1. Introduction. In this paper, we are interested in accurate numerical approximations to the nonlinear Camassa Holm (CH) equation:

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1 SIAM J. SCI. COMPUT. Vol. 38, No. 4, pp. A99 A934 c 6 Society for Indstrial and Applied Matematics AN INVARIANT PRESERVING DISCONTINUOUS GALERKIN METHOD FOR THE CAMASSA HOLM EQUATION HAILIANG LIU AND YULONG XING Abstract. In tis work, we design, analyze, and nmerically test an invariant preserving discontinos Galerkin metod for solving te nonlinear Camassa Holm eqation. Tis model is integrable and admits peakon solitons. Te proposed nmerical metod is ig order accrate, and preserves two invariants, momentm and energy, of tis nonlinear eqation. Te L -stability of te sceme for general soltions is a conseqence of te energy preserving property. Te nmerical simlation reslts for di erent types of soltions of te Camassa Holm eqation are provided to illstrate te accracy and capability of te proposed metod. Key words. stability discontinos Galerkin metod, Camassa Holm eqation, energy conservation, AMS sbject classifications. 65M6, 65M, 35Q53 DOI..37/5M75X. Introdction. In tis paper, we are interested in accrate nmerical approimations to te nonlinear Camassa Holm (CH) eqation: () m t + m +m =, m =, R, t>, were te sbscript t (or, respectively) denotes te di erentiation wit respect to time variable t (or ). By taking advantage of its Hamiltonian strctre to reformlate tis eqation, we develop an invariant preserving discontinos Galerkin metod for tis nonlinear CH eqation. Or proposed sceme is ig order accrate, and preserves two invariants, momentm and energy, of tis nonlinear eqation, ence prodcing wave soltions wit satisfying long time beavior. Te CH eqation was fond in a stdy by Camassa and Holm [4], as a bi- Hamiltonian model for waves in te sallow water. In tis contet, te eqation takes te form () t +k t +3 = +. If te parameter k is positive, te solitary wave soltions are smoot solitons. In te special case tat k =, eqation () redces to te CH eqation (), wic as peakon soltions: solitons wit a sarp peak, containing a discontinity at te peak in te wave slope profile. Eqation () can also be written as te system of eqations t + + p =, p ) k + + ( ), Sbmitted to te jornal s Metods and Algoritms for Scientific Compting section Jne, 5; accepted for pblication (in revised form) Marc, 6; pblised electronically Jly 6, 6. ttp:// Department of Matematics, Iowa State University, Ames, IA 5 (li@iastate.ed). Tis ator s work was spported by te National Science Fondation nder grant DMS-3636 and by NSF grant RNMS (Ki-Net) 79. Department of Matematics, University of California at Riverside, Riverside, CA 95 (ingy@cr.ed). Tis ator s work was spported by NSF grant DMS A99

2 A9 HAILIANG LIU AND YULONG XING wit p being te dimensionless pressre or srface tension. Tis nonlocal CH eqation is also one of tree eqations in te family t t + + c =(c + c + c 3 ), wic satisfies asymptotic integrability p to tird order, a necessary condition for te complete integrability. Te oter two cases in tis family are te Korteweg de Vries (KdV) eqation, t + + =, and te Degasperis Procesi (DP) sallow water eqation, t t +3 = +. Te CH eqation and DP eqation also belong to te -class of dispersive models: t t + = +( ), wit =/3and /4, respectively. Te -class was first identified in [], as a sbclass of tose introdced in [9] and te analysis on its global eistence verss finite wave breaking was given by Li and Yin [3] wit te following reslts: for apple apple, initial smootness is sown to persist for all time; if apple <, strong soltions of te -eqation may lose certain reglarity in finite time, and traveling waves sc as periodic, solitary, peakon, peaked periodic, csped periodic, or csped soliton are all permissible [8, 5]. Te two special eqations, te CH and DP eqations, in te -class sare several interesting properties inclding te bi-hamiltonian strctre, te complete integrability, and te soliton-like peakon soltions. Bot eqations can be viewed as models of sallow water waves [3, 4, 5, 7,, 6, ]. However, tere are also important differences between tese eqations: from te perspective of PDE teory, te soltions of te CH eqation basically belong to H (R) (te first order Sobolev space), wile te DP eqation can develop socks (jmp discontinities) wic may be nderstood as entropy soltions [8, 9, 4]. For te CH eqation wit any initial data H (R), varios papers ave stdied te global eistence of its soltions, conservative or dissipative. Uniqeness is a delicate isse becase in general te flow map as less reglarity tan sally needed to jstify te niqeness of te soltion. Recently, Bressan, Cen, and ang [] proved tat te Cacy problem of te CH eqation wit general initial data H (R) as a niqe, globally in time, conservative soltion. Te energy conservation property is essentially sed in teir niqeness proof to overcome certain di clty. We plan to compte tis niqe conservative soltion nmerically by te discontinos Galerkin (DG) metod. Te DG metod is a class of finite element metods sing discontinos piecewise polynomial spaces for bot te nmerical soltions and te test fnctions (see [7] for a istoric review). It combines advantages of bot finite volme and finite element metods, and as been sccessflly applied to a wide range of applications. Many DG metods ave been designed for varios water wave eqations. Some eamples can be fond in te review paper [8]. An energy dissipative DG metod for te CH eqation as been presented in [7]. Recently, tere ave been some stdies on invariant preserving DG metods, wic can preserve te mass and energy eactly in te discrete sense. Nmerical evidence sows tat sc metods often provide more accrate reslts in long time simlations. Invariant preserving DG metods ave been designed for varios wave eqations inclding te KdV eqation

3 DG METHOD FOR THE CH EQUATION A9 [3, 9], second order wave eqation [6, 6], and te DP eqation []. We also refer te interested reader to [5, 4, ] for some related metods for second order acostic wave eqations. Te goal of tis paper is to develop a novel DG metod for te CH eqation, to preserve bot te momentm and energy, wic are given by E = )d, E = R( R + d or E = md, E = md. R R wit m =. Te local DG metod introdced in [7] ses te following reformlation: m t + f() p +(r /) =, p (r) =, were f() =3 / and r =. Te La Friedrics fl for f() sed in [7] is eld accontable for te dissipation of te energy E. In tis work we eplore yet anoter reformlation m t + q + mr =, q m =, wic takes advantage of te first compatible Hamiltonian description (3) m t for D m + m@. Tis, togeter wit simple central nmerical fles, makes it possible tat te reslting sceme preserves bot te momentm and energy. Besides te ig order of accracy, one main featre of te sceme is its capability to prodce wave soltions wit satisfying long time beavior. Some comparison sows tat te sceme wic conserves tese two qantities as smaller pase error tan tose prodced by te energy dissipative sceme. Wit P elements, te DG discretization in te present work may be viewed as a finite di erence sceme sc as te one stdied in [], ts etending te reslts for te scalar CH eqation obtained in []. Finally, we point ot tat te CH eqation also admits anoter compatible Hamiltonian description m for E 3 = R + Tis work sows tat te sceme conserving bot E and E is clearly better tan tat wic conserves only E. One wold intitively tink it is even better to preserve E,E, and E 3. However, it appears a rater di clt task to preserve all tree conservation laws, yet still maintain te optimal order of convergence for smoot soltions. Te rest of tis paper is organized as follows. In section, we formlate or DG metod in bot semidiscrete and flly discrete setting, and prove te desired invariant preserving properties of te proposed DG metod. In section 3, we present a set of nmerical eamples to illstrate te capacity of te DG sceme to captre varios peakons and teir interactions. Some conclsion remarks are given in section 4. d.

4 A9 HAILIANG LIU AND YULONG XING. Te discontinos Galerkin metods... Reformlation. Te Hamiltonian description (3) sets te basis to constrct an E preserving discretization for (), and it leads to te following reformlation: (4) m t +(m) + m =, m =. By introdcing te ailiary variables r and q, we rewrite tis form into te following system: (5a) (5b) (5c) (5d) m t + q + mr =, q m =, m + r =, r =. Two conservation laws for te CH eqation can be epressed as (6) E = md, E = md = ( + r )d. R R R.. DG formlation. We develop an invariant preserving DG metod for te CH eqation sbject to initial data () and periodic bondary conditions. Let s denote te comptational mes of te domain I by I j =( j /, j+/ ) for j =,...,N. Te center of te cell is j =( j / + j+/ )/, and j = j+/ j /. We denote by w + j+/ te vale of w at j+/ evalated from te rigt element I j+, and w j+/ te vale of w at j+/ evalated from te left element I j.[w] =w + denotes te jmp of w at cell interfaces, and {w} = (w+ + w ) denotes te average of te left and rigt interface vales. We ten define te piecewise polynomial space V as te space of polynomials of degree k in eac cell I j,i.e., V = {w : w P k (I j ) for I j,j =,...,N}. Te DG metod for (5) is formlated as follows: for all te test fnctions,,, V,findm,q,r, V sc tat (7a) (7b) (7c) I j (m ) t d I j (q m ) d =, I j m d q ( ) d + ˆq j+/ ˆq + j / + m r d =, I j I j r ( ) d + ˆr j+/ ˆr + j / (7d) r d + ( ) d û j+/ + û + =. j / I j I j d =, w

5 DG METHOD FOR THE CH EQUATION A93 Te at terms, known as nmerical fles, take only single vales on cell interfaces given by (8) (ˆq, ˆr, û )=({q }, {r }, { }). Te global form of te DG formlation may be obtained by smming over all te cells I j, (9a) (9b) (9c) (9d) (m ) t d q ( ) d (q m ) d =, m d r d + r ( ) d ( ) d + j= (ˆq [ ]) j+/ + (ˆr [ ]) j+/ j= (û [ ]) j+/ =, j= m r d =, d =, were R = P N R j= I j, and te periodic bondary conditions ave been sed..3. Stability and conservative properties. We now trn to establis te conservation of E and E. Teorem.. Let (m,q,,r ) be te nmerical soltions of te DG formlation (7) (8). Ten we ave te following: (i) Bot invariants E and E are preserved in te sense tat d dt (ii) Te DG sceme is L stable: m d =, d ( + r dt )d =. k (t)k applek k H 8 t>. Proof. (i) To sow tat te sceme preserves E, we simply take = in (9a), = r in (9c), and = in (9d), and sm te reslting relations to obtain d dt m d = = = j= r (r ) d apple r ˆr [r ] j= (ˆr [r ]) j+/ + apple +û [ ] j+/ ( ) d + (({r } ˆr )[r ] ({ } û )[ ]) j+/ =. j= (û [ ]) j+/ j= To sow te conservation of E, we take = in (9a), = r in (9b), =

6 A94 HAILIANG LIU AND YULONG XING in te time derivative of (9c), =(r ) t in (9d), and = q in (9d) to obtain (m ) t d q ( ) d (ˆq [ ]) j+/ + m r d =, q r d m r d =, j= (m ) t d + (r ) t ( ) d + r (r ) t d + r q d (r ) t d + (q ) d j= ((ˆr ) t [ ]) j+/ + ( ) t d =, (û [(r ) t ]) j+/ =, j= (û [q ]) j+/ =. j= Adding te above togeter and integrating te complete derivative ot, we get d + r d = (ˆq [ ]+û [q ] [ q ]) dt j+/ j= + ([ (r ) t ] (ˆr ) t [ ] û [(r ) t ]) j+/ =. j= (ii) From te conservation of energy E,weave k k = E (t) kr k = E () kr k apple E (). Hence k k apple E () as desired, and we ave te L stability..4. Time discretization. In tis sbsection, we develop flly discrete metods tat maintain te invariant preserving property of te semidiscrete metod (7). To acieve tis, we will employ time stepping metods tat preserve te discrete invariants. A family of symplectic temporal integrators wic preserve te invariants p to rond-o error is te implicit Rnge Ktta collocation type metods associated wit te diagonal elements of te Padé table for e z []. In tis paper, we consider te second order midpoint rle and te fort order two-stages Gass Legendre metods. Let {t n },n =,,...,M, be a niform partition of te time interval [,T], and t n = t n+ t n. Let = be te piecewise L projection of te initial condition (). We pdate te soltion n+ from n sing te midpoint rle in te following manner: for n =,...,M, n+ V is given by () n+ = n, n, were n, is te soltion of te eqation () n, n t n F( n, )=, were F( ) is te spatial operator; see (3) for te flly discrete metod. In Teorem., we ave sown tat te semidiscrete DG metod preserves te continos invariants E (t) and E (t). Along te same line of analysis, we can prove tat te discrete invariants, as defined in te following teorem, is conserved by te flly discrete metod.

7 DG METHOD FOR THE CH EQUATION A95 Teorem.. Te flly discrete midpoint rle DG metod () () conserves te discrete invariants () E n = m n d, E n = ( n ) +(r) n d for all n. Proof. Introdce te notation Dm n = mn+ m n t n = mn, t n / Combined wit te DG formlation (9), te midpoint rle DG metod () () can be rewritten as Dm n d q n, dq ( n, (3a) ) d [ ] + m n, rn, d =, j= j+/ (3b) (q n, m n, n, ) d =, m n, d r n, dr ( n, (3c) ) d [ ] n, d =, j= j+/ r n, d + n, d ( n, (3d) ) d + [ ] =. j= m n. j+/ To sow te conservation of E n, we take te test fnctions = in (3a), = r n, in (3c), and = n, in (3d), and sm te reslting relations to obtain Dm n d = r n, (rn, ) d j= d n, + [ ] j= j+/ = {r n, } d r n, [r ] j= dr n, [r ] + j+/ n, (n, ) d { n, } d n, [ ] =. j+/ Since R Dm n d =(En+ E n )/ t n,weavee n+ = E n. To sow te conservation of E n, we first consider (9c) at time levels t n and t n+, and let te test fnction be n, /( tn ). Sbtraction of tese two eqations yields (4) Dm n n, d + Dr n ( n, ) d + j= ddr n [n, j+/ ] + D n n, d =. We ten take = n, in (3a), = r n, in (3b), = Dr n q n, add te reslting eqations wit (4) to obtain (5) D n n, + Drn r n, d =. in (3d), and

8 A96 HAILIANG LIU AND YULONG XING Combined wit te fact tat D n n, = n+ n n+ t n + n = n+ ( n ) t n, Dr n r n, = rn+ (r n ) t n, te relation (5) leads to E n+ = E n. In some of te nmerical eperiments wit very ig accracy in spatial discretization, te following fort order two-stages Gass Legendre metods, wic also preserve te discrete invariants E n and E n, are condcted as te time approimations: (6) n+ = n + p 3 n, n,, were n, and n, are given as soltions of te copled system of eqations (7) (8) n, n t n a F( n, )+a F( n, ) =, n, n t n a F( n, )+a F( n, ) =, wit a = a =/4, a =/4 p 3/6, a =/4+ p 3/6..5. Algoritm. In tis section, we give details related to te implementation of te proposed invariant preserving DG metod.. First, from (7d) and (7c), we obtain r and m in te following matri form: (9) R = AU, M = U AR, were U (or R, M ) denotes te vectors containing te degree of freedom for te piecewise polynomial soltion (or r, m ). Tis leads to te following relation between M and U : () M = BU, wit te matri B = I A.. Eqation (7b) leads to q defined by () Q = (m ), were is te standard L projection onto piecewise P k polynomials. 3. Eqation (7a) and () leads to t M = AQ (m r ). 4. We ten combine () and () to obtain t U = B ( AQ (m r )). 5. We apply a symplectic temporal discretization metod, for eample te second order midpoint metod () or te fort order Gassian Legendre metod (6), to advance te obtained system (3) in time. Te di erential matri A is a sparse block matri, ence its mltiplication wit coefficient vectors can be implemented e ciently. Step 5 involves a linear solver wit te matri B, were we can perform an LU decomposition for B at te beginning to save comptational cost.

9 DG METHOD FOR THE CH EQUATION A97 3. Nmerical reslts. In tis section we provide some nmerical eamples to illstrate te accracy and capability of or metod. Te midpoint rle is sed as te temporal discretization in most of te nmerical reslts, ecept te accracy test (Eample ) were te fort order Gass Legendre metods are sed. Wit te aid of sccessive mes refinements we ave verified tat, in all cases, te reslts sown are nmerically convergent. Eample (accracy test). (4) (, t) =ce ct, Te peaked traveling wave soltion takes te form were c =.5 is te wave speed. Te domain is set as [ 4, 4], and te accracy is measred in te smoot parts of te soltion, / of te comptational domain away from te peak. We test te P k polynomial approimation on niform meses. Te L errors and te nmerical orders of accracy for k =,, at time t = are reported in Table. Te CFL condition is picked as.. We can clearly see tat te invariant preserving DG metods acieve te optimal convergence rates for P k elements. Te order of accracy is a little oscillating arond te optimal rate. Tis oscillating beavior of nmerical accracy is commonly observed for energy conserving metods, and a least sqare fitting of te order as been done in [6] and give te optimal convergence rate of. In [6], careflly cosen nmerical initial conditions are employed and optimal convergence rates are clearly observed tere, wic sows tat te energy conserving metod is more sensitive to te error in te initial conditions. Table Accracy test of DG metod for te CH eqation wit te eact soltion (4). Te periodic bondary condition, c =.5, niform meses wit N cells at time t =. P N L error Order N L error Order N L error Order.59e-3 -.9e e e e e e e e e e e e e e-3.9 P 4.6e e e e e e e e e e e e-3.9 P 4.3e e e e e e e e-3-3.7e e e e-4.94 In te net for eamples, te initial data will be generated from te following profiles: 8 >< c i cos(a/) cos( i), i applea/, i() = >: c i cos(a/) cos(a ( i)), i >a/, wit c i, i and a to be frter specified. Eample (peakon soltion). In tis eample, we present te wave propagation of a single peak soltion. Te initial condition is given by () = (),

10 A98 HAILIANG LIU AND YULONG XING wit te parameters c =, a = 3, and = 5. We set te comptational domain as [,a]. Te soltion profile at time t =, 5,, and are plotted in Figre. Te lack of smootness at te peak of peakons introdces ig-freqency dispersive errors into te calclation, wic will case nmerical oscillation near te peak. To resolve it, we se te P 4 polynomial wit cells Fig.. Eample, peakon interaction: Te soltion at time T =5,, 5, and. In Figre, we also provide te comparison of te conservative metods, compared wit te energy dissipative LDG metods introdced by X and S [7] and te eact soltions. We can see a smaller pase error compared wit te soltions from te energy dissipative metods. Te performance of conservative metods is epected to be better tan dissipative metods for long time simlations, as observed for te DP eqation in []. However, for te CH eqation te improvement in terms of te pase error is not as good as tat for te DP eqation. Tis obvios pase error between te nmerical soltion and te eact soltion is also observed in [] wen sing te invariant preserving finite di erence metod. Eample 3 (two-peakon interaction). In tis eample, we consider te twopeakon interaction of te CH eqation wit te initial condition () = +. wit c =, c =, = 5, = 5, and a = 3. We set te comptational domain

11 DG METHOD FOR THE CH EQUATION A99.. Conservative Dissipative Eact Conservative Dissipative Eact Conservative Dissipative Eact Conservative Dissipative Eact Conservative Dissipative Eact Conservative Dissipative Eact Fig.. Eample, Peakon interaction: Te comparison of nmerical soltions of energy conservative and energy dissipative DG metods at time T =,, 5, 5, 74, and. as [,a]. Te soltion profile at time t = 5,,, and 8 are plotted in Figre 3. We se te P 4 polynomial wit 3 cells to resolve te peakon. Eample 4 (tree-peakon interaction). In tis eample, we consider te treepeakon interaction of te CH eqation wit te initial condition () = wit c =, c =, c 3 =.8, = 5, = 3, 3 =, and a = 3. We set te comptational domain as [,a]. Te soltion profile at time t =,, 3, 4, and 6 are plotted in Figre 4. Again, we se te P 4 polynomial wit 3 cells to resolve te peakon.

12 A93 HAILIANG LIU AND YULONG XING Fig. 3. Eample 3, two-peakon interaction: Te soltion at time T =5,,, and8. Eample 5 (peakon-antipeakon interaction). In tis eample, we consider te interaction of peakon and antipeakon soltions of CH eqation, wit te initial condition given by () = +. wit c =, c =, = 7.5, =7.5, and a = 3. We set te comptational domain as [,a]. Since tese two peakon soltions ave te same magnitde bt wit opposite signs, te total momentm will remain zero, wic is observed nmerically dring te simlations. Te soltion profile at time t =, 4, 6, 7, 8, and are plotted in Figre 5. We se te P 4 polynomial wit 3 cells to resolve te peakon. Eample 6 (nonpeakon soltion). Te initial data fnction contains a discontinos derivative: () = (3 + ). We set te comptational domain as [ 3, 3]. Te soltion profile at time t =, 5,, and are plotted in Figre 6. In tis eample, we se te P polynomial wit 6 cells for te simlation. 4. Conclsions. We ave developed an invariant preserving DG sceme for te Camassa Holm eqation arising in te contet of sallow water wave teory. Te flly discrete sceme is sown to preserve bot te momentm and total energy of te system. Tese ensre tat te nmerical soltion provides a good long time

13 DG METHOD FOR THE CH EQUATION A Fig. 4. Eample 4, tree-peakon interaction: Te soltion at time T =,,, 3, 4, and6. beavior, ts revealing te desired dispersal wave pattern of te nderlying eqation. Nmerical eamples are presented to illstrate bot te accracy and e ciency in simlating te CH eqation. One ftre work is to investigate ways to frter redce

14 A93 HAILIANG LIU AND YULONG XING Fig. 5. Eample 5, peakon-antipeakon interaction: Te soltion at time T =, 4, 6, 7, 8, and. te pase error in long time simlations, and nderstand te di erent beaviors of invariant preserving DG metods for te CH and DP eqations.

15 DG METHOD FOR THE CH EQUATION A Fig. 6. Eample 6, soltion wit a discontinos derivative: Te soltion at time T =, 5,, and. Acknowledgment. Te second ator is gratefl to Yan X for providing te DG code in te paper [7]. REFERENCES [] D. Appelö and T. Hagstrom, A new discontinos Galerkin formlation for wave eqations in second order form, SIAM J. Nmer. Anal. 53 (5), pp , doi:.37/ [] A. Bressan, G. Cen and Q.-T. ang, Uniqeness of conservative soltions to te Camassa Holm eqation via caracteristics, DiscreteContin.Dyn.Syst.,35(5),pp.5 4. [3] J. L. Bona, H. Cen, O. Karakasian, and Y. Xing, Conservative discontinos Galerkin metods for te Generalized Korteweg de Vries eqation, Mat. Comp., 8 (3), pp [4] R. Camassa and D. D. Holm, An integrable sallow water eqation wit peaked solitons, Pys. Rev. Lett., 7 (993), pp [5] R. Camassa, D. D. Holm, and J. M. Hyman, A new integrable sallow water eqation, Adv. Appl. Mec., 3 (994), pp. 33. [6] C.-S. Co, C.-W. S, and Y. Xing, Optimal energy conserving local discontinos Galerkin metods for second-order wave eqation in eterogeneos media, J.Compt.Pys.,7 (4), pp [7] B. Cockbrn, G. Karniadakis, and C.-W. S, Te development of discontinos Galerkin metods, DiscontinosGalerkinMetods: Teory,ComptationandApplications,Lect. Notes Compt. Sci. Eng., Springer, Berlin,, pp. 3 5.

16 A934 HAILIANG LIU AND YULONG XING [8] G. Coclite and K. Karlsen, On te well-posedness of te Degasperis Procesi eqation, J. Fnct. Anal., 33 (6), pp [9] G. Coclite and K. Karlsen, On te niqeness of discontinos soltions to te Degasperis Procesi eqation, J.Di erentialeqations,34(7),pp.4 6. [] A. Constantin and D. Lannes, Te ydrodynamical relevance of te Camassa Holm and Degasperis Procesi eqations, Arc.Ration.Mec.Anal.,9(9),pp [] A. Degasperis, D. Holm, and A. Hone, A new integrable eqation wit peakon soltions, Teoret. and Mat. Pys., 33 (), pp [] K. Dekker and J. G. Verwer, Stability of Rnge-Ktta Metods for sti nonlinear di erential eqations, Nort Holland,Amsterdam,984. [3] B. Fcssteiner and A. S. Fokas, Symplectic strctres, teir bäcklnd transformations and ereditary symmetries, Pys.D,4(98),pp [4] M. J. Grote, A. Scneebeli, and D. Scötza, Discontinos Galerkin finite element metod for te wave eqation, SIAM J. Nmer. Anal., 44 (6), pp , doi:.37/56394x. [5] T. G. Ha and H. Li, On traveling wave soltions of te -eqation of dispersive type, J. Mat. Anal. Appl., 4 (5), pp [6] D. D. Holm and M. F. Staley, Wave strctre and nonlinear balances in a family of evoltionary PDEs, SIAM J. Appl. Dyn. Syst., (3), pp , doi:.37/ S4943. [7] R. S. Jonson, Camassa Holm, Korteweg de Vries and related models for water waves, J. Flid Mec., 455 (), pp [8] J. Lenells, Traveling wave soltions of te Degasperis Procesi eqation, J. Mat. Anal. Appl., 36 (5), pp [9] H. Li, Wave breaking in a class of nonlocal dispersive wave eqations, J. NonlinearMat. Pys., 3 (6), pp [] H. Li, On discreteness of te Hopf eqation, ActaMat. Appl. Sin. Engl. Ser., 4(8), pp [] H. Li, Y.-Q. Hang, and N.-Y. Yi, A direct discontinos Galerkin metod for te Degasperis Procesi eqation, MetodsAppl.Anal.,(4),pp [] H. Li and T. Pendleton, On invariant-preserving finite di erence scemes for te Camassa Holm eqation, Commn.Compt.Pys.,9(6),pp.5 4. [3] H. Li and. Yin, Global reglarity, and wave breaking penomena in a class of nonlocal dispersive eqations, Contemp.Mat.56,AMS,Providence,RI,,pp [4] H. Lndmark, Formation and dynamics of sock waves in te Degasperis-Procesi eqation, J. Nonlinear Sci., 7 (7), pp [5] B. Riviere and M. F. Weeler, Discontinos finite element metods for acostic and elastic wave problems, Contemp.Mat.39,AMS,Providence,RI,3,pp.7 8. [6] Y. Xing, C.-S. Co, and C.-W. S, Energy conserving local discontinos Galerkin metods for wave propagation problems, InverseProbl.Imaging,7(3),pp [7] Y. X and C.-W. S, A local discontinos Galerkin metod for te Camassa Holm eqation, SIAM J. Nmer. Anal., 46 (8), pp. 998, doi:.37/ [8] Y. X and C.-W. S, Local discontinos Galerkin metods for ig-order time-dependent partial di erential eqations, Commn.Compt.Pys.,7(),pp. 46. [9] N.-Y. Yi, Y.-Q. Hang, and H. Li, A direct discontinos Galerkin metod for te generalized Korteweg de Vries eqation: Energy conservation and bondary e ect, J.Compt. Pys., 4 (3), pp