An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs

Transcription

1 Commn. Comp. Phys. doi:.48/cicp.34.46a Vol., No., pp. -7 An Energy-Preserving Wavele Collocaion Mehod for General Mli-Symplecic Formlaions of Hamilonian PDEs Yezheng Gong and Yshn Wang Jiangs Key Laboraory of NSLSCS, School of Mahemaical Sciences, Nanjing Normal Universiy, Nanjing 3, P.R. China. Received XXX; Acceped (in revised version) XXX Absrac. In his paper, we develop a novel energy-preserving wavele collocaion mehod for solving general mli-symplecic formlaions of Hamilonian PDEs. Based on he aocorrelaion fncions of Dabechies compacly sppored scaling fncions, he wavele collocaion mehod is condced for spaial discreizaion. The obained semi-discree sysem is shown o be a finie-dimensional Hamilonian sysem, which has an energy conservaion law. Then, he average vecor field mehod is sed for ime inegraion, which leads o an energy-preserving mehod for mli-symplecic Hamilonian PDEs. The proposed mehod is illsraed by he nonlinear Schrödinger eqaion and he Camassa-Holm eqaion. Since differeniaion mari obained by he wavele collocaion mehod is a cyclic mari, we can apply Fas Forier ransform o solve eqaions in nmerical calclaion. Nmerical eperimens show he high accracy, effeciveness and conservaion properies of he proposed mehod. AMS sbjec classificaions: 65M6, 65M7, 65T5, 65Z5 Key words: Energy-preserving, average vecor field mehod, wavele collocaion mehod, nonlinear Schrödinger eqaion, Camassa-Holm eqaion. Inrodcion A nmerical mehod which can preserve one or more physical/geomeric properies of he sysem eacly is called geomeric or srcre-preserving inegraors. As geomeric inegraion has gained remarkable sccess in he nmerical analysis of ODEs [, ], i is believable o eend he idea of geomeric inegraion o significan PDEs. Many conservaive PDEs, for insance he sine-gordon (SG) eqaion, he nonlinear Schrödinger Corresponding ahor. addresses: gyz884@63.com (Y. Gong),wangyshn@njn.ed.cn (Y. Wang) hp:// c Global-Science Press

2 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 (NLS) eqaion, he Koreweg-de Vries (KdV) eqaion, he Camassa-Holm (CH) eqaion, he Mawell s eqaions and so on can be rewrien as mli-symplecic Hamilonian sysem, which has he properies of mli-symplecic srcre, energy and momenm conservaion laws [3 5]. To inheri he mli-symplecic srcre, many mlisymplecic mehods [6 9] are developed in recen years. For more deails in applicaions, please refer o review aricles [, ] and references herein. Ecep he mli-symplecic conservaion law, mli-symplecic Hamilonian sysem also has he energy conservaion law. The conservaion of energy is a crcial propery of mechanical sysems and plays an imporan role in he sdy of properies of solions [, 3]. I is valable o epec ha energy-preserving discreizaions for conservaive PDEs will prodce richer informaion on he discree sysems. Li and V-Qoc [4] gave a hisorical srvey of energy-preserving mehods for PDEs and heir applicaions, especially o nonlinear sabiliy. Energy-preserving mehods [5 9] have sccessflly made many applicaions. However, hese mehods have an ad hoc characer and are no compleely sysemaic eiher in heir derivaion or in heir applicabiliy; in conras, or mehod discssed here is compleely sysemaic, applied o a hge class of conservaion PDEs. Recenly, wavele-based nmerical mehods become increasingly poplar as hey combine he advanages of boh specral mehod and finie difference mehod (FDM) [ ]. Compared wih specral mehod, wavele-based mehods have good spaial localizaion and generae a sparse space differeniaion mari, and compared wih FDM, wavele-based mehods have good specral localizaion and higher order of accracy. The wavele-based algorihms can be roghly classified ino wo caegories: wavele-galerkin and wavele collocaion. In [3] and [4], Dabechies compacly sppored orhogonal waveles and second-generaion waveles are proposed o combine wih symplecic schemes o consrc mliresolion symplecic solvers for wave propagaion problems and he mehod is of wavele-galerkin ype. However, i is very difficl o deal wih nonlinear problems, as i needs he passage beween wavele coefficiens and physical space. In order o overcome his difficly, a wavele collocaion mehod is proposed in [5], in which he aocorrelaion fncions of Dabechies compacly sppored scaling fncions are shown o have he meris of symmery and nice inerpolaion properies, and so no era compaion is reqired for he passage beween wavele coefficiens and physical space. Based on he aocorrelaion fncions of Dabechies compacly sppored scaling fncions, he symplecic wavele collocaion mehod and he mli-symplecic wavele collocaion mehod are developed for Hamilonian PDEs wih periodic bondary condiions in [6] and [7], respecively, in which nmerical eperimens illsrae he remarkable behavior of he mehods. The average vecor field (AVF) mehod is firs wrien down in [8] and idenified as energy-preserving and as a B-series mehod in [9]. For ordinary differenial eqaion ẏ= f(y), y R d, (.)

3 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp he AVF mehod is he map y y defined by y y τ = f ( ( ξ)y+ξy ) dξ, (.) where τ is he ime sep. The AVF mehod (.) is affine-covarian, self-adjoin, and of second order [3]. When f is Hamilonian wih respec o a consan symplecic srcre, i.e., f = Ω H wih Ω a nonsinglar, skew-symmeric mari, he AVF mehod preserves he Hamilonian H. Noe ha he key of he AVF mehod is o inegrae he inegraion of (.) eacly. As we know, he above inegraion can be inegraed eacly when f is a polynomial. In [3] and [3], ahors apply high-order Rnge-Ka mehods and collocaion mehods o approimae he inegraion of (.) and obain energypreserving mehods. By sing he AVF mehod, Ref. [3] gave a class of sysemaic energy-preserving mehods based on symplecic formlaion of Hamilonian PDEs. Basic idea of he mehods is o firs discreize PDEs wih an niform spaial-discreizaion resling in a large sysem of Hamilonian ODEs. Then, he resling ODEs are inegraed by he AVF mehod. Recenly, we sared from general mli-symplecic formlaions of Hamilonian PDEs and ilized he AVF mehod o consrc a local energy-preserving mehod and a local momenm-preserving mehod [33]. The prpose of his paper is o consrc a new class of sysemaic energy-preserving mehods for mli-symplecic formlaions of Hamilonian PDEs. Based on he aocorrelaion fncions of Dabechies compacly sppored scaling fncions, he wavele collocaion mehod is condced for spaial discreizaion of mli-symplecic PDEs. The semi-discree sysem is shown o be a finie-dimensional Hamilonian sysem, which has an energy conservaion law. Then, he AVF mehod is sed for ime inegraion, which leads o an energy-preserving wavele collocaion mehod (EPWCM) for mlisymplecic Hamilonian PDEs. To or knowledge, i has no been repored ha he wavele collocaion mehod is applied o consrc energy-preserving schemes for he mli-symplecic PDEs. The proposed mehod is illsraed by he NLS eqaion and he CH eqaion. Since differeniaion mari obained by he wavele collocaion mehod is a cyclic mari [6, 34], we can apply Fas Forier ransform (FFT) o solve eqaions in nmerical calclaion. Nmerical eperimens show he high accracy, effeciveness and conservaion properies of he proposed mehod. The proposed mehod can be sed for general Hamilonian PDEs. Ths, he SG eqaion, he NLS eqaion, he KdV eqaion, he CH eqaion, he Mawell s eqaions and so on can be inegraed by he proposed mehod. The res of his paper is organized as follows: in Secion, mli-symplecic PDEs and aocorrelaion fncions are briefly inrodced, hen an EPWCM is developed for mli-symplecic PDEs. The proposed mehod is applied o derive energy-preserving algorihm for he NLS eqaion and he CH eqaion in Secions 3 and 4 respecively. To demonsrae he accracy and capabiliy of he mehod, nmerical eperimens for he NLS eqaion and he CH eqaion are repored in Secions 5 and 6 respecively. Conclsions are given in Secion 7.

4 4 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 Energy-preserving wavele collocaion mehod for mli-symplecic PDEs In his secion, mli-symplecic PDEs and aocorrelaion fncions are briefly inrodced, hen a novel energy-preserving wavele collocaion mehod is developed for mlisymplecic PDEs.. Mli-symplecic PDEs A wide range of PDEs can be wrien as a mli-symplecic sysem Mz +Kz = z S(z), z R d, (,) R, (.) where M and K are wo skew-symmeric marices and S:R d R is a scalar-valed smooh fncion. The above sysem has hree local conservaion laws [35, 36], namely mlisymplecic conservaion law local energy conservaion law (LECL) ω+ κ=, ω= dz M + dz, κ= dz K + dz, (.) E +F =, E=S(z)+z T K + z, F= z T K + z, (.3) local momenm conservaion law (LMCL) where M + and K + saisfy I +G =, G=S(z)+z T M +z, I= z T M +z, (.4) M=M + M T +, K=K + K T +. Noe ha he spliing of M and K is o dedce efficienly correc epressions of energy and momenm in physics. For periodic bondary condiions, he local conservaion laws can be inegraed in -direcion o obain global conservaion of energy and momenm.. Aocorrelaion fncions A Dabechies scaling fncion φ() wih M non-vanishing filer coefficiens (in shor, DM) saisfies he scaling relaion: φ()= M k= h k φ( k), (.5)

5 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp where M is a posiive even nmber and {h k } M k= are non-vanishing filer coefficiens. Define he aocorrelaion fncion θ() of φ() (in shor, ADM) as θ()= ( φ φ( ) ) ()= φ()φ( )d. (.6) Sppose ha V j is he linear span of {θ j,k ()= j/ θ( j k),k Z}, hen (V j ) j Z forms a mliresolion analysis where θ() plays he role of (nonorhonormal) scaling fncion. The vales of he fncion θ() and is derivaives a ineger poins can be obained by solving an eigenvale problem, see [, ] for more deails. The fncion θ() has inerpolaion propery: θ(l)= φ()φ( l)d=δ,l, l Z. (.7) An inerpolaion operaor on V j (wih he space sep h= j ) can be defined as I j ()= j ( j k)θ j,k ()= ( j k)θ( j k). (.8) k Z k Z.3 Energy-preserving wavele collocaion mehod for mli-symplecic PDEs Wavele collocaion mehod based on he aocorrelaion fncion θ() is sed for spaial discreizaion. The obained semi-discree sysem is shown o be a finie-dimensional Hamilonian sysem, which has an energy conservaion law. Then, he AVF mehod is employed for ime inegraion, which leads o an energy-preserving mehod for mlisymplecic Hamilonian PDEs..3. Wavele collocaion mehod for spaial discreizaion In his secion, wavele collocaion mehod is sed for spaial discreizaion for he mlisymplecic PDEs (.) wih periodic bondary condiions. We se he aocorrelaion fncion ADM in he framework of a collocaion mehod. Consider he mli-symplecic PDEs (.) wih he spaial domain being[ L, R ], where L and R are inegers. We se z=(,,, d ) T and se he inerpolaion operaor (.8) o approimae i (,) (i=,,,d). For a fied scale J = consan, we approimae fncion i (,) by an inerpolaion operaor I J i (,), which inerpolaes i (,) a collocaion poins l = L +l/ J, for l =,,,N, N =( R L ) J. The inerpolaion operaor I J i (,) has he form I J i (,)= im θ ( J ( L J +m) ), (.9) m Z where im = i ( m,) and im = im+n, m Z. According o he inerpolaion propery (.7), we have I J i ( l,)= i ( l,).

6 6 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 Applying he wavele collocaion mehod for spaial discreizaion of mli-symplecic sysem (.), we obain M(z l ) +K(I J z) =l = z S(z l ), (.) where I J z=(i J,I J,,I J d ) T and z l =( l, l,, dl ) T. To obain he eqaions for z l, he crcial sep is o epress he firs-order spaial parial derivaives a collocaion poins l in erms of he vales z l. Making once differenial wih (.9) and evalaing he resling epressions a collocaion poins, we obain I J i (,) = l l+m m=l (M ) im J θ (l m)=(b U i ) l, (.) where U i =( i, i,, in ) T and B is an N N mari wih elemens J θ (l m), M+ l m M ; J θ ( k), l m= N k, k M ; (B ) l,m = J θ (k), m l=n k, k M ;, oherwise. (.) The properies of space differeniaion mari have already been analyzed in deail in [6]. Here we noe ha B is skew-symmeric. Combining (.) wih he differeniaion mari B, we obain he wavele collocaion discreizaion for he mli-symplecic PDEs (.) M d N d z l+k (B ) l,m z m = z S(z l ), l=,,,n, (.3) m= which is eqivalen o he following compac form (I N M) d d Z+(B K)Z= Z S(Z), (.4) where I N is he N N ideniy mari, means Kronecker inner prodc, Z = (z T,zT,,zT N )T, and S(Z)= N l= S(z l). Since B and K are skew-symmeric, B K is symmeric, he semi-discree sysem (.4) can be wrien as (I N M) d d Z= ZH(Z), (.5) where H(Z)=S(Z) ZT (B K)Z. The discree oal energy of mli-symplecic sysem (.) can be defined as E()=hH(Z). (.6)

7 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp Remark.. H(Z) can also be wrien as H(Z)=S(Z) Z T (B K + )Z=S(Z)+Z T (B K T +)Z, (.7) which is corresponding o he discree oal energy of Eq. (.3). Theorem.. The wavele collocaion semi-discreizaion (.5) has he oal energy conservaion law d E()=. (.8) d Proof. Since I N M is anisymmeric, aking inner prodc wih d dz on boh sides of (.5), we obain =( d d Z ) T Z H(Z)= d d H(Z). Therefore, he oal energy conservaion law (.8) holds..3. Average vecor field mehod for emporal discreizaion Since mari I N M is sally singlar, he semi-discree sysem (.5) is algebraic differenial eqaions, and i is also a gradien sysem. Therefore, he key of consrcing energy-preserving mehod is o apply a discree gradien mehod for he semi-discree sysem (.5). Here he AVF mehod, as a special discree gradien mehod [8], is employed for emporal discreizaion. We discreize (.5) wih he AVF mehod and obain he energy-preserving wavele collocaion mehod (EPWCM) for he mli-symplecic PDEs (.) (I N M) Zn+ Z n = Z H ( ( ξ)z n +ξz n+) dξ. (.9) τ Noe ha he inegraion of (.9) can be inegraed eacly when H is a polynomial. In pracical applicaion, he scheme (.9) can be simplified by eliminaing some vales (see Secion 3 and Secion 4). Theorem.. The EPWCM (.9) possesses he discree oal energy conservaion law where E n = hh(z n ). E n+ =E n, (.) Proof. Since I N M is anisymmeric, aking inner prodc wih Z n+ Z n on boh sides of (.9), we have =(Z n+ Z n ) T Z H ( ( ξ)z n +ξz n+) dξ d = dξ H( ( ξ)z n +ξz n+) dξ = H(Z n+ ) H(Z n ). Therefore, he discree oal energy conservaion law (.) holds.

8 8 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 Remark.. Since differeniaion mari B is a cyclic mari [6, 34], we have B = F N diag(f Nb)F N, (.) where F N is discree Forier ransform wih elemens(f N ) j,n =W jn N, W N= e iπ/n, FN is inverse discree Forier ransform wih elemens(f N ) j,n= N Wjn N and b is he firs colmn of B. Remark.3. By he relaion (.), we can evalae he derivaives by sing he FFT algorihm insead of differeniaion mari B ino(nlog N) operaions raher hano(n ) operaions. Remark.4. The discree sysems for he nonlinear Hamilonian PDEs will be solved by sing he simple fied-poin ieraion mehod. For every ieraion sep, we need o solve linear eqaions A=b. Here A is a cyclic mari, so we can apply FFT for solving he eqaions. 3 Energy-preserving wavele collocaion mehod for he NLS eqaion The nonlinear Schrödinger eqaion iψ +ψ +a ψ ψ=, a>, (3.) can be wrien in mli-symplecic form by leing ψ= p+iq and inrodcing a pair of conjgae momena v= p, w=q [3]. Separaing (3.) ino real and imaginary pars, we obain he sysem q v = a(p +q )p, p w = a(p +q )q, (3.) p = v, q = w, which is eqivalen o he mli-symplecic form (.) for he NLS eqaion wih p z= q v, M=, K=, w and Hamilonian S(z)= ( v +w + a (p +q ) ). The corresponding energy and momenm conservaion laws are [ ( a (p +q ) v w )] + (vp +wq )=, (3.3)

9 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp and (pw qv)+ [ a ] (p +q ) +v +w pq +qp =. (3.4) Addiionally, he NLS eqaion (3.) also has local normal conservaion law [ ] (p +q ) + (pw qv)=. Under periodic bondary condiions, he above local conservaion laws can be inegraed in -direcion, respecively, o obain he oal energy conservaion law E()= L oal momenm conservaion law I()= and oal normal conservaion law B P n+/ = V n+/, B Q n+/ =W n+/. N()= ( a (p +q ) v w ) d=cons, L (pw qv)d = Cons, L (p +q )d=cons. Applying he EPWCM (.9) for he mli-symplecic formlaion (3.) of he NLS eqaion, we obain [( ) Q n+ Q n τ B V n+/ = a (P n+ ) +(P n ) + (Qn+ ) +(Q n ) Pn+ +P n ] + 6 (Qn+ Q n ) (P n+ Q n P n Q n+ ), Pn+ P n τ [( B W n+/ = (P a n+ ) +(P n ) + (Qn+ ) +(Q n ) ) Qn+ +Q n ] + 6 (Pn+ P n ) (Q n+ P n Q n P n+ ) Here τ is he ime sep, P n =(p n,pn,,pn N )T, P n+/ = (Pn +P n+ ), (P n ) = P n P n, P n Q n =(p nqn,pn qn,,pn N qn N )T, p n l is he approimaion o p( l, n ), l=,,,n, ec. By eliminaing he vales V and W, and sing Ψ= P+iQ, he scheme (3.5) is eqivalen o i Ψn+ Ψ n τ, (3.5) { Ψ +B n+ + Ψ n Ψn+/ +a Ψn+ +Ψ n + } 6 (Ψn+ Ψ n ) (Ψ n+ P n Ψ n P n+ ) =, (3.6)

10 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 where P n = Ψn +Ψ n. The residal of LECL for he EPWCM (3.6) is and he residal of LMCL is RE l = En+ l El n +(B F n+/ ) τ l, El n= { a ( (p n l ) +(q n l )) (v n l ) (w n l )}, F n+/ = V n+/ Pn+ P n +W n+/ Qn+ Q n, τ τ RM l = In+ l Il n +(B G n+/ ) τ l, Il n = p n l wn l qn l vn l, G n+/ = a ( (P n+/ ) +(Q n+/ ) ) +(V n+/ ) +(W n+/ ) P n+/ Qn+ Q n +Q n+/ Pn+ P n. τ τ Moreover, he discree oal energy ise n =h N l= En l, he oal momenm isi n =h N and he oal normal is N n = ( h N l= (p n l ) +(q n l )). l= In l 4 Energy-preserving wavele collocaion mehod for he Camassa-Holm eqaion The Camassa-Holm (CH) eqaion +3 =, (4.) is a model eqaion which represens he nidirecional propagaion of shallow waer waves over a fla boom [37]. This eqaion is rich in geomeric srcres and is an infinie-dimensional compleely inegrable Hamilonian sysem for a large class of iniial daa [38]. A grea deal of nmerical mehods have been sed o solve he CH eqaion. Finie difference scheme was proposed and proved o be convergen in [39, 4]. Psedospecral scheme is sed o invesigae several aspecs of he periodic raveling wave solions of he CH eqaion in [38] and he convergence of a specral projecion of he periodic CH eqaion was given in [4]. A local disconinos Galerkin mehod was developed for solving he CH eqaion in [4]. Recenly, David Cohen proposes wo new mli-symplecic formlaions for he CH eqaion and make eperimens for each formlaion by Eler bo scheme in [44]. Mli-symplecic wavele collocaion mehod was proposed for discreizing he CH eqaion in [7]. In his paper, we propose an energypreserving wavele collocaion mehod for he CH eqaion and simlae he smooh periodic wave and peaked wave solions.

11 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 Eq. (4.) can be rewrien in he form ( 3 ( ) + + ) =, (4.) which is eqivalen o a mli-symplecic PDEs φ p v = w 3 p, +w =, φ =, = p, = p+v, (4.3) wih z=(,φ,w,v,p) T, Hamilonian S(z)= w 3 / p /+vp, and he pair of skewsymmeric marices M and K, M=, K= The corresponding energy and momenm conservaion laws are. ( ) (3 + ) + (wφ v )=, (4.4) and ( ) ( + ) + ( w 3 p +vp p ) φ =. (4.5) Under periodic bondary condiions, he above local conservaion laws can be inegraed in -direcion, respecively, o obain he oal energy conservaion law E()= and oal momenm conservaion law I()= ( 3 + )d=cons, ( + )d=cons. Addiionally, he CH eqaion has a conserved qaniy H = d.

12 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 We apply he wavele collocaion mehod o discreize he mli-symplecic formlaion (4.3) of he CH eqaion in space and se he AVF mehod o inegrae he semidiscree sysem in ime. The resling scheme is δ Φ n δ P n B A V n = A W n δ U n = B A W n, B A Φ n = A U n, 6 ((U n ) +U n U n+ +(U n+ ) ) ((P n ) +P n P n+ +(P n+ ) ), B A U n = A P n, δ U n = 6 (Un P n +U n P n+ +U n+ P n +U n+ P n+ )+ A V n, (4.6) where δ U n = Un+ U n τ and A U n = Un+ +U n, ec. Making iniial condiion P = B U, we have P n = B U n. Eliminaing he variables Φ,W,V,P, we can obain he following EP- WCM for he CH eqaion (4.) A {δ U n δ B Un ( 6 B U n B U n +U n B U n+ +U n+ B U n +U n+ B U n+) +B [ ( (U n ) +U n U n+ +(U n+ ) ) + 6( (B U n ) +B U n B U n+ +(B U n+ ) )]} =. (4.7) Becase he scheme (4.7) is a hree level scheme, we need a wo ime levels scheme o give he iniial dam for he second level vales of he scheme (4.7). Since eqaion φ =, = p in (4.3) conain no ime derivaives, we can omi he operaor A in he corresponding eqaions o ge more accrae discreizaions in (4.6), namely B Φ n = U n, B U n = P n. Then, we can obain a wo ime levels scheme for (4.) δ U n δ B Un ( 6 B U n B U n +U n B U n+ +U n+ B U n +U n+ B U n+) [ ( +B (U n ) +U n U n+ +(U n+ ) ) + ( (B U n ) +B U n B U n+ +(B U n+ ) )] =. (4.8) 6

13 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp The hree discree conserved qaniies are H n N = E n = I n = l= n l h, N ( ( n l ) 3 + n l (B U n ) ) l h, l= N ( ( n l ) +(B U n ) ) l h. l= 5 Nmerical simlaion for he NLS eqaion For convenience, he EPWCM based on he aocorrelaion fncion of Dabechies scaling fncion ADM is called EPWCM wih ADM (abbr. EPWCM-ADM). To gain insigh ino he performance of he proposed EPWCM, he following nmerical eperimens for he NLS eqaion (3.) wih he consan a= are performed. In each ime sep, he scheme (3.6) is comped by he following fied-poin ieraion mehod i Ψn+,(s+) Ψ n τ +B Ψ n+,(s+) +Ψ n { Ψ n+,(s) + Ψ n +a Ψn+,(s) +Ψ n + } 6 (Ψn+,(s) Ψ n ) (Ψ n+,(s) P n Ψ n P n+,(s) ) =. (5.) Taking he iniial ieraion vecor Ψ n+,() = Ψ n, we condc he ieraions nil convergence, i.e., when he following crieria is saisfied Ψ n+,(s+) Ψ n+,(s) < 4. Here he discree L and L norms of comple-valed fncion ψ are defined as ψ = ( h ) / N ψ l l= and ψ = ma l N ψ l. I is noed ha he linear eqaions (5.) can be solved efficienly by he FFT algorihm. All he simlaions are comped by a compiled code of MATLAB 7.. Eample 5.. We consider he NLS eqaion (3.) wih he one-solion solion ( ( ψ(,) = sech ( 4)ep i c 3 ) ), (5.) where c=.

14 4 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 Table : Space accracy es of he EPWCM-AD for (3.) wih iniial condiion (5.) (τ =.e 6). J N Error Order L L L L e-3.653e e-6.45e e e Table : Time accracy es of he EPWCM-AD for (3.) wih iniial condiion (5.) (J= 4, N= 6). τ Error Order L L L L.4.93e e e e e-5.97e e e Firsly, he problem is solved in [ 5,5] ill ime = for accracy es. The space and ime accracy of he EPWCM (3.6) wih AD are esed. The L and L errors and he nmerical order of accracy are given in Tables and. Noice ha boh errors decrease very qickly when he space grid nmber N becomes larger. As shown in Table, he nmerical orders measred by he L error are.8977, , and 9.757, which means ha he error decays a an eponenial rae wih respec o he space grid nmber N. So does he L error. The nmerical resls are saisfacory, b he error analysis needs o be considered in he fre work. Noe ha in Table, he spaial error (N = 6) is negligible and he error is dominaed by he ime discreizaion error. Table clearly indicaes ha he AVF mehod is of second-order in ime. Table 3 shows he nmerical errors and CPU ime of he EPWCM wih differen ADM. The errors of he EPWCM decrease in an eponenial rae in space for each ADM and he decreasing rae is increasing when M becomes larger. Therefore, he EPWCM has high accracy in space. However, for a fied grid nmber N, CPU imes sed by he EPWCM wih differen ADM are almos he same lile, which is de o he FFT algorihm. Second, o es he conservaion properies of EPWCM, he problem is solved in [ 3,] ill ime =5 by sing he EPWCM-AD wih grid nmber N= and ime sep τ=.. As can be seen from Fig. (a), he moion of he solion is simlaed very well. The errors in global conservaion laws are shown in Fig. (b)-(d). If he scheme (3.6) is solved eacly, hen he oal energy is conserved eacly. The global energy is conserved p o he accracy of he EPWCM, b he errors of he global energy look like o show linear decrease in Fig. (c). The following nmerical eperimens also

15 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp GN.6 ψ (a) (b) GE GM (c) (d).5 R E R M (e) 5 5 (f) Figre : The EPWCM-AD (N = and τ =.) for (3.) wih he iniial condiion (5.). (a) Propagaion of soliary wave ψ ; (b) error in oal norm; (c) error in oal energy; (d) error in oal momenm; (e) error in LECL; (f) error in LMCL.

16 6 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 Table 3: Space accracy of he EPWCM wih AD, AD and AD3 wih iniial condiion (5.) (τ=.e 6, = ). EPWCM-AD EPWCM-AD EPWCM-AD3 N= L errors. 5.55e-.54e- L errors e- CPU(s) N=4 L errors.8e-3.7e-5 7.e-6 L errors.65e e e-5 CPU(s) N=8 L errors.4e-6 7.9e- 3.98e- L errors.45e-6 8.6e-.6e- CPU(s) appear similar siaions. So we sspec ha i may be de o solve approimaely he discree nonlinear sysem by sing he ieraive mehod. The residals in he local conservaion laws are also ploed as a fncion of he spaial grid locaion and he ime sep in Fig. (e)-(f). The errors are mainly concenraed arond he moving fron. Eample 5.. In his eample we consider he collision of doble solions of he NLS eqaion (3.) wih he iniial condiion [43] ψ(,) = j= ) sech( j )ep( ic j( j ). (5.3) Here c = 4, =, c = 4, =. For he prpose of nmerical, we consider a mli-symplecic wavele collocaion mehod (MSWCM) [7]. The problem is solved in [ 5,5] ill ime =5 by sing he EPWCM-AD and MSWCM-AD wih N =4 and τ =.. The nmerical solions of he wo mehods are he same, so we only show he nmerical solions obained sing he EPWCM. As shown in Fig. (a), he doble solions collide a =±5 and =. The shapes and velociies of he doble solions keep nchanged afer collision. The nmerical errors in he conserved qaniies can be fond in Fig. (b)-(d). I is shown ha he EPWCM conserves oal energy eacly, while he MSWCM conserves oal norm eacly. In his eample, he oal momenm of he EPWCM is conserved well as ha of he MSWCM. These resls show ha he EPWCM is effecive and has good conservaion properies.

17 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp GN 4 MSWCM EPWCM ime (a) (b) GE GM 3 MSWCM EPWCM ime (c) MSWCM EPWCM ime (d) Figre : The EPWCM-AD and MSWCM-AD (N= 4 and τ=.) for (3.) wih he iniial condiion (5.3). (a) Propagaion of soliary wave ψ ; (b) error in oal norm; (c) error in oal energy; (d) error in oal momenm. 6 Nmerical simlaion for he Camassa-Holm eqaion In his secion, smooh periodic ravelling wave, periodic peakon ravelling wave and collision of peakons of he CH eqaion are simlaed in Eamples Long ime simlaions are made for smooh periodic wave in Eample 6.. Accracy es and long ime simlaion are made for peakon wave in Eample 6.. Collision of wo peakons are given in Eample 6.3 and collision of hree peakons are given in Eample 6.4. Eample 6. (Smooh solion). As poined in [38], he solion of he Camassa-Holm eqaion can be described by hree parameers m,m,z R, where z=c M m. The eqaion has a smooh periodic ravelling wave when hree parameers m,m,z flfill he relaion z<m< M<c. By choosing m=.3, M=.7 and c=, we sdy a smooh periodic ravelling wave wih period of L The iniial daa is consrced by performing a spline inerpolaion o obain as a fncion of, see [38] for deails. Noe ha he eac

18 8 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp =.75 = = = Figre 3: Time evolion of he smooh periodic ravelling wave over he ime inerval [,], sing he EPWCM-AD (N= 8, h=.5, τ=.4). solion is obained by periodic eension of he iniial fncion. By aking a coordinae ransform = Lξ, he CH eqaion (4.) is eqivalen o L ξξ L 3( ξ) ξξ + ( 3 L + ) L ξ =, ξ [,]. Then, he problem is solved ill ime = by sing he EPWCM-AD wih grid nmber N =8 and ime sep τ =.4. As shown in Fig. 3, he proposed mehod simlaes he smooh periodic ravelling wave very well. The L and L errors a = are.37 5 and 6. 6 respecively. Fig. 4 shows ha he invarians are conserved beer han he MSWCM (see [7]). Eample 6. (Peakon solion). In his eample we consider periodic peaked ravelling waves wih he iniial condiion c cosh(a/) (,) = cosh( ), a/, c cosh(a/) cosh( a ( ) ) (6.), > a/, where c is he wave speed, a is he period and is he posiion of he rogh. ξ

19 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp error in H GE GM Figre 4: The errors in H, energy and momenm of he EPWCM-AD wih he smooh periodic iniial daa. Firsly, we presen he wave propagaion of he CH eqaion wih parameers c=, a= and =. The compaional domain is [,a]. Table 4 shows he nmerical errors and CPU ime of he EPWCM wih differen ADM. Compared wih he Eler bo scheme Table 4: Space accracy of he EPWCM wih AD, AD and AD3 wih iniial condiion (6.) (τ=.e 5, = 3). EPWCM-AD EPWCM-AD EPWCM-AD3 N=8 L errors 3.e-3.5e-3.95e-3 L errors 5.57e e-4.99e-4 CPU(s) N=56 L errors.53e-3.4e e-4 L errors.44e-4.4e-4.4e-4 CPU(s) N=5 L errors 8.39e-4 6.6e-4 5.e-4 L errors.8e e e-5 CPU(s)

20 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7. = Figre 5: A periodic peakon comped by he EPWCM-AD3 (N= 56, τ=.). Dashed crves are he iniial daa while solid crves are comped resls. 4 3 The L error of error in H GE.5 GM Figre 6: The L error of nmerical solion and he errors in H, energy, momenm of he EPWCM-AD3 wih he non-smooh periodic iniial daa (6.). in [44] and he specral mehod in [4], he errors obained by he EPWCM are smaller. As he MSWCM in [7], he errors obained by he EPWCM are he same order of magnide, b CPU ime here is less. Second, we solve he problem ill ime = 5 by he EPWCM-AD3 wih N = 56 and τ=.. The comped resl is shown in Fig. 5. The L and L errors a =5 are and respecively. The L error of nmerical solion and he errors of invarians are shown in Fig. 6. The errors of H, energy, momenm are bonded

21 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7. =3. = =7. = Figre 7: A periodic peakon comped by he EPWCM-AD3 (N= 8, τ=.). by.96,.464, , so he invarians are approimaely conserved. Third, we consider he peakon solion wih he parameers c=, a= and =. The problem is solved in [,a] ill ime = by he EPWCM-AD3 wih N= 8 and τ=.. As can be seen from Fig. 7, he moving peak profile is resolved very well. Eample 6.3 (Two-peakon ineracion). In his eample we consider he wo-peakon ineracion of he CH eqaion wih he iniial condiion (,)=φ ()+φ (), (6.) where c i cosh(a/) φ i ()= cosh( i), c i cosh(a/) cosh( a ( i ) ), i a/, i > a/, i=,. (6.3) The parameers are c = 3, c =, = 8, =, a=5. The compaional domain is [,a]. The EPWCM-AD3 is sed o simlae his ineracion wih N=6 and τ=.. As can be seen from Fig. 8, he aller wave overakes he shorer one a ime = 4 and

22 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 = = =4 = =8 = Figre 8: Ineracion of wo peakons comped by he EPWCM-AD3 (N= 6, τ=.). aferwards boh waves reain heir original shapes and velociies. The errors of invarians are ploed in Fig. 9. These resls show ha he EPWCM simlaes he ineracion of wo peakons well. Eample 6.4 (Three-peakon ineracion). In his eample we show he ineracion of hree peakons of he CH eqaion wih he iniial condiion (,)=φ ()+φ ()+φ 3 (), (6.4) where φ i, i=,,3, are defined as in (6.3). The parameers are c =, c =, c 3 =.8, = 5, = 3, 3 =, a=3. The compaional domain is [,a]. EPWCM-AD3 is sed

23 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp error in H GE GM Figre 9: The errors in H, energy and momenm of he EPWCM-AD3 wih he wo peakons. o simlae his ineracion wih N =9 and τ =.. As can be seen from Fig., he moving peak ineracion is resolved very well. In addiion, he resls obained here are similar wih ha obained by a local disconinos Galerkin mehod in [4]. The above nmerical resls for he CH eqaion demonsrae he EPWCM can simlae peakons very well and has good conservaion properies. 7 Conclsions In his paper, we develop a novel energy-preserving wavele collocaion mehod for solving general mli-symplecic formlaions of Hamilonian PDEs wih periodic bondary condiions. We apply he proposed mehod o derive energy-preserving scheme for he NLS and CH eqaions. Since differeniaion mari obained by he wavele collocaion mehod is a cyclic mari, we can apply Fas Forier ransform o solve eqaions in nmerical calclaion. Nmerical eperimens confirm ha he proposed mehod cold conserve energy conservaion law and have ecellen performance in long ime comping. The proposed mehod can be sed for general Hamilonian PDEs. Therefore, he sine-gordon eqaion, he Mawell s eqaions and so on can be inegraed by he proposed mehod.

24 4 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp. -7 = = = = =4 = =8 = Figre : Ineracion of hree peakons comped by he EPWCM-AD3 (N= 9, τ=.).

25 Y. Gong and Y. Wang / Commn. Comp. Phys., (), pp For mli-symplecic Hamilonian PDEs, he semi-discree sysem obained by discreizing firs-order differenial operaor wih anisymmeric differenial mari is a large sysem of Hamilonian ODEs, which has an energy conservaion law. Many mehods can be sed for spaial discreizaion, for eample Forier psedospecral mehod, wavele collocaion mehod and compac finie difference mehod. In his paper, he key of consrcing energy-preserving mehod is o apply a discree gradien mehod for he gradien sysem (.5). In fac, ofen he sysem (.5) can be wrien as a energy sysem ẏ = S H(y) (S is anisymmeric) by eliminaing some vales. Therefore, one can replace he average vecor field mehod by higher-order energy-preserving B-series mehod [9], while reaining energy conservaion law. I will be considered in he fre work. Acknowledgmens Yezheng Gong s work is parially sppored by China Posdocoral Science Fondaion hrogh Grans 6M5954. Yshn Wang s work is sppored by he Jiangs Collaboraive Innovaion Cener for Climae Change, he Naional Naral Science Fondaion of China (Gran Nos. 795, 4373) and he Prioriy Academic Program Developmen of Jiangs Higher Edcaion Insiions. References [] E. Hairer, C. Lbich and G. Wanner, Geomeric Nmerical Inegraion: Srcre Preserving Algorihms for Ordinary Diffenenial Eqaions, Berlin: Springer-Verlag,. [] K. Feng and M. Z. Qin, Symplecic Geomeric Algorihms for Hamilonian Sysems, Berlin/Hangzho: Springer-Verlag/Zhejiang Pblishing Unied Grop, Zhejiang Science and Technology Pblishing Hose, 3. [3] T. J. Bridges, Mli-symplecic srcres and wave propagaion, Mah. Proc. Cambridge Philos. Soc., (997), [4] J. Marsden, G. Parick and S. Shkoller, Mli-symplecic Geomery, Variaional Inegraors and Nonlinear PDEs, Comm. Mah. Phys., 99 (998), [5] S. Reich, Mli-symplecic Rnge-Ka collocaion mehods for Hamilonian wave eqaions, J. Comp. Phys., 57 (), [6] S. Reich, Backward error analysis for nmerical inegraors, SIAM J. Nmer. Anal., 36 (999), [7] J. Frank, B. E. Moore and S. Reich, Linear PDEs and nmerical mehods ha preserve a mli-symplecic conservaion law, SIAM J. Sci. Comp., 8 (6), [8] B. N. Ryland and R. I. Mclachlan, On mli-sympleciciy of pariioned Rnge-Ka mehods, SIAM J. Sci. Comp., 3 (8), [9] Y. Z. Gong, J. X. Cai and Y. S. Wang, Mli-symplecic Forier psedospecral mehod for he Kawahara eqaion, Commn. Comp. Phys., 6 (4), [] T. J. Bridges and S. Reich, Nmerical mehods for Hamilonian PDEs, J. Phys. A: Mahemaical and General, 39 (6), [] Y. S. Wang and J. L. Hong, Mli-symplecic algorihms for Hamilonian parial differenial eqaions, Comm. Appl. Mah. Comp., 7 (3), 63-3.

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