L.A. Wazzan and M.S. Ismail- Finite Element Solution of K(2,2) Equation

8/3/2019 L.A. Wazzan and M.S. Ismail- Finite Element Solution of K(2,2) Equation

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P lar . I 'akistan Aratl. Sri. 44(1):21-26.2007

FINITE ELEMENT SOLUTION OF K(2 ,2) EQUATIONL.A. Wazzan' and M.S. Ismail2Depnrl~i~enff Mafh~,117(1/ics,ing Ab duIAziz Unive~ .sify. ed(Iu11.Sn t fd iArnbiaReceived Sept emb er 2006 , accepted December 2006Communicated by P rof . Dr. M. I q b a l C h o u d h a r y

Abstract: In this paper a Petrov-Galerkin method is used to derive a schem e for the K(2.2) equation,where we have ch osen cubic B-splines as test functions and linear function s as trial funct ions. Productapproximation technique is applied for the nonlinear terms. A Crank-Nicolson Scheme is used todiscretize in time. Allonlinea r peuta-diagonal sys tem is obtained and we s olve this system by Newton'smethod and by a linearization techniq ue. Accuracy and stability of the sche me have been investigated.The single compacton solution and the conserved quantities are used to assess the accuracy of thescheme. The interaction of two compactons are displayed and the numerical results have show n that.these compacton so lution s exhibits true soliton solution.Kej'words: Petrov-Galerkin method, Cubic b-splines, trial functions, product approximation, Crank-Nicolson sch eme, non-linea r terms, compacton so lution, soliton solution

1 . Introduction which has the exact solutionNumer ica l and ana ly t i ca l so lu t ions o fnonlinear partial diff eren tial equ ation s with u,(x,t) = [%os2(:), ~ ~ ~ = ~ x - v t ~ 5 2soliton solutions has been studied extensively otherwue. (3)0,during the recent years [1,2,3,4]. Well-known

partial differential equatio ns ( PD Es) with so liton Eq. (2) has the following conserved quantities:solutions include Si ne G ord on (SG) Equations,cubic nonlinear Schrodinger (1JLS) Equations co mI, = Judn, I, = J u 3 d x ,a n d K o r t e w e g - d e V r i e s ( K d V ) e q u a t i o n s .-m -m

(4 )Rosenan and Hyman [5 ] reported a class ofP DE s, Direct calculation o f Eq. (4) gives the exact val-

u,(l~"l),+(u'l)~,,=O,> l , lSn53 ( 1 ) uesrn mwhich is a generalization of the KdV equation. 871I, = Ju(x, t )dx = J u ( x , ~ ) d r -vThese equations with the values of m and n ar e -m -co 3denoted by K ( I ~ , I ~ ) .

andIn this paper we will describe numericalm msolut io~~sf one of these PD Es; nam ely, K(2,2). I, = I u 3 ( x , t ) & = Iu3(x ,0 )dx =-+0n. ,The K(2,2) equa tion is g iven by, -- -- 27

l i t + (u l ) , + (ul),,, = 0 , (2 ) for all t 2 0.

I S. 2 . De p a i~ m u n t1. Mptliematics, King A b d u l A r i z University, Jcddall, Saud i ArabiaE-m;i>l: [email protected]


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An overview o fth e paper is as follows: Sec-tlon 2 is devoted to the num erical metho d andlini te element method. The methods are thentcsted by m eans of two test problem s in Section3, and conclud ~ng emarks are discussed in Sec-tion 4.2. Numerical methods

We will describe a finite elem ent method tosolve the .K(2,2) equation. We assume that thesolut lon o fth e K ( 2 , 2 ) equation is negligible out-side an interval [x,, x,], so we replace Eq. ( 2 )by

with initial con dition

and boundary conditions

A standard weak fomiu lation of this problem isobtained b y multiplying (5) by a so-called testf i~nct ionY' t H,'(x,,.u,), integrating by parts theterm ( t r ' )< r r (usin s the fact that Y(.xL)= Y(xJ= O).This leads to the form ulation:

Find u t H,:(xL,x,), such that

Now w e will de scribe the methods.2.1 Finite element method

The space interval [ x , , ~ , ] s discretized b yunifomi ( N + l )grid points

X R - Lwhcrc the grid spa cing 17 is given by h = N '

Let U ,, t ) denote the approximate solution tothe exact solution u(x,,,,t).

Following the nictliod used by Sanz-Semaand Chrisite [ 6 ] o solve K dV equation by finiteelement m ethod, w e mo dify their method to solveth e K ( 2 , 2 ) equation

Using Petrov-Galerkin method, we assume ih csolution of Eq. ( 6 ) s

where b i ( x ) , =O,l, ...,N are the usual piecewiselinear "hat" function given by

The unknown functions U,,,(t), n = 1,2,...,N aredetermined from the system of ODES,

4; =

where

x - hI+-- ij" ( i - l ) h < x < ihx - i h1 if i < x i ( i + l ) hh

1 x - i h14(h+2)3 f ( i - 2 ) h < x < ( i - 1 ) h

v j ( X ) = '3 x - i h 3 x - i hI--(-)' --(-I3 if ( i - l ) h < x < i2 h 4 h3 x - i h 3 x - j h ,( ) ( f i < x S ( i + l ) h2 h 4 h


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j = 1, 2 ,....,N are the t es t functions which are 1' selected as cubic B-spline with com pac t suppo rt . F ( U , ) = 4h(u:+2 +10U:+, -lO U i-l U , i - 2 )The P I -oduc t app r ox i m a t i on t echn i que i s

1 ,+ ? ( U , n , , -2u:.,, +2 u:- , -U:.,),used for the non lineal- tcrm s in th e follo win g 2h

rnannel- (se e [7]) andN

u 2 ( x , t ) = C u : ( t N , ( x ) , (9 ) U = [ U , , U,, .....,U,] ' .m = l

Th e scheme in E q. (10) has a t runcation er ror ofPerforming thc integrat ion on Eq . (a), will give order O ( h 9 ) .the following syste mNo w t o solve this ordinary di r rercnt ial sys-

' ' i tern (I I; . W e as s um e t hat U,;, o he th c fully tlis-- rn,> t 26U,+1 +66U,+26Um-,+U, .Z120 crete approximation to the exact solut ion rc(.v,,,.t,,).w h e r e ttZ nk, and k is the tirne step size ant11+ -(qn,+2 + l04,,+1 0qrn-,- 4 4 ) us i ng i m p l i c i t m i d po i n t r u l e Lo ge t the f u l l24h discret izat ion, the fol low ing nonlinear sys tem is

1 1 0 obtained+- i (9, . .2 -2qm+,+ 2qrn-i- qm-2)= 02 7whcre q,,,= U,,,?and

U , = U , = O , u , = u , + , = o . At the nodal points Eq.(12) isThis scheme can bc writ ten in a matrix vectorform as 1-[v-*, +26Vm+, 66Vm+261/,,., + V m ,].20

k+,,[w.*, + l o w m , , -~0Wn,-,wn,-21 ( I ? )( 1 1) k+-(wm,,-zw,,,+zwm.,-wm.,)=oh3where Mis apenta-diago nal matrix with ele men ts U r ' + U i""re yn= (url-u: ,wm= q( 1.

\

Now this system is a nonline;irpent;ldiagon:~lsys tem and ca n he solved by any i terative schemesuch as Newton method or Predictol - -Cot- rccm

, method. Newton ' s m ethod i s ado pted in th iswork. I t can be sho wn that the schc me ( 13 )satis-f i es the conserved q uant i t i es g iven in Eq. (4).

'i To study the stabili ty of the l inearized vcr-sion of Eq. (13) , we use the von Ncum ann analy-


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I: io~$c lc~ncnlolution of K(2.2) cqoalion 24

sis . and af ter some manipulat ions, the amplif i - Th e cho i ce of & is a delicate matter which wccation factor is given by can choose as small as possible in such a waynot to loose the accuracy and the propert ies of

A - i ( B + C ) the differential equations. E xperime ntat ion withenk - A + i ( B + C ) ' (14 ) the va lues of E in the spatial filtering term is Ire-quired.whcl-e 2.2 Linearization Technique

To avoid solvin g a nonlinear system at eacht ime s tep we use Taylor ' s sel-ies expansion ofu:"+u;

U () about the nth t ime level as fol-

; ~ n d = 2u, here i~ is considered to represent lows:local ly the maximum value of u.From Eq . (14). we d educe tha tleZk= 1, for all 0S o this schem e is uncondit ional ly s table in The terms i n Eq, ( 13 ) can be app rox i -thc l inearized sense. We havc noticed that this mated byscheme b lows up af te r eel-tain t ime s teps . Even

i f we I-educe he time size, this can delay the blowLI D but i t wil l not prev ent i t . S o in order to ove r- w, = UilJ:" +0 ( k 2 ) .com e this diff iculty, an anif icial dissipation termi s a d d e d t o t h e p r o p o s e d s c h e m e ( n o n d i s s i - Hence the d iscre te sys tem E q. (13) becomes. ~pativc) . We ad d a foul-th order qu anti ty

1-(v,., +26V.,, +66Vm+26V.~, +Vm~ , )12 0"821 " k~f i :U; ,+ '=E(u:: - 4 ~ " "+ 6~: -4uni' +u:T:) ( I 6 ) +-(ur;>u:.2 + IOU::;~:~, -1ou::;u: , -u :~ ;u ;~~)24 h (181

ki n ortiel- to stabilize it , an d hen ce the amplifica. + ~ ( U : ~ ~ U : . ~ - ~ U ~ : ~ U : , ~h + Z u ~ ~ ~ u ~ ~ ,uz::u:t ion facto r sat isf ies the fol lowing condit ion

This is a pentadiagonal l inear system which canah A ~ + ( B + c ) ~ be so lved by Crouts method.1 = ( A + D ) ' + ( B + c ) 2 (17)

3. Numerical resultsTo examine and compare the accuracy 01'

the methods, two ini t ial condit ions are consid-ered . The conserved quant i t i es a re ca lcu la tedusing Simpson's rule.


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L.A. Weeen& M.S. immtl

1 SingleCompactonPFor this testwe choose a s m initial condition

Table 2.Single Compacton using linearization method.

The following parameters k = 0.05, h =0.1, v = 1 , ~=0.0033,ave been chosen. Two con- Table 3.served quantitiesI,, , and Lm,which is the value Single Compacton using difference methodof (uc-u:( ith the maximum modulus (n =0,1,.. .,N) at time level t = nk for n = 0,1,. . arecalculated. The results are given in the follow-ing Tables. In Table 1,we present the results forthe single compacton using Newton's method forsolving the resulting nonlinear system. In Table2, we present the numerical results using thelinearization techniques. In Table 3, we presentthe numerical solution using an implicit finitedifference method reported in [ 2 ] .We have foundthe results are in very good agreement with theexact solution. Fig.1presents a single compactonatt=0,5,10 ..., 0.

Table 1 .Single Compacton using Newton's method.

Figure 1. Single Cornpaton for K(2,2) equation.


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Fini te clemenl solulion of K(2,Z) equation

3.2 Two Com pactons interactionThe initial condition is to display the inter-

action scenario given by

where

otherwise (19)where

In this test we choose the following param-eters h = 0 . 2 , k = 0 . 0 1 , x , = 1 0 , ~ , = 3 0 , v , = 1 . 0 v' T0.5 and E = 0.01. The interaction scenario is dls-played in Fig. 2, and shows that how the two wellseparated compactons interact and emerge afterthe interaction is unchanged in shape and veloci-ties but we have noticed birth of compacton-anticompacton pair with small amplitude and thisbehavior has been noticed before in [2].

4. Concluding remarksFrom the numerical experiments using the

finite element method and the solution of theresulting system by Newton's method, we haveseen that thls method is the most accurate one Icomparing to the linearization technique andfinite difference method obtained before. Thenumerical solution agreed very closely with theexact solution. The schemes conserve theconsented quantities almost exactly.We concludethat the finite element and solution obtained usingNewton's method is the best among the methodsmentioned in this paper. We refer this to the fourthorder accuracy in the space direction and secondorder in the time direction.References1. Ismail , M.S. a n d Taha, T.R. 1997. A nrrmericnl

stu& uof'Korteweg-de ieies like erjzrofioi~s.roceed-ings of the 15Ih macs World Congress on ScientificComputation Modeling an d Applied Mathematics,pp. 131-136.2. Ismail , M.S. an d Taha, T.R. 1998. A numericalstudy of compactons. Math. Comp. Simt f i .47:519-

530.3. Ismail , M.S. 2000. A finite difference method ofKorteweg-de Vries like equation with nonlinear

dispersion. In fernad. J. Cornprater Maths 73. '-2.4. Ismail , M.S. a n d Al-Solamy, F.R. 2001. A numeri-cal study of K ( 2 , 3 ) equations. Internall. J. Compu-

ter Math . 76549.560.5 . Rosenau, P. and Hym an , J . M . 1993. Compactonssolitons with finite lengths. P1z.y Rev Letter70:564-

567.6. Sanz-Serna, J .M. an d Chri s ti e , I. 1981. Pctrov.Galerk inmetho ds for nonlinear dispersive. J Comp.

Phys. 39:94-103.7. Christ ie, I., Griffi ths, D., Mitchel1,A. an d Sanz-Serna, J .M. 1981. Produc t approximation for non-linear urohlems in the finite element method. IMA I

Figure 2. Two Com patons Interaction ofK ( 2 , 2 ) equation.

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L.A. Wazzan and M.S. Ismail- Finite Element Solution of K(2,2) Equation