 3, we hve where α 3 iα 5 V m+ α 3 +iα 5 V m, α 5 α sinwh+α sinwh. Therefore, he growh fcor g for he proposed preserving scheme is g V m+ V m α 3 +iα 5 α 3 iα 5. Therefore, i mees he uncondiionlly sble crierion g nd we conclude h he proposed -preserving scheme is uncondiionlly sble. V. NMERICAL EXPERIMENTS In his secion, we will es he proposed schemes numericlly. Through hese numericl emples, we will nlyze he ccurcy nd conservive properies of he proposed schemes, nd furher illusre he dvnges of he preserving scheme. A. Single soliry wve Here we consider he KdV equion, nd se A, d 6 nd g 9.8, hen we cn obin α, c, β nd γ by. According o ], Eq. hs he following soliry wve soluion: ] 3A u, A sech d 3 κ, where κ c + d A. This soluion corresponds o soliry wve of mpliude A. Here c denoes he velociy of he rveling wve. Here we dop he following iniil soluion ] 3A u, A sech d 3, nd he periodic boundry condiion u, ub,. On he oher hnd, in order o vlide he efficiency of he proposed mehods, in he sequel, we will use L m i N i, n u n i nd order log u n n h / u n n h o evlue he ccurcy nd he convergence orders of he mehods. In ddiion, we will use Ii n Ii /I i o denoe he relive errors of he invrins, where I i i,,3 respecively corresponds o he globl 6 nd mss, momenum he discree level. Firsly, we es he ccurcy nd he convergence orders of he proposed schemes. Here we ssume he problem is solved on he inervl,]. In order o mesure he error in spce, relively smll ime sep. is chosen such h he error from he ime direcion cn be negligible, nd he spil seps re respecively chosen s h, h, h / nd h /4. The spil L errors nd corresponding convergence res of he proposed Advnce online publicion: 4 My 7

6 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_47 3 TABLE I: Spil L errors nd convergence orders of he proposed mehods wih N,.,. h MFVEM order EFVEM order FVEM order.694e e e e e e 4.98 /.934e e e 5. /4.7368e e e 6.98 TABLE II: Temporl L errors nd convergence orders of he proposed mehods wih T, h /6,. MFVEM order EFVEM order FVEM order / 8.796e 3 9.e e 3 /4.366e e e 3.93 / e e e 4.97 /6.6364e e e 4.97 TABLE III: L errors nd convergence orders of he proposed mehods wih T 3, h,. h MFVEM order EFVEM order FVEM order 9.775e 9.97e 9.58e /.956e e e.76 / e e e 3.93 /8.94e e e 3.98, n u, n EFVEM FVEM MFVEM,3 u, h /6 h /8 h /4 h / Fig. : The errors of he proposed schemes: he mimum errors of he hree proposed schemes T 3 nd wih h /6, b he numericl errors corresponding o differen seps of he -preserving scheme T 3. b mehods re presened in Tble I, which clerly shows h he -preserving scheme nd he finie volume elemen scheme hve higher ccurcy hn he momenum-preserving scheme. Similrly, for he ime direcion, relively smll spil sep h /6 is chosen such h he error from he spil direcion cn be negligible. The emporl L error nd corresponding convergence res of he proposed mehods re presened in Tble II, which clerly shows h he convergence res of hree mehods re pproimely equl o. I is lso noed h he error of he momenumpreserving scheme cese o decrese cerin poin, which is becuse he error from he ime direcion become very smll such h i cn no be disinguished from he spil error. On he oher hnd, he L errors nd he convergence res of hree mehods wih h nd T 3 re presened in Tble III, which clerly shows h he convergence res of hree mehods re ll pproimely equl o in spce nd ime. Secondly, in order o compre he ccurcy of he proposed mehods, we plo he vriion of he L errors of he proposed mehods, when h /6 nd T 3, in Figure, which clerly shows h he growh of he errors of hree mehods re liner, nd he momenum-preserving scheme hs he lrges error. On he oher hnd, Figure b presens he numericl errors corresponding o differen seps of he -preserving scheme T 3. A ls, we es he propgion of he soliry wve nd he conservive properies of he proposed schemes. In he sequel, we se spil sep h.5 nd emporl sep., nd he problem is solved over he inervl 5, 5]. Figure presens he numericl resuls of he -preserving scheme for in, 4]. The surfce plo Advnce online publicion: 4 My 7

7 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_ lg I i /Ii 8 4 mss momenum Fig. : The numericl resuls of he -preserving scheme: numericl soluion, b he relive errors of invrins, when h.5,.,, d 6, T 4 nd 5 5. b lg I i /Ii 8 mss momenum lg I i /Ii mss momenum Fig. 3: The relive errors of he invrins of he proposed schemes: momenum-preserving scheme, b finie volume elemen scheme, when h.5,.,, d 6, T 4 nd 5 5. b 5.5 lg I i /Ii mss momenum Fig. 4: The numericl resuls of he -preserving scheme: numericl soluion, b he relive errors of invrins, when h.5,.,, d 6, T nd 5 5. b of numericl soluion ime T 4 is presened in Figure, which shows he soliry wve moves o he righ consn speed, nd he wve shpe nd mpliude lmos unchnged wih ime increse. Figure b shows h he relive errors of he invrins he discree level. I is clerly seen h he -preserving mehod cn Advnce online publicion: 4 My 7

8 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_ lg I i /Ii 8 mss momenum lg I i /Ii 8 9 mss momenum Fig. 5: The relive errors of he invrins of he proposed schemes: momenum-preserving scheme, b finie volume elemen scheme, when h.5,.,, d 6, T nd 5 5. b precisely preserve he discree mss nd o wihin mchine precision. The relive errors of he invrins of he momenum-preserving scheme nd he finie volume elemen scheme re presened in Figure 3, which shows h he momenum-preserving scheme cn precisely conserve he discree mss nd momenum o wihin mchine precision, nd he finie volume elemen scheme cn precisely conserve he discree mss o wihin mchine precision. In view of Figure nd Figure 3, we conclude h hree schemes ll cn be used o simule he propgion of he soliry wve, bu in he spec of ccurcy nd conservion properies, he -preserving mehod is beer mehod. On he oher hnd, in order o es he long ime behvior of he preserving scheme, we lso presen he numericl resuls of he -preserving mehod for in, ] in Figure 4, which illusres he scheme hs good sbiliy nd long ime compuion biliy. In ddiion, for comprison, we lso presen he relive errors of he invrins of he momenumpreserving scheme nd he finie volume elemen scheme in Figure 5. B. Inercion of wo soliry wves In order o furher illusre he effeciveness of he proposed scheme, here we discuss he inercion of wo soliry wves. Seing α, β, γ , hen we obin he following KdV equion, u +uu u, 4. 5 Here we consider he KdV equion 5 wih he following periodic boundry condiion u, u4,, >. On he oher hnd, ccording o 8], 4], Eq. 5 hs he following ec soluion, u, γlogf, 6 where l l F +epη +epη + epη +η, l +l η i l i li 3 γ+m ii,,.3. l γ, l γ, m.48l, m.7l. For ll he compuions, we hve used he sepsizes h.5,.5, nd he compuions re done up o ime T 6. The numericl resuls of he -preserving scheme re presened in Figure 6, which shows h he -preserving scheme cn ecly preserve he globl mss nd he discree level. On he oher hnd, Figures 7-9 disply he inercion process of wo soliry wves. I is noed h he ller wve iniilly loced on he lef of he lower wve. Then,.5, he ller wve cched up he lower wve nd occurred inercion, nd 3 nd 3.5, he ller wve nd he lower wve overlpped, s is noed in Figure 8. I is lso noed h he ller wve nd he lower wve sred o leve wy when > 3.5, nd he ller wve nd he lower wve inerchnged heir posiions 6, s is shown in Figure 9. On he oher hnd, he numericl resuls lso shows h he preserving scheme hs beer performnces hn he ones of 4], i.e., he mehod of 4] needs smller sepsizes nd is no suible for long ime compuion. VI. CONCLSIONS In his pper, n -preserving scheme is proposed for he Koreweg-de Vries equion. We invesige he ccurcy nd he conservive properies of he proposed mehod nd compre is performnces wih he ones of momenum-preserving scheme nd finie volume elemen scheme. The numericl resuls show h hree schemes ll cn simule he KdV equion, bu in he spec of ccurcy nd conservive properies, he -preserving scheme hs smller ccurcy nd beer conservive properies hn oher wo schemes. Thus he -preserving scheme is beer choice for he KdV equion. Besides, he preserving scheme is uncondiionlly sble nd hs beer long ime compuion biliy. Advnce online publicion: 4 My 7

9 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_ lg I i /Ii mss momenum Fig. 6: The numericl resuls of he wo soliry wves obined by he -preserving scheme: he surfce plo of numericl soluion, b he relive errors of invrins, when α, β, µ , h.5, T 6, nd 4. b Fig. 7: The plos of he wo soliry wves obined by he -preserving scheme:, b.5, when α, β, µ , h.5, T 6, nd 4. b c 3 4 Fig. 8: The plos of he wo soliry wves obined by he -preserving scheme: c 3, d 3.5, when α, β, µ , h.5, T 6, nd 4. d Advnce online publicion: 4 My 7

10 IAENG Inernionl Journl of Applied Mhemics, 47:, IJAM_ e 3 4 Fig. 9: The plos of he wo soliry wves obined by he -preserving scheme: e 3.75, f 6, when α, β, µ , h.5, T 6, nd 4. f REFERENCES ] D. J. Koreweg nd G. de Vries, On he chnge of form of long wves dvncing in recngulr chnnel nd new ype of long sionry wves, Philosophicl Mgzine, vol. 39, no. 5, pp , 895. ] P. G. Drzin nd R. S. Johnson, Solions: n Inroducion. Cmbridge niversiy Press, ] R. M. Miur, C. S. Grdner, nd M. D. Krusl, Koreweg-de vries equion nd Generlision. II. Eisence of Conservion Lws nd Consns of Moion, Journl of Mhemicl Physics, vol. 9, pp. 4 9, ] D. Furih nd T. Msuo, Discree vriionl derivive mehod: srucure-preserving numericl mehod for pril differenil equions. CRC Press,. 5] D. Furih nd M. Mori, A sble finie difference scheme for he Chn-Hillird equion bsed on Lypunov funcionl, Journl of Applied Mhemics nd Mechnics, vol. 76, no., pp , ] S. Koide nd D. Furih, Nonliner nd liner conservive finie difference schemes for regulrized long wve equion, Jpn Journl of Indusril nd Applied Mhemics, vol. 6, no., pp. 5 4, 9. 7] T. Msuo nd D. Furih, Dissipive or conservive finie difference schemes for comple-vlued nonliner pril differenil equions, Journl of Compuionl Physics, vol. 7, no., pp ,. 8] T. Yguchi, T. Msuo, nd M. Sugihr, An eension of he discree vriionl mehod o nonuniform grids, Journl of Compuionl Physics, vol. 9, no., pp ,. 9] T. Msuo nd H. Kurme, An lerning discree vriionl derivive mehod, AIP Conference Proceedings, vol. 479, no., pp. 6 63,. ] W. Hcbusch, On firs nd second order bo schemes, Compuing, vol. 4, pp , 989. ] R. H. Li, Z. Y. Chen, nd W. Wu, Generlized Difference Mehods for Differenil Equions: Numericl Anlysis of Finie Volume Mehods. Mrcel Deer, Inc.,. ] Q. X. Wng, Z. Y. Zhng, X. H. Zhng, nd Q. Y. Zhu, Energypreserving finie volume elemen mehod for he improved Boussinesq equion, Journl of Compuionl Physics, vol. 7, pp , 4. 3] Z. Y. Zhng, Error esimes of finie volume elemen mehod for he polluion in groundwer flow, Numericl Mehods for Pril Differenil Equions, vol. 5, no., pp , 9. 4] J. L. Yn nd Z. Y. Zhng, Two-grid mehods for chrcerisic finie volume elemen pproimions of semi-liner Sobolev equions, Engineering Leers, vol. 3, no. 3, pp , 5. 5] L. Z. Qin nd H. P. Ci, Two-grid mehod for chrcerisics finie volume elemen of nonliner convecion-domined diffusion equions, Engineering Leers, vol. 4, no. 4, pp , 6. 6] S. Li nd L. Vu-Quoc, Finie difference clculs invrin srucure of clss of lgorihms for he nonliner Klein-Gordon equion, SIAM Journl on Numericl Anlysis, vol. 3, no. 6, pp , ] Z. Fei, V. M. Pérez-Grci, nd L. Vázquez, Numericl simulion of nonliner Schrödinger sysems: new conservive scheme, Applied Mhemics nd Compuion, vol. 7, no. -3, pp , ] T. R. Th nd M. I. Ablowiz, Anlyicl nd numericl specs of cerin nonliner evoluion equions. III. Numericl, Koreweg-de Vries equion, Journl of Mhemicl Physics, vol. 55, no., pp. 3 53, ] S. Zhng nd Z. Y. Wng, Improved homogeneous blnce mehod for muli-solion soluions of Grdner equion wih ime-dependen coefficiens, IAENG Inernionl Journl of Applied Mhemics, vol. 46, no. 4, pp , 6. ]. M. Ascher nd R. I. McLchln, Mulisymplecic bo schemes nd he Koreweg-de Vries equion, Applied Numericl Mhemics, vol. 48, no. 3-4, pp , 4. ] D. D. Bh nd M. I. Bhi, Numericl soluion of KdV equion using modified Bernsein polynomils, Applied Mhemics nd Compuion, vol. 74, no., pp , 6. ] M. T. Drvishi, S. Kheybri, nd F. Khni, A numericl soluion of he Koreweg-de Vries equion by pseudospecrl mehod using Drvishis precondiionings, Applied Mhemics nd Compuion, vol. 8, no., pp. 98 5, 6. 3] I. Dğ nd Y. Dereli, Numericl soluions of KdV equion using rdil bsis funcions, Applied Mhemicl Modelling, vol. 3, no. 4, pp , 8. 4] A. J. Kh nd. I. Sirj, A comprive sudy of numericl soluions of clss of KdV equion, Applied Mhemics nd Compuion, vol. 99, pp , 8. Jin-ling Yn ws born in Pingyo, Shni Province, chin, in 979. The uhor respecively received his Ph.D. nd Mser s degrees in compuionl mhemics Nnjing Norml niversiy, Nnjing, Jingsu Province, Chin, in July 6 nd July. He holds eching posiion Wuyi niversiy, Informion nd Compuion Science in he deprmen of Mhemics nd Compuer. The Wuyi niversiy is loced in Wu Yi Shn, Fujin Province of Chin. His curren reserch ineress include srucure-preserving lgorihms, numericl soluion of pril differenil equions nd compuing sciences. Ling-hong Zheng ws born in Nnping, Fujin Province, Chin, in 983. The uhor received her bchelor s degree in compuer science nd echnology from Minnn Norml niversiy, Zhngzhou, Fujin Province, Chin, in July 9. Also, she is echer No. Middle School in he deprmen of Informion nd Technology. The No. Middle School is loced in Nnping, Fujin Province of Chin. Her curren reserch ineress include lgorihm design, rificil inelligence, robo compeiion nd video producion. Advnce online publicion: 4 My 7