Hf(x) := 1 π p.v. f(y)

Transcription

1 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION arxiv: v3 [mah.na] 22 May 214 VASSILIOS A. DOUGALIS, ANGEL DURAN, AND DIMITRIOS MITSOTAKIS Absrac. In his paper we consider he Benjamin eqaion, a parial differenial eqaion ha models one-way propagaion of long inernal waves of small amplide along he inerface of wo flid layers nder he effecs of graviy and srface ension. We solve he periodic iniial-vale problem for he Benjamin eqaion nmerically by a new flly discree hybrid finie-elemen / specral scheme, which we firs validae by pinning down is accracy and sabiliy properies. Afer esing he evolion properies of he scheme in a sdy of propagaion of single - and mli-plse soliary waves of he Benjamin eqaion, we se i in an eploraory mode o illminae phenomena sch as overaking collisions of soliary waves, and he sabiliy of single-, mli-plse and depression soliary waves. 1. Inrodcion In his paper we will consider he Benjamin eqaion +α +β γh δ =, (1.1) where = (,), R,,α,β,γ,δ are posiive consans, and H denoes he Hilber ransform defined on he real line as Hf() := 1 π p.v. f(y) y dy or hrogh is Forier ransform as Ĥf(k) = isign(k) f(k),k R. The Benjamin eqaion, cf. [5, 6, 2], is a model for inernal waves propagaing nder he effec of graviy and srface ension in he posiive -direcion along he inerface of a wo-dimensional sysem of wo homogeneos layers of incompressible, inviscid flids consising a res of a hin layer of flid 1 of deph d 1 and densiy ρ 1 lying above a layer of flid 2 of very large deph d 2 d 1 and densiy ρ 2 > ρ 1. The pper layer is bonded above by a horizonal rigid lid and he lower layer is bonded below by an impermeable horizonal boom, as in Figre 1. I is frher assmed ha he following physical regime of ineres is o be modelled: Le a be a ypical amplide and λ a ypical wavelengh of he inerfacial wave. The parameers ǫ = a/d 1 and µ = d 2 1 /λ2 are assmed o be small and saisfy µ ǫ 2 1; i is also assmed ha capillariy effecs along he inerface are no negligible. Under hese assmpions (1.1) was derived in [5] from he wodimensional, wo-layer Eler eqaions in he presence of inerface srface ension 21 Mahemaics Sbjec Classificaion. 76B15 (primary), 65M6, 65M7 (secondary). Key words and phrases. Benjamin eqaion, Soliary waves, Hybrid Finie Elemen-Specral mehod. 1

2 2 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS ρ 1 z = d 1 z = ρ 2 z = d 2 Figre 1. Inerfacial graviy-capillary waves by dispersion relaion argmens. The variables in (1.1) are nondimensional and scaled, and he coefficiens are given by ρ2 ρ 1 α =, β = 3 ρ 1 2 αǫ, γ = 1 2 α µ ρ 2 αt, δ = ρ 1 2gλ 2 (ρ 2 ρ 1 ), where T is he inerfacial srface ension and g he acceleraion of graviy. The variables and are proporional o disance along he channel and ime, respecively, and (, ) denoes he downward verical displacemen of he inerface from is level of res a (,). The inerfacial srface ension T is assmed o be mch larger han g(ρ 2 ρ 1 )d 2 1. (For a frher discssion of he physical regime of validiy of (1.1) cf. [2].) Noe ha if he parameer δ is aken eqal o zero, (1.1) redces o he Benjamin-Ono (BO) eqaion, [4, 22], while, if we p γ = we obain he KdV eqaion wih negaive dispersion coefficien. I is well known, cf. [5], ha sfficienly smooh solions of (1.1) ha vanish siably a infiniy preserve he fncionals m() = I() = 1 2 E() = d, (1.2) 2 d, (1.3) ( β γh + 1 ) 2 δ2 d. (1.4) Globalwell-posednessin L 2 forhe Cachyproblemandalsoforheperiodic iniialvale problem for (1.1) was esablished in [19]. In his paper we will sdy (1.1) nmerically, paying pariclar aenion o properies of is soliary-wave solions. These are ravelling-wave solions of he form (,) = ϕ( c s ),c s >, sch ha ϕ and is derivaives end o zero as ξ = c s approaches ±. Sbsiing his epression in (1.1) and inegraing once we obain (α c s )ϕ+ β 2 ϕ2 γhϕ δϕ =, (1.5) where = d/dξ, and he operaor H is defined by H := H, i. e. by Ĥf(k) = k f(k),k R. We will assme ha α c s >.

3 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION 3 If we perform he change of variables ϕ(ξ) = 2(α c s) ψ(z), z = β α cs ξ, δ in (1.5), we see ha he soliary-wave profile ψ(z) saisfies he ordinary differenial eqaion (ode) where ψ 2 γhψ ψ zz ψ 2 =, z R, (1.6) γ = γ 2 δ(α c s ). (1.7) This change of variables and he resling eqaion (1.6) was sed in [5, 6], and [2]. (In hese references γ is denoed by γ.) In his papers Benjamin showed, sing degree heory, ha for each γ [,1), here eiss a solion ψ of (1.6) which is an even fncion of z wih ψ() = ma z R ψ(z) >. He also arged by formal asympoics ha for each γ [,1) here is a bonded inerval cenered a z =, in which ψ oscillaes (wih he nmber of oscillaions increasing as γ approaches 1), while oside his inerval he conclded in [6] ha ψ decays like 1/z 2. In addiion, in he same paper he olined an orbial sabiliy heory for hese soliary waves for small γ. In [2] a complee heory of eisence and orbial sabiliy of he soliary waves for small γ was presened, based on he implici fncion heorem, perrbaion heory of operaors, and he fac ha γ = corresponds o soliary waves of he KdV eqaion. Frher isses of eisence and rigoros asympoics of he soliary waves of (1.1) and relaed eqaions were eplored in [12]. In [3] concenraion compacness argmens were sed o esablish eisence and a weaker version of sabiliy of he soliary waves of (1.1) for < γ < 1. In his paper we will employ he soliary-wave eqaion in he form (1.5). As a resl, normally he soliary waves will have negaive maimm ecrsions from heir level of res. Since eplici formlas for he soliary waves of he Benjamin eqaion are no known (ecep when one of γ or δ is se eqal o zero), one ms resor o approimae echniqes for heir consrcion. The presence of he nonlocal erms in (1.1) and (1.5), which have a handy Forier represenaion in he periodic case as well, narally sggess sing specral-ype mehods for approimaing heir solions. The preceding discssion of he Benjamin eqaion applies o is associaed Cachy problem on R. Solving i nmerically reqires posing i on a finie -inerval [ L, L] wih, say, periodic bondary condiions, assming 2L-periodic iniial daa. In case soliary waves, heir generaion and ineracions, are he focs of ineres, one shold ake ino accon ha hey decay qadraically. Conseqenly, he inerval [ L, L] shold be aken sfficienly large in some eperimens o ensre ha he nmerical solion in he emporal range of ineres remains sfficienly small a he endpoins so ha he simlaions give valid approimaions of he solions of he Cachy problem. In [2] he eqaion (1.6) was discreized in space by a psedospecral echniqe andhereslingnonlinearsysemofeqaionsforheforiercoefficiensofψ = ψ γ for a desired vale of γ (,1) was solved by an incremenal coninaion mehod. This enailed defining a homoopic pah γ = < γ 1 <... < γ M = γ, saring from he known profile of a soliary wave ψ γ of he KdV eqaion wih a given speed c s, and comping ψ γj+1, given ψ γj, by Newon s mehod. Wih his echniqe

4 4 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS he ahors of [2] were able o consrc approimae solions of (1.6) ha were even fncions wih a posiive absole maimm a z =. As γ approached 1 he oscillaing ails of he soliary wave became more prominen and he maimm vale of he wave decreased. I was fond ha he lengh of he inervals beween consecive zeros of he oscillaing ails was qie close o he vale prediced by he asympoic analysis of [6]. In [17] he ahors solved nmerically he periodic iniial-vale problem for he Benjamin eqaion sing a psedospecral (collocaion) mehod in space copled wih a second-order ime-sepping procedre. They confirmed ha resolion of siable general iniial profiles ino a nmber of soliary waves pls a dispersive ail (a phenomenon ha has been observed in oher nonlinear dispersive wave eqaions) also occrs in he case of he Benjamin eqaion. They specifically sdied he resolion of iniial Gassian profiles ino soliary waves conrasing i wih he analogos resolion observed in he case of wo BO-ype eqaions. In some cases hey observed, in addiion o deached soliary waves, he emergence of clsers (pairs, riples, ec.) of orbiing soliary waves ha ineraced among hemselves. They conjecred ha hese srcres wold evenally separae ino disinc soliary waves. They also consrced approimae soliary waves, sing he resolion propery, by rncaing and ieraively cleaning a separaed soliary wave as has been freqenly done in nmerical sdies of oher nonlinear dispersive wave eqaions. (Of corse in his manner one does no have in general a priori knowledge of he speed c s or he vale of γ of he emerging soliary wave.) They sed wo sch approimae soliary waves of differen speeds o sdy heir overaking collision and observed ha he ineracion was no elasic, a fac indicaing ha he Benjamin eqaion is no inegrable. In [9], he ahors considered soliary waves of he Benjamin eqaion and compared hem o soliary waves of he fll Eler eqaions for inerfacial flows in he presence of srface ension when he parameers of he problem are close o he Benjamin eqaion regime of validiy and also farher from i. The nmerical scheme hey sed for approimaing soliary waves of he Benjamin eqaion was based on a hybrid spaial discreizaion ha employed forh-order finie differences on a niform grid for he derivaives, and he discree Forier ransform for he nonlocal erm. The resling nonlinear sysem of eqaions was solved again by a coninaion-newon echniqe. The emporal discreizaion of he periodic iniial-vale problem for he Benjamin eqaion was effeced by an eplici predicor-correcor scheme. They idenified anoher branch of soliary wave solions of he Benjamin eqaion, he depression soliary waves (resembling analogos solions of he Eler eqaions), and esed heir sabiliy by sing hem as iniial vales in heir flly discree scheme for he ime-dependen eqaion. They observed ha he iniial profile propagaed wiho change for some ime, gradally developed an insabiliy de o he perrbaive effec of he nmerical scheme, and resolved iself ino wo plses resembling sal ( elevaion ) soliary waves of he Benjamin eqaion pls small-amplide dispersive oscillaions. (A linearized sabiliy analysis, also performed in [9], yields ha he depression soliary waves are linearly nsable.) In a recen paper [15], we made a sdy of several incremenal coninaion echniqes for approimaing soliary waves of he Benjamin eqaion ha saisfy

5 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION 5 (1.5). (The vales of α,β,δ and c s were fied, and γ was sed as coninaion parameer.) A sandard psedospecral (collocaion) mehod yielded he nderlying discree nonlinear sysem. We fond ha Newon s mehod, combined wih a siably precondiioned conjgae gradien echniqe for solving he aendan linear sysem a each Newon ieraion, was he generally mos efficien echniqe of implemening he incremenal sep and prodced very accrae approimaions of he soliary waves for γ < 1. Wih his mehod we also comped oher branches of solions of (1.5), namely mli-plse soliary waves, by saring he homoopy pah from linear combinaions of soliary waves of he KdV eqaion. We verified he accracy of hese profiles as ravelling waves of he Benjamin eqaion by sing hem as iniial vales in a fll discreizaion of he periodic iniial-vale problem for (1.1) and inegraing forward in ime. The solver combined he psedospecral spaial discreizaion wih he hird-order accrae wo-sage DIRK ime-sepping echniqe, modified o preserve discree analogs of he invarians (1.2) and (1.3). I was fond ha several qaniies of ineres, sch as he speed, he amplide and he hird invarian (1.4) of he discree ravelling waves, were preserved o very high accracy, lending confidence in he validiy of his echniqe for comping soliary waves. In he paper a hand we conine or nmerical sdy of he Benjamin eqaion. We consrc and es nmerically a new, efficien ime-sepping mehod based on a specral-finie elemen hybrid spaial discreizaion combined wih a forh-order implici Rnge-Ka scheme for ime-sepping. This mehod is sed o eplore properies of soliary-wave solions of (1.1), sch as heir generaion, ineracion and sabiliy. Mch of nmerical work wih specral-ype mehods for one-dimensional, nonlocal, nonlinear dispersive wave eqaions has been cenered arond he Benjamin- Ono (BO), [4, 22], and he Inermediae Long Wave (ILW) eqaion, [16, 1]. Early compaional work was reviewed in [23]; here we menion only he rigoros convergence resls known o s. In [24] L 2 error esimaes were derived for he sandard Forier-Galerkin semidiscreizaion of he BO and ILW eqaions. If he nmber of Forier modes is 2N +1 and he iniial vale is 2L periodic and belongs o he periodic Sobolev space Hp, r he L 2 -error bonds derived in [24] are of O(N 1 r ). In addiion, he fll discreizaion of he semidiscree sysem of ode s wih he eplici leap-frog scheme is shown in [24] o have an L 2 error bond of O(N 1 r + 2 ) nder he sabiliy resricion ha N 2 C for a sfficienly small consan C; here is he ime sep. For a class of eqaions wih he same nonlocal erms and more general nonlinear erms i was sbseqenly shown in [13] ha he error of he Forier-Galerkin semidiscreizaion is of opimal order O(N 1/2 r ) in Hp 1/2. In he same paper he semidiscree problem was discreized in ime in he manner sggesed in [11], i. e. sing as a basis he leap-frog mehod copled wih implici Crank-Nicolson differencing of he linear dispersive erm. This eplici-implici ime-sepping scheme may be implemened efficienly in Forier space and does no reqire solving linear sysems of eqaions; as shown in [13] i has an error bond of O(N 1/2 r + 2 ) in Hp 1/2 nder he mild sabiliy condiion N 1/2 C for some sfficienly small consan C. In addiion, in [14] he ahors analyze he more efficien specral collocaion mehod (ha was sed in acal compaions in [23] and elsewhere,) for he BO and ILW eqaions, and prove ha he associaed semidiscree problem converges wih an Hp 1/2 error bond of O(N 3/2 r ).

6 6 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS A differen ype of mehod for he BO eqaion was consrced and analyzed in [25]. I consiss of a Crank-Nicolson ime-sepping scheme ha is copled wih a spaial discreizaion in which he nonlinear erm is approimaed by conservaive differencing and he nonlocal erm is discreized in physical space by he midpoin qadrare formla, which is hen inerpreed as a discree convolion and comped by he discree Forier ransform. Since he flly discree scheme is implici, a nonlinear sysem of eqaions has o be solved a each ime sep. This sysem is linearized by a simple ieraive scheme in which he nonlinear erm is lagged backwards in ime and he linear par is rivial o inver in Forier space, as e. g. in [11]. The overall mehod is shown o be of second-order accracy in L 2 in space and ime. In he presen paper he nmerical scheme ha we se is a hybrid finie elemenspecral mehod. We consider he periodic iniial-vale problem for (1.1) and discreize i in space by he Galerkin mehod sing smooh periodic splines of order r 3 on a niform mesh wih meshlengh h. (Cbic splines, i. e. r = 4, are mainly sed in he compaions.) The nonlocal erm is comped sing a specral approimaion as described in Secion 2. Then, he sysem of ode s represening he semidiscree problem is discreized in ime; we se as a base ime-sepping scheme he wo-sage, forh-order accrae, Gass-Legendre implici Rnge-Ka mehod. This scheme has high accracy and good sabiliy properies and has previosly been eensively sed for he emporal discreizaion of siff parial differenial eqaions wih a KdV erm, cf. e. g. [7] and is references. We describe in deail he implemenaion of his flly discree hybrid mehod and make a compaional sdy of is accracy and sabiliy properies when i is applied o he Benjamin and Benjamin-Ono (i. e. when δ is se o zero) eqaions. In addiion, we validae he hybrid scheme by making a deailed comparison of he solions ha i prodces wih hose of a sandard flly discree psedospecral scheme in he case of hree nmerical eperimens involving he propagaion of soliary waves of he Benjamin and Benjamin-Ono eqaions. In Secion 3 we review he coninaion-conjgae gradien-newon echniqe of [15] for generaing single and mli-plse soliary-wave solions (i. e. solions of (1.5)) of he Benjamin eqaion for varios vales of γ wih pariclar aenion o vales close o 1. We se hese nmerical profiles as iniial condiions in nmerical evolion eperimens wih he hybrid scheme and invesigae wih varios merics heir accracy as ravelling wave solions of he Benjamin eqaion. Or conclsion from he nmerical eperimens of Secions 2 and 3 is ha he hybrid scheme yields very accrae and sable approimaions of solions of he Benjamin eqaion, and in pariclar of he soliary waves for vales of γ (,1) ha can be aken qie close o 1. In Secion 4 we make a deailed compaional sdy of overaking ( one-way ) collisions of soliary waves of he Benjamin eqaion and compare he inelasic characer of hese ineracions wih he analogos, clean ineracions in he case of he inegrable BO eqaion. Finally, in Secion 5 we eplore isses of sabiliy and insabiliy of single-and mli-plse soliary waves of he Benjamin eqaion nder small and large perrbaions. Or compaional sdy confirms he sabiliy of he single-plse soliary waves for small and moderae vales of γ b is inconclsive for cases of γ very close o 1. The mli-plse waves appear o be nsable and or eperimens sgges ha afer an iniial orbiing or dancing phase, hey prodce

7 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION 7 separaed soliary waves. This confirms he conjecre of [17] ha was menioned previosly. Finally, we eamine he sabiliy of he depression soliary waves and confirm he resls of [9] regarding heir insabiliy. In smmary, he main conribions of he paper a hand are The consrcion of a novel, highly accrae, sable and efficien hybrid scheme ha combines he accracy of he specral approimaion of he nonlocal erm and he accracy of he spline discreizaion of he res of he erms of he Benjamin eqaion wih an accrae, ncondiionally sable ime sepping procedre which is effecive in approimaing highly siff problems sch as semidiscreizaions of he Benjamin eqaion in he presence of he KdV erm. The validaion of he accracy of he nmerically generaed single- and mli- plse soliary wave solions by showing ha when sed as iniial vales of he hybrid scheme hey prodce highly accrae approimaions o ravelling wave solions of he evolion problem. These approimae soliary waves were comped by a Forier spaial discreizaion of he soliary wave ode (1.5) copled wih a coninaion conjgae gradien- Newon nonlinear sysem solver ha was proposed by he ahors in [15] and can prodce accrae soliary waves for any desired vales of he speed c s > α and γ [,1), avoiding he drawbacks of he ieraive cleaning. The illminaion, by compaional means, of imporan phenomena associaed wih soliary waves of nonlinear dispersive wave eqaions, sch as heir one-way ineracion (overaking collision) and sabiliy properies in he case of he Benjamin eqaion. In he paper, we denoe, for ineger r, by C r p he periodic fncions, on [ L,L] or [,2π] as he case may be, ha belong o C r. The inner prodc for real or comple-valed fncions in L 2 is denoed by (, ) and he associaed norm by. 2. The hybrid specral-finie elemen scheme We consider he periodic iniial-vale problem for he Benjamin eqaion, i. e. for we seek a 2L periodic real fncion = (,) sch ha +α +β γg δ =, [ L,L], >, (2.1) (,) = (), [ L,L] where is a given smooh 2L periodic fncion and α,β,γ,δ posiive consans. The operaor G is he Hilber ransform acing on 2L periodic fncions; for he prposes of his secion i will be represened by is principal-vale inegral form [1] Gf() := 1 L 2L p.v. co L ( π( y) 2L ) f(y)dy, (2.2) where f is 2L periodic. In he seqel we will assme ha he solion of (2.1) is sfficienly smooh. For simpliciy, we assme ha he problem (2.1) has been ransformed ono he spaial inerval [, 2π].

8 8 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS 2.1. The semidiscree hybrid scheme. For ineger r 3 and an even ineger N, le h = 2π/N, j = jh,j =,...,N, and consider he finie dimensional spaces and S N = span { e ik : k Z, N/2 k N/2 1 }, S h = { } φ Cp r 2 : φ P [j, j+1 ] r 1, j N 1. The hybrid specral-finie elemen approimaion h of he solion of (2.1) is a real S h -valed fncion h () of defined by he ode iniial-vale problem ( h,χ)+(α h +β h h,χ)+γ(p N G h,χ )+δ( h,χ ) =, χ S h,, h () = P h, (2.3) where P h, P N are he L 2 projecionsono S h and S N, respecively, given for w L 2 as (P h w,χ) = (w,χ), χ S h and (P N w,φ) = (w,φ), φ S N, where (, ) is he L 2 (,2π) inner prodc. For f L 2, P N f is represened by P N f() = N/2 1 k= N/2 ˆf k e ik, where ˆf k = 1 2π 2π f()e ik d,k Z are he Forier coefficiens of f. Noe ha (Gf) k = isign(k)ˆf k and ha G is anisymmeric in L The flly discree hybrid scheme. We define or flly discree hybrid scheme following he derivaion of he analogos scheme of [7] in he case of he generalized KdV eqaion. (This scheme was also sed in [8].) Denoing again by (, ) he L 2 (,2π) innerprodc, wedefine, foreach [,T], he mapf : S h S h by he eqaion (F( h ),χ) = [(α h +β h h,χ)+γ(p N G h,χ )+δ( h,χ )], χ S h. Then, he iniial-vale problem (2.3) may be wrien as h = F( h ), T, h () = P h. (2.4) In addiion o F we define he maps B : S h S h S h, Θ 1 : S h S h and Θ 2 : S h S h ha saisfy for v,w S h and for all χ S h (B(v,w),χ) = 1 2 (βvw,χ ) = 1 2 (β(vw),χ), (Θ 1 v,χ) = (αv δv,χ ), and (Θ 2 v,χ) = (γp N Gv,χ ). If we p F(v,w) := B(v,w)+Θ 1 v +Θ 2 v, we see ha F(v) := F(v,v) = B(v)+Θ 1 v +Θ 2 v, where B(v) = B(v,v). The iniial-vale problem (2.4) is siff. I is discreized in he emporal variable by he 2-sage Gass-Legendre implici Rnge-Ka mehod,

9 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION 9 which is forh-order accrae and has good nonlinear sabiliy properies. I corresponds o he Bcher able a 11 a 12 τ 1 a 21 a 22 τ 2 b 1 b 2 = The flly discree scheme is now specified more precisely. Le n = nk, n =,1,...,M, where T = Mk. We seek U n approimaing h ( n ), and U n,i in S h, i = 1,2, as solions of he sysem of nonlinear eqaions U n,i = U n +k 2 a ij F(U n,j ), i = 1,2, n M 1, (2.5) j=1 and se 2 U n+1 = U n +k b j F(U n,j ), n M 1, (2.6) j=1 where U = h (). A each ime sep we solve he nonlinear sysem (2.5) sing Newon s mehod as follows. Given n, le U n,i S h, i = 1,2 be an accrae enogh (see below) iniial gess for U n,i, he solion of (2.5). Then he ieraes of Newon s mehod (called he oer ieraes for reasons ha will become clear presenly) U n,i j, j = 1,2,... (U n,i j approimaes U n,i ) saisfy he 2 2 block linear sysem in S h S h, [ ] [ ] I +ka 11 J(U n,1 j ) ka 12 J(U n,2 j ) U n,1 [ ] ka 21 J(U n,1 j ) I +ka 22 J(U n,2 j+1 U n j ) U n,2 = U n (2.7) j+1 [ ] [ ] a11 a k 12 B(U n,1 j ), a 21 a 22 where, for ψ,φ in S h B(U n,2 j ) J(φ)ψ = J 1 (φ)ψ +J 2 (φ)ψ, and J 1 (φ)ψ = 2B(φ,ψ) Θ 1 ψ, J 2 (φ)ψ = Θ 2 ψ. The eqaions (2.7) represen a 2N 2N linear sysem for he coefficiens of he new Newon ieraes U n,i j+1, i = 1,2, for each j, wih respec o a basis of S h. The wo operaor eqaions in (2.7) are ncopled as follows: We evalae he enries of he mari in he lef-hand side of (2.7) a a poin U S h, defined by U = 1 2 (Un,1 +U n,2 ), (2.8)

10 1 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS (which makes he operaorsin he enriesof his mari independen ofj and allows hem o comme wih each oher). We may hen wrie (2.7) eqivalenly as [ ] [ ] I +ka11 J 1 (U ) ka 12 J 1 (U ) U n,1 ka 21 J 1 (U ) I +ka 22 J 1 (U j+1 = ) [ ] [ ] [ U n a11 a U n k 12 a 21 a 22 [ ] [ a11 a +k 12 a 21 a 22 B(U n,1 j ) B(U n,2 j ) ] U n,2 j+1 J 1 (U ) J(U n,1 j ) J 1 (U ) J(U n,2 j ) ] [ U n,1 j+1 U n,2 j+1 ], (2.9) for j, a form ha immediaely sggess an ieraive scheme for approimaing U n,i j+1, i = 1,2. This scheme generaes inner ieraes denoed by Un,i,l j+1 for given n,i,j, and l =,1,2,...(U n,i,l j+1 approimaesun,i j+1 ) ha arefond recrsivelyfrom he eqaions [ ] [ ] [ ] I +ka11 J 1 (U ) ka 12 J 1 (U ) U n,1,l+1 ka 21 J 1 (U ) I +ka 22 J 1 (U j+1 r n,1,l ) U n,2,l+1 j+1 = j+1 r n,2,l, (2.1) j+1 for l, where r n,i,l j+1 = Un k 2 m=1 a im B(U n,m j )+k 2 m=1 a im (J 1 (U ) J(U n,m j ))U n,m,l j+1. The linear sysem (2.1) can be solved efficienly as follows: Since a 12 a 21 <, i is possible, pon scaling he mari on he lef-hand side of he sysem by a diagonal similariy ransformaion, o wrie i as [ I kj 1(U ) kj 1 (U )/4 3 kj 1 (U )/4 3 I kj 1(U ) ] [ U n,1,l+1 j+1 µu n,2,l+1 j+1 ] = [ r n,1,l j+1 µr n,2,l j+1 ], (2.11) where µ = 2 3. The sysem (2.11) is eqivalen o he single comple N N sysem (I +kζj 1 (U ))Z = R, (2.12) where ζ = 1 4 +i/4 3, and where Z and R are comple-valed fncions wih real and imaginary pars in S h which depend pon n, l and j and are given by Z = U n,1,l+1 j+1 +iµu n,2,l+1 j+1, R = r n,1,l+1 j+1 +iµr n,2,l+1 j+1. (2.13) In pracice only a finie nmber of oer and inner ieraes are comped a each ime sep. Specifically, for i = 1,2, n, we compe approimaions o he oer ieraes U n,i j for j = 1,...,J o, for some small posiive ineger J o. For each j, j J o 1, U n,i j+1 is approimaed by he las inner ierae Un,i,Jinn j+1 of he seqence of inner ieraes U n,i,l j+1, l J inn ha saisfy linear sysems of he form (2.12). J inn and J o are sch ha ( 2 k=1 U n,k,l+1 j+1 U n,k,l j+1 2 l 2 ) 1/2 ε,

11 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION 11 and ( 2 k=1 U n,k j+1 Un,k j 2 l 2 ) 1/2 ε, where v l2 denoes he Eclidean norm of he coefficiens of v S h wih respec o is basis, and ε is sally aken o be 1 1. Given U n, he reqired saring vales U n,i for he oer (Newon) ieraion are comped by erapolaion from previos vales as U n,i = α,i U n +α 1,i U n 1 +α 2,i U n 2 +α 3,i U n 3, (2.14) for i = 1,2, where he coefficiens α j,i are sch ha U n,i is he vale a = n,i of he Lagrange inerpolaing polynomial of degree a mos 3 in ha inerpolaes o he daa U n j a he for poins n j, j 3. (If n 2, we se he same linear combinaion, ping U j = U if j <.) The inegrals involving he local erms are comped in general sing he 5- poin Gass-Legendre qadrare rle in each spaial inerval. The inner prodc (P N G h,χ ) involving he nonlocal erm is comped as he inner prodc (I N G h,χ ) where he Forier inerpolan I N is defined as I N v() = N/2 1 k= N/2 ˆv k e ik, (2.15) where by ˆv k we denoe he discree Forier coefficiens of v, comped by he Fas Forier Transform. The inner prodc (, ) is approimaed by he rapezoidal qadrare rle, which is very accrae for periodic fncions. In he seqel, we shall se he flly discree scheme described above wih he C 2 cbic splines (r = 4) as he finie elemen sbspace S h. We shall refer o his mehod as he hybrid scheme/mehod. We checked nmerically he orders of convergence of he hybrid scheme as follows. De o lack of analyical formlas for solions of he Benjamin eqaion we considered he nonhomogeneos eqaion + +G = f(,), (,) [ 1,1] [,T], (2.16) wih periodic bondary condiions and f(,) = e (sin(π)+ π 2 e sin(2π)+ ) ) (π 2 π3 cos(π). 2 The specific eqaion has a solion (,) = e sin(π). We solved i nmerically p o T = 1 and we comped he discree maimm erroron he qadrare nodes and he normalized L 2 error defined as e h (, n ) / e h (,), where e h = U. The nmerical mehod appears o converge wih an opimal rae in space (r = 4) b wih a sbopimal rae eqal o hree in ime. Tables 1 and 2 show he nmerical spaial and emporal raes of convergence of he error for his eperimen comped in he discree maimm norm and he normalized L 2 norm a = T = 1. Here N is he nmber of spaial inervals and M = T/k. We observe ha he spaial rae is pracically opimal (for) and ha he emporal rae approimaes he vale p = 3 as N,M increase. (For his eperimen, wih he olerance se a ǫ = 1 1, he nmber of Newon ieraions J o came o o be always one and J inn varied in general beween one and for

12 12 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS N M L Error Rae L 2 Error Rae Table 1. Spaial raes of convergence (hybrid scheme) N M L Error Rae L 2 Error Rae Table 2. Temporal raes of convergence (hybrid scheme) provided k and h were sfficienly small.) The heoreical order of accracy of he wo-sage Gass-Legendre RK mehod is of corse eqal o for and his vale is observed eperimenally for he KdV eqaion, i. e. when he nonlocal erm G is no presen, see e. g. ([7], Table 3). In or case, he loss of one order of emporal accracy is apparenly cased by he presence of he nonlocal erm: Observe ha in he Jacobian J 1 (U ) in he mari of operaors in he lef-hand side of (2.9) we did no inclde he par of he Jacobian J 2 = Θ 2 corresponding o he nonlocal erm b ransferred i o he righ-hand side, in order o reain sparsiy in he operaors on he lef when a basis of small sppor is chosen for S h. This efficiency consideraion renders he scheme eplici wih respec o he nonlocal erm and linearly implici wih respec o he res of he erms in he eqaion, and cases he loss of emporal accracy by one order. We did no deec any need for a sabiliy bond on k/h for hese compaions. (Vales as high as k/h = 8 were ried.) Of corse accracyis redced as k increases and so in he nmerical eperimens of secions 3-5 k/h was aken mch smaller. In he seqel, we shall also on occasion compe solions of he Benjamin-Ono (BO)eqaion, mainlyinorderoesornmericalschemes. (BOisagoodesing grond for or prposes since i has soliary-wave solions ha are known in closed form and are no rivial o simlae on a finie inerval as hey decay like O( 2 ) as. In addiion, heir ineracions are clean de o he inegrabiliy of he BO.) For his reason, we briefly repor on he performance of he hybrid mehod in he case of he BO. I is easy o verify, o begin wih, ha he spaial rae of convergence is again eqal o 4. However, we fond ha he eplici way ha he Newon solver reas he nonlocal erm cases he hybrid mehod o converge nder a sabiliy condiion of he form k = αh 2. (In he case of he eample (2.16) wih no KdV erm, α =.6 was sfficien.) In he case of he Benjamin-Ono eqaion, de o he resricive sabiliy condiion k = αh 2, if we ake a fied nmber N of spaial inervals, we observe ha

13 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION 13 he errors cease o decrease a a cerain poin becase he emporal error becomes mch smaller han he spaial error. I is hs no easy o compe he asympoic rae of he emporal error. To accomplish his we did he following: For a fied vale of h, we solved he problem in he domain [ 15,15] wih he hybrid mehod p o T = 1 for varios vales of k. We chose h =.5 (i.e. N = 6) o ensre ha he spaial errors will be larger han he emporal errors. We also chose a reference vale of k = k ref = 1 4 (M = 1) and we comped he solion U ref. We hen chose vales of k larger han k ref b small enogh so as o saisfy he sabiliy condiion and comped U k and he normalized errors E (T) = U ref(t) U k (T). () I rns o ha for small vales of k, which are neverheless considerably larger han k ref, he epeced emporal rae of convergence is visible becase sbracing U ref (T) from U k (T), essenially cancels he spaial error of he laer approimaion. The resls of hese compaions are presened in Table 3. N M L Error Rae L 2 Error Rae Table 3. Temporal raes of convergence for BO (hybrid scheme) 2.3. A flly discree psedospecral scheme. In addiion o he hybrid mehod, we shall se for checking prposes a specral mehod. For coninos 2π periodic comple-valed fncions,v we le (,v) N := 2π N 1 N j= ( j)v( j ). We consider he following semidiscree Forier-collocaion (psedospecral) scheme, cf. [2, 1], ha approimaes he solion of (2.1) on [,2π] by N S N defined by he eqaions ( N +[α N +(β/2)( N ) 2 γg N δ N ],χ) N =, χ S N,, N (,) = I N, (2.17) where I N is given by (2.15). By choosing χ = e ik for k = N/2,...,N/2 1, we obain he following sysem of ode s for he Forier coefficiens û k of N for k = N/2,...,N/2 1: where d dûk + β 2 ik(û û) k +ω(k)û k =,, û k () = Î N k, (2.18) ω(k) = αik γi k k +δik 3. Mliplying he ode s by e ω(k) and seing Ûk = e ω(k) û k we may wrie hem as d + β ] dûk 2 ikeω(k)[ (e ω(k) Û) (e ω(k) Û) =. (2.19) k

14 14 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS To compe he convolion we se he formla F([F 1 (e ω(k) Û)] 2 ), where F is he discree Forier ransform. The resling ode sysem is discreized by he eplici classical forh-order Rnge-Ka mehod in ime. Hence, his flly discree scheme belongs o he class of he so-called inegraing facor schemes, [11, 21, 18], having improved sabiliy properies, as hey aemp o redce siffness. (The las-qoed paper has a sefl review of relaed schemes.) We verified he forh order of emporal accracy of his scheme by comping is errors in he case of he nonhomogeneos problem (2.16) a = 1 for N = 1 and an increasing nmber of ime seps. The resls are shown in Table 4. (The nmerical emporal rae in he case of he analogos nmerical eperimens for he BO eqaion was also fond o be 4.) N M L Error Rae L 2 Error Rae Table 4. Temporal raes of convergence (specral scheme). We shall henceforh refer o his flly discree psedospecral scheme as he specral mehod Validaion of he hybrid mehod. We now presen he resls of some nmerical ess ha we performed wih boh schemes in order o validae frher he hybrid mehod and compare is resls wih hose of he specral scheme. In or firs eperimen we simlae he propagaion of a periodic ravellingwave solion of he Benjamin-Ono eqaion ha was sed in [25]. This solion resembles a soliary wave and is given by he formla (,) = 2c s A A 2 cos(c s A( c s )), (2.2) where A = π c sl. This is a 2L periodic solion of he BO wih coefficiens α = δ =, β = γ = 1 in (1.1). We approimaed i by he specral mehod wih N = 124,k =.2 and he hybrid mehod in wo rns wih N = 256 and k =.1 and wih N = 124 and k = 5 1 4, respecively, on he inerval [ L,L] wih L = 15 and c s =.25 for 1, sing (2.2) a = as iniial condiion. The nmerical solion is shown in Figre 2 a =,1 and 1. (All hree nmerical profiles coincided wihin graph hickness.) In his eample, he errors of he specral mehod were all in he range 1 9 o In he wo rns of he hybrid scheme, he normalized L 2 error, defined as ma n ( n ) U n U, was of O(1 7 ) for N = 256 and of O(1 11 ) for N = 124. In boh cases, he L 2 norm of he nmerical solion was eqal o while he Hamilonian (invarian E() given by (1.4)) was eqal o (Boh were preserved for 1 p o he welve significan digis shown.) In addiion, for he hybrid scheme we comped for each n several oher ypes of errors ha are relevan in assessing he accracy of approimaion of soliary-ype waves, cf. [7, 8]. These were: (i) The (normalized) amplide error AE( n ) =

15 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION = =1 = ma Un ( ) ma Figre 2. Nmerical evolion of he periodic-ravelling wave solion (2.2) of he Benjamin-Ono eqaion., where ma is he maimm vale of he eac solion and is he poin where he approimae solion U n achieves is maimm, fond by applying Newon s mehod o compe he roo of he eqaion d d Un () = ha corresponds o he maimm of U n. (ii) The L 2 (normalized) shape error defined as SE( n ) = inf τ U n (,τ) /, comped as SE( n ) = ξ(τ ), where τ is he poin near n (fond by Newon s mehod) where d dτ (ξ2 ) =, wih ξ(τ) = U n (,τ) /. (iii) The associaed phase error PE( n ) = τ n. Figre 3 shows hese errors as fncions of n p o T = 1, for N = 256 and N = 124. The speed c s =.25 of he ravelling wave was preserved for N = 256 o 6 digis p o = 5 and o 5 digis p o = 1, while for N = 124 p o a leas 7 digis p o = 1. In a second validaion eperimen we comped he evolion of a soliary wave for he Benjamin eqaion (2.1) wih γ =.5 (all oher coefficiens being eqal o one) wih L = 128 p o T = 1. The iniial soliary-wave profile was generaed wih high accracy by nmerical coninaion wih he CGN mehod as eplained in [15] and in Secion 3 of he presen paper. We solved he problem by he hybrid and he specral schemes. Table 5 presens he resls of wo rns wih comparable errors for his problem. The specral mehod is faser by a facor of wo b he Hybrid Specral N k L 2 error H 1 error SE PE H cp ime (sec) 59 3 Table 5. Errors a T = 1 and parameers for he hybrid and specral mehods. Soliary wave, Benjamin eqaion, γ =.5 hybrid mehod conserves he Hamilonian H = I + E p o 1 digis, for more

16 16 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS N= N=124 AE() SE() PE() 1 1 Figre 3. Amplide (AE( n )), Shape (SE( n )) and Phase (PE( n )) errors of he hybrid scheme for N = 256,124, approimaing he solion (2.2) of he BO eqaion han in he case of he specral mehod. In he able he L 2 and shape errors are normalized as eplained earlier. The (normalized) H 1 error, defined analogosly, is a sefl error meric for oscillaory profiles sch as he soliary waves of he Benjamin eqaion. In or hird eperimen we solved he Benjamin eqaion in he form + + G + = for [ 3,3] p o T = 1 sing as iniial condiion he Gassian (,) = 2e (/4)2. As epeced, [17], he iniial profile resolves iself ino a series of soliary waves. As Figre 4 shows, by T = 1 hree soliary waves have appeared, followed by a dispersive ail. We sed he solion obained by he specral scheme wih N = 6,k =.1 as he benchmark and recomped he solion wih he hybrid scheme for varios vales of he discreizaion parameers h and k saring from h =.1,k =.1 and redcing h and/or k. Some of he profiles prodced by he hybrid rns are shown in Figre 4; hey all coincide wihin graph hickness wih he specral solion. (I shold be menioned ha he specral scheme wih k = 6/N blew p and needed k = O((6/N) 2 ) for sabiliy.)

17 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION 17 3 Hybrid, d=.1, d=.1 Hybrid, d=.1, d=.1 Hybrid, d=.5, d=.5 Psedospecral, d=.1, d= Figre 4. Resolion of he Gassian 2e (/4)2 ino soliary waves. Benjamin eqaion, T=1. The profile on he boom is a magnificaion of ha on he op. 3. Generaion and propagaion of soliary waves In his secion we firs review he nmerical echniqe ha we sed o generae soliary-wave solions of he Benjamin eqaion. These soliary-wave profiles were aken as iniial vales for he hybrid ime-sepping mehod and inegraed forward in ime. We presen in some deail he emporal evolion of varios error merics siable for assessing he accracy of hese nmerically generaed ravelling waves. As was already menioned in he Inrodcion, he soliary waves of he Benjamin eqaion are ravelling-wave solions of (1.1) of he form (,) = ϕ( c s ),c s >, sch ha ϕ and is derivaives end o zero as ξ = c s approaches ±. Conseqenly, ϕ saisfies he eqaion (1.5), from which, aking Forier ransforms, we obain ( c s +α γ k +δk 2 ) ϕ+ β 2 ϕ 2 =, k R, where ϕ(k) is he Forier ransform of ϕ. If we discreize his eqaion assming periodic bondary condiions on [ L, L] and sing he discree Forier ransform o compe he convolion as in secion 2.3, we obain he N N nonlinear sysem of eqaions ( c s +α γ k +δk 2 ) ϕ N k + β ( ϕn ϕ 2 )k N =, k = N 2,..., N 1, (3.1) 2

18 18 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS where ϕ N is he approimaion of ϕ in S N and ϕ N k denoes is kh Forier coefficien. To solve (3.1) we se an incremenal coninaion echniqe wih respec o he parameer γ, following e. g. [2]. For a fied se of consans α,β,δ,c s in (3.1) we consider a homoopic pah γ = < γ 1 <... < γ M = γ and solve (3.1) sccessively for γ,γ 1,...,γ M wih an ieraive nonlinear solver, sing for each j he nmerical solion for γ = γ j 1 as an iniial gess in solving for γ = γ j. (The saring vale γ = of he pah corresponds o he KdV eqaion for which eac soliary-wave solions are available.) The incremenal coninaion echniqe has he added advanage ha i prodces a series of soliary waves for varying vales of γ wih a fied speed c s. The nonlinear sysem solver ha we sed o generae he solion of (3.1) for each γ j was Newon s mehod, wherein he aendan linear sysems were solved by an inner ieraion performed by he precondiioned conjgae gradien echniqe. The resling ieraive scheme, called CGN in he seqel, was described in deail in [15], where i was also compared wih several oher nonlinear solvers and fond o be more efficien, wih respec o a variey of merics, for approimaing solions of (3.1). We refer he reader o [15] for he implemenaion of CGN; le s js menion ha for he compaions in he presen paper he Newon ieraion was erminaed when he qaniy ϕ N [ν] ϕn [ν 1] / ϕn [ν] became less han (Here ϕ N [ν] is he ν-h Newon ierae approimaing ϕn ). The precondiioned conjgaegradien inner ieraion was erminaed when R (i) M / R () M became less han 1 2. Here R (i) is he residal defined in he sandard way in he conjgaegradien algorihm, and he norm M is he weighed L 2 norm (,M 1 ) 1/2, where M = ci is he precondiioning operaor ha we sed; is acion in Forier variables is c + k 2 and he vale c =.275 was fond o be opimal in compaions. The nmber of CG inner ieraions needed o reach he hreshold defined above varied beween 3 and 1 ypically..2.2 (a) γ = (b) γ = (c) γ = (d) γ = Figre 5. Soliary waves of he Benjamin eqaion for varios vales of γ, c s =.45.

19 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION (a) γ = (b) γ = (c) γ = (d) γ = Figre 6. Soliary waves of he Benjamin eqaion for varios vales of γ, c s =.75. Using his algorihm we prodced soliary waves of he Benjamin eqaion in [ 256,256] wih N = 496 sing γ j = j γ,j = 1,...,99, wih γ =.1 and an eac soliary wave of he KdV eqaion a γ =. In all compaions we ook α = β = δ = 1. Figre 5 shows he comped profiles of he soliary waves for c s =.45 and γ =,1,.5,.9,.99, while Figre 6 shows he soliary waves corresponding o c s =.75 for he same vales of γ. As is well-known, he nmber of oscillaions increases wih c s and γ. We also consrced wih he same echniqe mli-plse soliary waves by saring a γ = wih a sperposiion of ranslaed KdV soliary waves as eplained in [15]. Two and hree plse sch soliary waves are shown for γ =.1,.5, and.9 and c s =.75 in Figre 7. As a measre of he accracy of he CGN mehod for approimaing he solion of (3.1) for each vale of γ we comped he L 2 norm of he residal r, whose k-h Forier componen is defined as he lef-hand side of (3.1) wih φ N replaced by is nmerical approimaion. The vale of r for single and wo and hree plse soliary waves as a fncion of γ remained smaller han b in general he residal increases as γ approaches one, a fac ha reflecs he difficly in solving he nonlinear sysems wih γ close o one. The above-described echniqe for generaing soliary waves of he Benjamin eqaion was fond o be more accrae, compared o ieraive cleaning, cf. e. g. [17], wherein one isolaes and cleans ieraively soliary waves ha are prodced by resolion of siable iniial daa, and which works well in case he soliary waves decay eponenially. In he case of he Benjamin eqaion, for which he soliary waves are known o decay qadraically, [6, 12], we fond ha even for large spaial compaional inervals i was very hard o make he vales a he bondaries of he soliary waves prodced by ieraive cleaning less han O(1 5 ). This small rncaion error prodced dispersive oscillaions of he same order of magnide ha very fas polled he ensing solion when sch soliary-wave profiles were sed as iniial vales in evolion sdies. Of corse, for soliary waves prodced

20 2 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS.2 Two plse soliary waves.2 Three plse soliary waves (a) γ = (d) γ = (b) γ = (e) γ = (c) γ = (f ) γ = Figre 7. Two-plse(a,b,c) and hree-plse(b,d,f) soliary waves of he Benjamin eqaion for γ =.1,.5,.9,c s =.75 by ieraive cleaning one does no have a priori knowledge of heir speed, so i is no easy o design sysemaic eperimens wih families of soliary waves of varying speed. We sed he nmerical soliary waves ha we consrced as iniial vales and inegraed in ime he Benjamin eqaion sing he flly discree hybrid scheme implemened as in Secion 2. As a frher es of he accracy of he nmerical soliary waves and he ime-sepping echniqe we comped several invarians of he evolion and varios perinen error measres. In all cases we sed he spaial inerval [ 256,256] and N = 496 and we inegraed he eqaion p o T = 3. Table 6 shows he vales of he L 2 norm, of he invarian H = I +E, where I and E are discree versions of he qaniies defined in (1.3) and (1.4), respecively, and of he amplide of he nmerically propagaed single-plse soliary waves wih c s =.75 for varios vales of γ. The digis shown for each qaniy were conserved p o T = 3. Table 7 shows he conserved digis of he same qaniies for he analogos propagaion eperimen wih wo- and hree-plse soliary waves wih γ =.5. In hese compaions he qaniy H was defined a n as 1 2 L L (U 2 + β3 U3 +δu 2 γui NGU ) d,

21 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION 21 γ L 2 -norm H amplide Table 6. Conserved qaniies for nmerical evolion p o T = 3 of single-plse soliary waves of speed c s =.75 for varios vales of γ. Nmber of plses γ L 2 -norm H amplide Table 7. Conserved qaniies for nmerical evolion p o T = 3 of mli-plse soliary waves of speed c s =.75 for γ =.5 where U = U n, he inegrals being evalaed by nmerical qadrare as described in Secion 3. InFigre8weshowheL 2 (normalized)shapeerrorofhepropagaingnmerical single-plse soliary wave for c s =.75 and varios vales of γ, as fncion of n. This qaniy is defined as SE( n ) = inf τ Un ϕ h ( c s τ) / ϕ h, where ϕ h = P h ϕ N = U is he L 2 -projecion on S h of he nmerically generaed iniial soliary wave ϕ N. As in secion 2, SE( n ) is again comped as ξ(τ ), where τ is he poin near n (fond by Newon s mehod) where d dτ ξ2 (τ ) =, wih ξ(τ) := U n ϕ h ( c s τ) / ϕ h. The shape errors increase wih γ and sabilize wih ecep in he case γ =.99 where a linear emporal growh is observed. (They range from O(1 8 ) o O(1 6 ).) Figre 9 shows he analogos graphs for he phase error, defined as PE( n ) = τ n. The phase errors increase linearly wih and wih γ for fied, ranging from O(1 7 ) o O(1 5 ) a = 3. Finally, we comped he relaive speed error of he simlaions, defined as (C n c s )/c s, where C n = ( ( n +δ) ( n ))/δ and an approimaion of he cener of he plse, i. e. he posiion of is mos negaive ecrsion. When we choose δ = 1 he absole vales of he specific error never eceeded for all γ; he mean vale of he speed remained consan dring he compaions. Finally, as a measre of he qaliy of he nmerically generaed ravelling mliplse soliary waves, we presen in Figres 1, he shape and phase errors dring he nmerical propagaion of wo plse and hree plse soliary waves wih c s =.75 and γ =.5. The shape errors are of O(1 7 ) while he phase errors of abo O(1 5 ) a = 3. In conclsion, he ocome of he nmeros ess performed in his and he preceding secion of he validiy and accracy of he nmerical echniqe for generaing iniial soliary-wave profiles and of he flly discree hybrid scheme ha was sed for heir nmerical evolion, give s enogh confidence o se hese schemes in he

22 22 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS SE() (a) γ = (b) γ = SE() (c) γ = (d) γ =.95 3 SE() (e) γ =.99 3 Figre 8. Shape error of he nmerical propagaion of singleplse soliary waves wih c s =.75 and varios vales of γ. sdy of ineracions and sabiliy of soliary waves of he Benjamin eqaion o be nderaken presenly. 4. Overaking collisions of soliary waves In his secion we sdy in some deail, by compaional means and sing he hybrid mehod, overaking collisions of soliary waves of he Benjamin eqaion. For a given vale of γ (,1) soliary waveswih smaller(absole) amplide (i. e. a smaller in absole vale maimm negaive ecrsion) have larger speed and will conseqenly overake soliary waves wih larger (absole) amplide, which are slower. The soliary waves inerac nonlinearly and emerge largely nchanged; heir ineracion is inelasic, i. e. i is accompanied by he prodcion of a small amplide dispersive ail since he Benjamin eqaion does no appear o be compleely inegrable, as already noed in [17] where resls of a simlaion of an overaking collision for soliary waves of he Benjamin eqaion have been shown. To se he sage we firs presen, as a benchmark, he resls of a simlaion wih he hybrid mehod of an overaking collision of wo soliary waves of he BO eqaion. The iniial soliary waves (cf. (2.2)) had amplides A 1 = 4,A 2 = 1 and corresponding speeds c s,1 = 2 and c s,2 = 1.25 and were cenered a,1 = 1 and,2 = 1, respecively. The compaion was effeced wih N = 496 and k = h/2 on [ 256,256], and prodced he evolion depiced in Figres a

23 NOTES ON THE NUMERICAL SOLUTION OF THE BENJAMIN EQUATION PE() (a) γ = (b) γ = PE() (c) γ = (d) γ =.95 3 PE() (e) γ =.99 3 Figre 9. Phase error of he nmerical propagaion of singleplse soliary waves wih c s =.75 and varios vales of γ Two plse soliary wave Three plse soliary waves SE() PE() γ =.5 3 Figre 1. Shape and phase error of he nmerical propagaion of mli-plse soliary waves wih c s =.75, γ =.5. seleced insances of [, 4]. The wo soliary waves inerac elasically arond = 265. Dring he ineracions here always are wo disinc peaks presen. No arificial oscillaions accompany he nmerical solion afer he ineracion

24 24 V. A. DOUGALIS, A. DURAN, AND D. MITSOTAKIS 4.5 (a) = 4.5 (e) = (b) = (f ) = (c) = (g) = (d) = (h) = Figre 11. Overaking collision of wo soliary waves of he Benjamin-Ono eqaion. We now rn o he simlaions of overaking collisions of pairs of soliary waves of he Benjamin eqaion. We sdied sch collisions for varios vales of γ; we presen here he resls for γ =.1 and γ =.99. For all cases we sed he hybrid mehod on he spaial inerval [ 512,512] wih h =.125 and k =.2 and consrced iniial soliary-wave profiles of varios speeds (cenered a 1 = 256 and 2 = 256) by he procedre described in Secion 3. Figre 13 shows several emporal insances of he overaking collision of wo soliary waves of speeds c s,1 =.45 and c s,2 =.75 in he case γ =.1. (Dring hissimlaionhel 2 normofhe solionwas = ,andhevale of he invarian qaniy H = I+E was H = p o T = 3.) The faser soliary wave overakes he slower and hey inerac nonlinearly wih wo peaks always presen dring he ineracion. The collision prodces a dispersive ail (see Figre 13(g)), a fac sggesing ha he Benjamin eqaion is no inegrable. Noe ha he dispersive ail precedes he soliary waves being of smaller amplide and hence faser in or framework. Figre 14 shows some deails of he ineracion: In (a) he maimm negaive ecrsion of he solion is ploed verss ime. In (b) a magnificaion of (a) one may observe how he maimm negaive ecrsion of he faser wave approaches asympoically is iniial vale. The pahs of he soliary waves are ploed in (c): The faser wave is shifed slighly forward and he slower backward afer he ineracion.