Numerical Simulations of the Complex Modied Korteweg-de Vries Equation. Thiab R. Taha. The University of Georgia. Abstract

Transcription

1 Numercal Smulatos of the Complex Moded Korteweg-de Vres Equato Thab R. Taha Computer Scece Departmet The Uversty of Georga Athes, GA 002 USA Tel e-mal Abstract I ths paper mplemetatos of three umercal schemes for the umercal smulato of the complex moded Korteweg-de Vres (CMKdV) equato are reported. The rst s a tegrable scheme derved by methods related to the Iverse Scatterg Trasform (IST). The secod s derved from the rst ad s called the local IST scheme. The thrd s a stadard te derece scheme for the CMKdV equato. Travellg-wave soluto as well as a double homoclc orbt are used as tal codtos. Numercal expermets have show that the stadard scheme s subject to stablty ad the umercal soluto becomes ubouded te tme. I cotrast the tegrable IST scheme does ot suer from ay stabltes. The ma derece amog the three schemes s the dscretzato of the olear term the CMKdV equato. Ths demostrates the mportace of proper dscretzato of olear terms whe a umercal method s desged for solvg a olear deretal equato. 1 Itroducto I 1991 Herbst et al. [1] derved a tegrable deretal derece equato, based o the IST, that has as ts lmtg form the CMKdV equato. Also, they derved aalytcal expressos for the homoclc orbts assocated wth the above equato, ad vestgated the eect of dscretzato of the equato the vcty of these orbts. They showed that a stadard te derece scheme s subject to a stablty. O the other had, they showed that the tegrable deretal derece scheme of the CMKdV equato does ot suer from ay stabltes. Recetly Taha derved a tegrable partal derece equato, based o the IST, that has as ts lmtg form the CMKdV equato [2] q t +jqj 2 q x + q xxx =0 (1.1) Here q s a complex valued fucto, ad jj deotes the modulus. I the preset paper ths partal derece equato s used as a umercal scheme for solvg Eq. (1.1). Ths scheme, call t the tegrable IST scheme,

2 as well as ts local verso, are mplemeted ad compared to a stadard te derece scheme for the umercal smulato of Eq. (1.1). Our umercal expermets have show that the stadard scheme suers from stablty, ad the umercal soluto becomes ubouded te tme. O the other had, our umercal expermets have show that the tegrable partal derece scheme does ot suer from ay stabltes. Ths result s agreemet of the result foud by [1] for the tegrable deretal derece scheme. Also, ths paper a tegrable partal derece equato that has as ts lmtg form the defocusg CMKdV equato q t ; jqj 2 q x + q xxx =0 (1.2) s derved. The method of dervato s smlar to the oe gve [2]. I secto 2 the partal derece equatos for (1.1) ad (1.2) are gve. 2 The Represetato of the CMKdV Equatos Usg Numercal Methods () The tegrable IST scheme s m Q m = Q m +2 A(4) ; ; Q m D ; (4) + Q m +1 S +1 ; Q m+1 +1 P ; Q m+1 ;2 A(4) ; + Q m ;2 ;2D ; (4) ; Q m+1 ;1 S ;2 + Qm ;1 P ;1 + Q m (A(0) ; where * deotes the complex cojugate, ad X l=;1 m Q m = Q m+1 ; Q m = T l ) ; Q m+1 (A (0) ; Y =;1 m+1 m m = m S = A (2) ; + A (4) ; F + D (4) ; +1 F = [Q m+1 +1 (Q ) m+1 ; H = f(q ) m Qm+1 +1 m+1 P = (D (2) ; + X [A (4) j=;1 X j=;1 ;1 X l=;1 =1jQ m j2 X H j j=;1 m ((Q ) m j Qm j+1 )] ; (Q ) m ;1 Qm+1 m g ;1 ; E j + D ; (4) G j ] j ) T l ) (2.1)

3 h j = ;1 j E = Q m (Q ) m+1 ;1 m+1 ; Q m +1 (Q ) m+1 m j G j = (Q m+1 j+1 (Q ) m+1 j ; Q m j (Q ) m j;1 )m+1 j j;1 T l = Q m+1 l [(Q ) m+1 l;2 A(4) ; ; (Q ) m l;2 l;2d (4) ; +(Q ) m+1 l;1 S l;2 ; (Q ) m l;1 P l;1] ; (Q ) m l [Q m l+2 A(4) ; ; Q m+1 l+2 l+1d ; (4) + Q m l+1 S l+1 ; Q m+1 l+1 P l] m A (2) ; = ; 2 A(0) ; + 1 D(2) ; = ; 2 2 A(0) ; ; 1 2 A (4) ; = 1 A(0) ; ; 1 D(4) ; = 1 4 A(0) ; where = t, (x) A (0) ; s a arbtrary costat, Q m =xqm, jj <p(p s half the legth of the terval of terest), ad m>0. Ths scheme s cosstet wth the CMKdV equatos (1.1) ad (1.2) ad has a trucato error of order 0((t) 2 ) + 0((x) 2 ). It s mplemeted wth the value of A (0) ; =. 2 () A local IST scheme whch s derved from the tegrable IST scheme (2.1) wth A (0) ; = 2 s q m+1 t ; q m = qm+1 ;1 ; q m+1 +q m+1 +1 ; q m (x) + qm ;2 ; qm ;1 +qm ; qm +1 2(x) 1 h q m (x) fjqm +1 j2 + jq m j2 g;q m ;2 fjqm+1 j 2 + jq m+1 ;1 j2 g + qm f(q ) m qm +1 +(q ) m+1 q m qm (q ) m ;1 g ; qm ;1 2 fqm+1 ;1 (q ) m+1 + q m ;1 (q ) m +2qm+1 (q ) m+1 +1 g + qm g;qm+1 2 f(q ) m+1 q m (q ) m qm +1 2 f(q ) m+1 q m+1 ;1 +(q ) m qm ;1 g ; fjq m j2 q m+1 j 2 q m ;1 g +1 ;jqm+1 (2.2) Ths scheme has a trucato error of order 0((t) 2 ) + 0((x) 2 ).

4 () A stadard umercal scheme s q m+1 t ; q m = (qm+1 ;1 ; qm+1 +q m+1 +1 ; qm+1 +2 ) 2(x) + (qm ;2 ; qm ;1 +qm ; qm +1 ) 2(x) 2(x) h jq m+1 j 2 (q m+1 +1 ; qm+1 ;1 )+jqm j2 (q m +1 ; qm ;1 ) (2.) Ths scheme has a trucato error of 0((t) 2 ) + 0((x) 2 ). Numercal Implemetato The umercal methods descrbed the prevous secto are appled to the CMKdV equato (1.1) subject to the followg codtos (a) Travellg-wave soluto [1]. The exact soluto of (1.1) s q(x t) =a exp[(kx ;!t)] (.1) where! satses the dsperso relato,! =jaj 2 k ; k ad a s the complex ampltude. For tal codtos, Eq. (.1) s used at t = 0, wth a = 1 k =1. Perodc boudary codtos o the terval 2 [0 4] are mposed. (b) A double homoclc orbt [1]. The tal codto q(x 0) = a exp(kx)(1 + 0 cos x) (.2) wth a = =01 k = = 2 L,adL =4p 2. Perodc boudary codto o the terval [0,L] are mposed. The above parameters allow two ustable modes. The three schemes are mplemeted usg several values of x ad t. Oe way to mplemet the above schemes s to solve baded crculat Toepltz systems of equatos of the form

5 2 4 ; 1 ;1 ;1 ; 1 ;1 ; 1 ;1 ; 1 1 ;1 ; ; 1 ;1 = 2 4 B ;N +1 B ;N +2 B ;N + B N;2 B N;1 B N q ;N +1 q ;N +2 q ;N + q N;2 q N;1 q N 5 (.) where =+ = 2(x),ateach tme level. Note that the oly derece the three schemes s t the rght had sde of Eq. (.). There are several algorthms to ecetly solve the above system. A moded Gaussa elmato method [] s used ths paper. Also, t s worth otg that the above system s very sutable for parallel mplemetato [4]. 4 Coclusos Based o umercal expermets, we have draw the followg coclusos 1. For the Travellg-wave soluto, t s foud that the case of a coarse dscretzato of Eq. (1.1) by the stadard umercal scheme Eq. (2.), the umercal soluto becomes ubouded at t = 1 wth N =40 x = L. If the dscretzato s reed, the blow upspostpoed. The soluto becomes N ubouded at t =4215 wth N = 48. O the other had, the umercal soluto dd ot blow up whe the tegrable IST scheme Eq. (2.1) or ts local IST verso Eq. (2.2) are used the dscretzato of Eq. (1.1) (see Table I, ad Fgs. 1-).

6 Table I shows the values of L 1 ad L 2 at t =5 10 ad 20 for the umercal schemes utlzed solvg Eq. (1.1), t =00125 N =40 1 Table 1 Itegrable IST Local IST Stadard t =5 L L t =10 L L t =20 L blow upat L t = For the double homoclc orbt, t s foud that the case of the dscretzato of Eq. (1.1) by the stadard umercal scheme Eq. (2.), the umercal soluto becomes ubouded at t = 84 wth N = 40. If the dscretzato s reed, the blow up s postpoed. The soluto becomes ubouded at t =422 wth N = 5. O the other had, the umercal soluto dd ot blow up whe the tegrable IST scheme Eq. (2.1) or ts local verso Eq. (2.2) are used the dscretzato of Eq. (1.1). (see Fgs. 4-) It s to be oted that the oly derece amog the three umercal schemes used to solve Eq. (1.1) s the dscretzato of the olear term of ths equato. Ths shows that the dscretzato of a olear term a olear deretal equato s crucal. Also, t shows that the tegrable IST schemes, whch have the same qualtatve propertes as the assocated cotuous equatos, wll play a mportat role the proper dscretzato of olear terms olear deretal equatos. Refereces 1. B.M Herbst, M.J. Ablowtz ad E. Rya, \Numercal homoclc stabltes ad the complex moded Korteweg-de Vres equato", Comput. Phys. Commu. Vol. 5, pp , T. R. Taha, \A Partal-Derece Equato for the Complex Moded Korteweg-de Vres Equato", Advaces Computer Methods for Partal Deretal Equatos, VII (R. Vchevetsky, ed.) IMACS, (1992) T. Taha ad M. Ablowtz, \Aalytcal ad Numercal Aspects of Certa Nolear Evoluto Equatos II, Numercal Nolear Schrodger Equato", J. Comput. Phys. Vol. 55, 2, pp , L 1 = max(jj~qj ;jqjj) ~q s the umercal soluto ad q s the exact soluto at the pot (x t) for all x =the cremet x. L2 = p ((j~qj ;jqj) 2 for all.

7 4. T. R. Taha ad Peqg Jag, \Parallel Algorthms for solvg Baded Toepltz Lear Systems", J. Neural, Parallel ad Scetc Comput., Vol. 1, (199)