Linearly implicit structure-preserving schemes for Hamiltonian systems

Transcription

1 Linearly implici srucure-preserving schemes for Hamilonian sysems Sølve Eidnes, Lu Li, Shun Sao January 4, 9 arxiv:9.373v [mah.na] Jan 9 Deparmen of Mahemaical Sciences, NTNU, 749 Trondheim, Norway. solve.eidnes@nnu.no, lu.li@nnu.no (corresponding auhor). Graduae School of Informaion Science and Technology, The Universiy of Tokyo, Bunkyo-ku, Tokyo, Japan. shun_sao@mis.i.u-okyo.ac.jp. Absrac s mehod and a wo-sep generalizaion of he discree gradien mehod are boh linearly implici mehods ha can preserve a modified energy for Hamilonian sysems wih a cubic Hamilonian. These mehods are here invesigaed and compared. The schemes are applied o he Koreweg de Vries equaion and he Camassa Holm equaion, and he numerical resuls are presened and analysed. Keywords: Linearly implici mehods, Hamilonian sysem, energy preservaion, Camassa Holm equaion, Koreweg de Vries equaion. Inroducion The field of geomeric numerical inegraion (GNI) has garnered increased aenion over he las hree decades. I considers he design and analysis of numerical mehods ha can capure geomeric properies of he flow of he differenial equaion o be modelled. These geomeric properies are mainly invarians over ime; hey are conserved quaniies such as Hamilonian energy, angular momenum, volume or sympleciciy. Among hem he conservaion of energy is paricularly imporan for proving he eisence and uniqueness of soluions for parial differenial equaions (PDEs) []. Numerical schemes inheriing such properies from he coninuous dynamical sysem have been shown in many cases o be advanageous, especially when inegraion over long ime inervals is considered []. For general non-linear differenial equaions, one may use a sandard fully implici scheme o solve a problem numerically. Then a non-linear sysem of equaions mus be solved a each ime sep. Typically his is done by he use of an ieraive solver where a linear sysem is o be solved a each ieraion. This quickly becomes a compuaionally epensive procedure, especially since

2 he number of ieraions needed in general increases wih he size of he sysem; see a numerical eample comparing he compuaional cos for implici and linearly implici mehods in [3]. A fully eplici mehod on he oher hand, may over-simplify he problem and lead o he loss of imporan informaion, and will ofen have inferior sabiliy properies. The golden middle way may be found in linearly implici schemes, i.e. schemes where he non-linear erms are discreized such ha he soluion a he ne ime sep is found from solving one linear sysem. Our aim is o presen and analyze linearly implici schemes wih preservaion properies. We consider ordinary differenial equaions (ODEs) ha can be wrien in he form ẋ = f () = S H(), R d, () =, (.) where S is a consan skew-symmeric mari and H is a cubic Hamilonian funcion. The famous geomeric characerisic for equaions like (.) is ha he eac flow is energy-preserving, and symplecic, d d H() = H()T d d = H()T S H() =, Ψ y () T J Ψ y () = J, (.) where Ψ y () := ϕ (y ) y, wih ϕ : R d R d, ϕ (y ) = y() he flow map of (.) []. A numerical one-sep mehod is said o be energy-preserving if H is consan along he numerical soluion, and symplecic if he numerical flow map is symplecic. Boh he energy-preserving mehods and he symplecic mehods, he laer of which has he abiliy o preserve a perurbaion of he Hamilonian H of (.), have heir own advanages. In paricular, he energy-preserving propery has been found o be crucial in he proof of sabiliy for several such numerical mehods, see e.g [4]. However, here is no numerical inegraion mehod ha can be simulaneously symplecic and energy-preserving for general Hamilonian sysems []. In his paper we will focus on energy-preserving numerical inegraion. We will sudy schemes based on eising mehods wih geomeric properies. The firs one is s mehod for quadraic ODE vecor fields [6], which by consrucion is linearly implici, and for which he geomeric properies have been sudied in [7]. This is a one-sep mehod, bu we will also give is formulaion as a wo-sep mehod in his paper. The oher mehod o be sudied here, is based on he linearly implici mehod for PDEs presened by Furihaa, Masuo and coauhors in he papers [8, 9, ] and he monograph []. A generalizaion of his mehod, from wo-sep schemes o general mulisep schemes, is given by Dahlby and Owren in [3]. We presen here he wo-sep mehod as i looks for ODEs of he form (.), from which he schemes of he aforemenioned references may arise afer semi-discreizing he Hamilonian PDE in space o obain a sysem of Hamilonian ODEs. This paper is divided ino wo main pars. In he ne chaper, we presen he mehods in consideraion, and give some heoreical resuls on heir geomeric properies. In Chaper 3, we presen numerical resuls for he Camassa Holm equaion and he Koreweg de Vries equaion, including analysis of sabiliy and dispersion, comparing he mehods.

3 Linearly implici schemes We will presen he ODE formulaion of he linearly implici schemes presened by Furihaa, Masuo and coauhors in [8, 9,, ] and by Dahlby and Owren in [3]. Following he nomenclaure of he laer reference, we call hese schemes polarised discree gradien (PDG) mehods. Then we presen a special case of his polarizaion mehod in he same framework as s mehod, wih he goal of obaining more clariy in comparison of he mehods.. Polarised discree gradien mehods The idea behind he PDG mehods is o generalize he discree gradien mehod in such a way ha a relaed varian of he preservaion propery is inac, while nonlinear erms are discreized over consecuive ime seps o ensure lineariy in he scheme. Le us firs recall he concep of discree gradien mehods. A discree gradien is a coninuous map H : R d R d R d such ha for any, y R d The discree gradien mehod for (.) is hen given by H(y) H() = (y ) T H(, y). (.) n+ n = S H( n, n+ ), (.) which will preserve he energy of he sysem (.) a any ime sep. Here and in wha follows, n is he numerical approimaion for a = n and n is he numerical approimaion for he kh k componen of a = n. Resricing ourselves o wo-sep mehods, we define he PDG mehods as follows. Definiion. For he energy H of (.), consider he polarised energy as a funcion H : R d R d R saisfying he properies H(, ) = H(), H(, y) = H(y, ). A polarised discree gradien (PDG) for H is a funcion H : R d R d R d R d saisfying H(y, z) H(, y) = (z )T H(, y, z), (.3) H(,, ) = H(), (.4) and he corresponding polarised discree gradien scheme is given by n+ n = S H( n, n+, n+ ). (.) Proposiion. The numerical scheme (.) preserves he polarised invarian H in he sense ha H( n, n+ ) = H(, ) for all n. 3

4 Proof. H( n+, n+ ) H( n, n+ ) = (n+ n ) T H( n, n+, n+ ) = H( n, n+, n+ ) T S H( n, n+, n+ ) =, where he las equaliy follows from he skew-symmery of S. We remark here ha in he cases where we seek a ime-sepping scheme for he sysem of Hamilonian ODEs resuling from discreizing a Hamilonian PDE in space in an appropriae manner, e.g. as described in [], H will be a discree approimaion o an inegral H. Thus a wo-sep PDG mehod and a sandard one-sep discree gradien mehod, he laer in general fully implici, will preserve wo differen discree approimaions separaely o he same H. The ask of finding a PDG saisfying (.3) is approached differenly in our wo main references, [8, 9,, ] and [3]. Furihaa, Masuo and coauhors apply a generalizaion of he approach inroduced by Furihaa in [3] for finding discree variaional derivaives, while Dahlby and Owren sugges a generalizaion of he average vecor field (AVF) discree gradien [4], given by AVF H(, y, z) = H(ξ + ( ξ)z, y)dξ, where H(, y) is he gradien of H(, y) wih respec o is firs argumen. Provided ha he spaial discreizaion is performed in he same way, hese wo approaches lead o he same scheme for an H quadraic in each of is argumens, as does a generalizaion of he midpoin discree gradien of Gonzalez []. Based on his, we now propose a new, sraighforward approach for finding his specific PDG: Proposiion. Given an H(, y) ha is a mos quadraic in each of is argumens, define H(, y) as he gradien of H wih respec o is firs argumen. Then a PDG for H is given by Proof. We may wrie H(, y, z) = H( + z, y). (.6) H(, y) = T A(y) + b(y) T + c(y), for some symmeric A : R d R d R d, b : R d R d and c : R d R. Then H(, y) = A(y) + b(y), and Furhermore, H( + z, y)t (z ) = (A(y) + z + b(y)) T (z ) = z T A(y)z + b(y) T z T A(y) b(y) T = H(y, z) H(, y). H(,, ) = H(, ) = H(). 4

5 As remarked in Theorem 4. of [3]: if he polarised energy H(, y) is a mos quadraic in each of is argumens, he scheme (.) wih he PDG (.6) is linearly implici. An alernaive o (.6) could be a generalizaion of he Ioh Abe discree gradien [6], defined by is i-h componen where IA H(, y, z) i = { H(, y, z) i if i y i, H i ((z,..., z i, i,..., d ), y)) if i = z i, H(, y, z) i = H((z,..., z i, i+,..., d ), y) H((z,..., z i, i,..., d ), y) z i i. A symmerized varian of his, given by SIA H(, y, z) := ( IA H(, y, z)+ IA H(z, y, )) is again idenical o (.6), whenever H is quadraic in each of is argumens.. A general framework and s mehod For ODEs of he form (.), consider he wo-sep schemes of he form k+ k 3 = S α i j (H ( k +i ) k +j + β( k +i )), (.7) i,j = where H : R d R d R d is he Hessian mari of H and β() := H() H (). For cubic H, his scheme is linearly implici if and only if α 33 =. In his secion, we firs consider he case when he Hamilonian is a cubic homogeneous polynomial, in which case he erm β() in (.7) will disappear, and hen generalize he resuls o he non-homogeneous case. Theorem. The scheme (.7) wih α = α 3 = 4, α i j = oherwise, i.e. n+ n = 4 SH ( n+ )( n + n+ ), (.8) is s mehod composed wih iself over wo consecuive seps when applied o ODEs of he form (.) wih homogeneous cubic H. Proof. As shown in [7], s mehod can be wrien ino a Runge Kua form Two seps of his can be wrien as n+ n = f (n ) + f ( n + n+ ) f (n+ ). n+ n = 4 f (n ) f (n+ ) 4 f (n+ ) + f ( n + n+ ) + f ( n+ + n+ ). (.9) Using ha for homogeneous cubic H we have H() = H (), H ()y = H (y) and H ( + y) = H () + H (y), and insering f () = S H() in (.9), we ge (.8).

6 s mehod preserves he polarised invarian [7] H(, y) = 3 H()y = 3 H(y) = 6 T H ( + y )y. I can be shown ha many well known Runge Kua mehods composed wih iself over wo consecuive seps is a mehod in he class (.7) when applied o (.) wih cubic H. As wo eamples, he implici midpoin mehod over wo seps is (.7) wih α = α 33 = 6,α = α = α 3 = 8, α i j = oherwise, while he rapezoidal rule is (.7) wih α = α 33 = 8,α = 4, α i j = oherwise. The inegral-preserving average vecor field mehod [7] over wo seps is (.7) wih α = α = α 3 = α 33 =,α = 6, α i j = oherwise. A special case of he PDG mehod which preserves he same polarised Hamilonian as s mehod, can also be wrien on he form (.7): Theorem. For a homogeneous cubic H and he polarised energy given by H(, y) = 6 T H ( +y )y, he scheme (.) wih he PDG (.6) applied o (.) is equivalen o (.7) wih α = α = α 3 = 6, α i j = oherwise, i.e. Proof. and hus n+ n = 6 SH ( n+ )( n + n+ + n+ ). H(, y) = 6 H ( + y )y + 6 H ( y ) = H ( + y)y, H(, y, z) = H( + z, y) = 6 H ( + y + z)y = 6 H (y)( + y + z). Now, in he cases where H is non-homogeneous, one can use he echnique employed in [7], adding one variable o generae an equivalen problem o he original one, for a homogeneous Hamilonian H : R d+ R defined such ha H(,,..., d ) = H(,..., d ). Also consrucing he (d + ) (d + ) skew-symmeric mari S by adding a zero iniial row and a zero iniial column o S, we ge ha solving he sysem = S H( ), R d+ () = (, ), (.) is equivalen o solving (.). Following he above resuls for he homogeneous H and (.), we can generalize Theorem and Theorem for all cubic H. Generalizaions of he preservaion properies follow direcly; e.g., s mehod and he PDG mehod can preserve he perurbed energy H( n, n+ ) := 6 ( n ) T H ( n + n+ ) n+ also for non-homogeneous cubic H. 6

7 3 Numerical eperimens To have a beer undersanding of he above mehods, we will apply hem o sysems of wo differen PDEs: he Koreweg de Vries (KdV) equaion and he Camassa Holm equaion. We will compare our mehods o he midpoin mehod, which is a symplecic, fully implici mehod. We solve he wo PDEs by discreizing in space o obain a Hamilonian ODE sysem of he ype (.) and hen apply he PDG mehod (denoed by ), s mehod (Kanan) and he midpoin mehod () o his. 3. Camassa Holm equaion In his secion, we consider he Camassa Holm equaion u u + 3uu = u u + uu (3.) defined on he periodic domain S := R/LZ. I has he conserved quaniies H [u] = S(u + u )d, H [u] = ( u 3 + uu ) d. S Here we consider he variaional form of he Hamilonian H : ( )u δh = δu, δh δu = 3 u + u (uu ). (3.) We follow he approach presened in [] and semi-discreize he energy H of (3.) as H (u) = K k= ( u 3 k + u k (δ + u k) + (δ u k) ), (3.3) where he difference operaors δ + u k := u k+ u k, δ u k := u k u k. For laer use, we here also inroduce he noaion δ u k := u k+ u k, δ u k := u k+ u k +u k, µ + ( ) u k := u k++u k, µ u k := u k +u k, and he marices corresponding o he difference operaors δ +, δ, δ, δ, µ + and µ are denoed by D +, D, D, D, M + and M. Denoing he numerical soluion U = [u,...u K ] T, and by using he properies of he above difference operaors, we hus ge H (U ) = 3 U + M (D + U ) D U, where U is he elemenwise square of U. Then he semi-discreized sysem for he Camassa Holm equaion becomes U = S H (U ) = (I D ) D H (U ). (3.4) The above-menioned schemes applied o (3.4) gives us (I D ) U n+ U n = D H ( U n+ +U n ), () (3.) (I D ) U n+ U n = D H (U n )U n+, () (3.6) (I D ) U n+ U n = D H (U n,u n+,u n+ ), () (3.7) 7

8 where H (U ) = 3diag(U )+M diag(d + U )D + D diag(u ) is he Hessian of H (U ) and H (U n,u n+,u n+ ) is he PDG of Proposiion wih polarised discree energy H (U n,u n+ ) := K ( k= u n k un+ k u n k + un+ k + ( a) (µ+ un k )(δ+ un+ k for some a R, ypically beween and. + a(µ + ) + (µ + un+ k u n k + un+ k )(δ + un k )(δ+ un+ k ) )(δ + un k ) Remark. We performed numerical eperimens for finding a good choice of he parameer a in (3.7) and based on hese se a = in he following. 3.. Numerical ess for he Camassa Holm equaion Eample (Single peakon soluion): In his numerical es, we consider he same eperimen as in [8], where mulisymplecic schemes are considered for he Camassa Holm equaion wih u(,) = cosh( L L ), cosh(l/) [,L], L = 4, [,T ], T =, spaial sep size =.4 and ime sep size =.. All our mehods keep a shape close o he eac soluion ecep some small oscillaory ails, also observed in [8], resuling from he semi-discreizaion, see Figure (he righ wo). Following [3], we define shape and phase error by ɛ shape := min τ U n u( τ), ɛ phase := argmin U n u( τ) c n. τ The numerical simulaions show ha he global error is mainly due o he shape error, see Figure. In Figure (he lef one), we can see ha he numerical energy for all he mehods oscillae, bu appears o be bounded. Here we consider also coarser grids. We observe ha here appear some small wiggles for boh and s mehod for =. and long ime inegraion T =. However, he wiggles in he soluion by are much more eviden han hose in he soluion of s mehod, see Figure 3 (he lef wo plos). We keep on increasing o. and.; we observe ha he numerical soluion obained wih he PDG mehod wih =. suffers from eviden numerical dispersion, while s mehod seems o keep he shape well when comparing o he eac wave. Spurious oscillaions appear also in s mehod when he ime-sep is increased o he value =., see Figure 3 (righ). ), 8

9 Shape error Phase error Global error Figure : In his eperimen, space sep size =.4 and ime sep size =.. Lef: shape error, middle: phase error, righ: global error. relaive energy error H u(,) u(,) Figure : In his eperimen, =.4, =.. Lef: relaive energy errors. middle: propagaion of he wave by. righ: propagaion of he wave by s mehod. u(,). u(,). u(,) Figure 3: In his eperimen, space sep size =.4. Lef: propagaion of he wave by, =., middle: propagaion of he wave by s mehod, =., righ: propagaion of he wave by s mehod, =.. Eample (Two peakons soluion): Now we consider he iniial condiion u(,) = cosh( L 4 L ) cosh(l/) + 3 cosh( 3L 4 L ), cosh(l/) where [,L], L = 4, [,T ], T =, and =.4, =.. We observe ha all he mehods keep he shape of he eac soluion very well and he numerical energy appears 9

10 bounded, see Figure. The numerical simulaion shows ha he global error is mainly due o he shape error, see Figure 4. When a coarser ime grid and longer ime inegraion is considered, =. and T =, small wiggles appear in he soluion of and s mehod, see Figure 6 (he lef wo figures). We increase o., and observe ha fails o preserve he shape of he soluion, while s mehod can sill keep a shape close o he eac soluion even hough also for his mehod he numerical dispersion increases, see Figure 6 (righ). Shape error Phase error Global error Figure 4: In his eperimen, space sep size =.4, ime sep size =.. Lef: shape error, middle: phase error, righ: global error. relaive energy error H u(,) - 4 u(,) - 4 Figure : In his eperimen, =.4, =.. Lef: relaive energy errors, middle: propagaion of he wave by, righ: propagaion of he wave by s mehod. u(,) u(,) u(,) Figure 6: In his eperimen, =.4. Lef: propagaion of he wave by, =., middle: propagaion of he wave by s mehod, =., righ: propagaion of he wave by s mehod, =..

11 3. Koreweg de Vries equaion In he previous eample, he vecor field of he semi-discreized sysem based on he Camassa Holm equaion is a homogeneous cubic polynomial. In his secion, we deal wih he KdV equaion, for which he vecor field of he semi-discreized equaion is a non-homogeneous cubic polynomial. The KdV equaion u + 6uu + u = (3.8) on he periodic domain S := R/LZ has he conserved Hamilonians H (u()) = u d, H (u()) = S S ( u 3 + u ) d. In he following we consider he variaional form based on he Hamilonian H : 3.. Numerical schemes for he KdV equaion u = δh δu, δh δu = 3u u. (3.9) We discreize he energy H for he KdV equaion (3.9) as K H (U ) = ( u 3k + (δ+ u k) + (δ u k) ). 4 k= From simple calculaions, he corresponding gradien is given by H (U ) = ( 3U D U ), and hus we have he semi-discreized form for (3.9): Applying he schemes under consideraion o (3.) gives U = D ( 3U D U ). (3.) U n+ U n =D H ( U n +U n+ ), () (3.) U n+ U n = D ( H(U n ) + H(U n+ ) 4 H( U n +U n+ )), () (3.) U n+ U n =D H (U n,u n+,u n+ ), () (3.3) where H (U ) = 6diag(U ) D is he Hessian of H (U ) and H (U n,u n+,u n+ ) is found as in Proposiion, wih polarised discree energy d H (u n k,un+ k ) := ( u n k un+ k k= + a u n k + un+ k (δ + un k ) + (δ + un+ k + a (δ+ un k )(δ+ un+ k ) Remark. We perform several numerical simulaions o find a good choice of parameer a, and we ake a = for in he following numerical eamples for KdV equaion. ) ).

12 3.. Sabiliy analysis of he schemes To analyse he sabiliy of he above mehods, we perform he von Neumann sabiliy analysis for he and schemes applied o he linearized form of he KdV equaion (3.8) The equaion for he amplificaion facor for s mehod is and is roo is u + u =. (3.4) ( + iλ(cosθ )sinθ)g + iλ(cosθ )sinθ =, g = iλ(cosθ )sinθ + iλ(cosθ )sinθ, where λ := 3. Since g is a simple roo on he uni circle, s mehod is uncondiionally sable for he linearized KdV equaion. The equaion for he amplificaion facor for is The wo roos of he above equaion are hus g + iλ(3g g + 3)(cosθ )sinθ =. (3.) g = 3b + + 8b + ib(3 + 8b ) + 9b, g = 3b + 8b ib(3 + 8b + ) + 9b, where b = λ( cosθ)sinθ. We observe ha g = g =, and g g, herefore he PDG mehod is uncondiionally sable for he linearized KdV equaion Numerical ess for he KdV equaion Eample (One solion soluion): Consider he iniial value u(,) = sech ( L/), where [, L], L = 4. We apply our schemes over he ime inerval [, T ], T =, wih sep sizes =., =.. From our observaions, all he mehods behave well. The shape of he wave is well kep by all he mehods, also for long ime inegraion, see Figure 7. The energy errors of all he mehods are raher small and do no increase over long ime inegraion, see Figure 8 (lef). We hen use a coarser ime grid, =.3, and boh mehods are sill sable, see Figure 9 (lef wo). However we observe ha he global error of becomes much bigger han ha of s mehod. When an even larger ime sep-size, =.4, is considered, he soluion for blows up while he soluion for s mehod is raher sable. In his case, he PDG mehod applied o he nonlinear KdV equaion is unsable and he numerical soluion blows up a around =8. Even if we increase he ime sep-size o =., s mehod

13 Shape error.6.4. Phase error... Global error 4 3 Figure 7: Space sep size =., ime sep size =.. Lef: shape error, middle: phase error, righ: global error. relaive energy error H - - u(,) 4 4 u(,) 4 4 Figure 8: Wih =., =.. Lef: relaive energy errors, righ wo: propagaion of he wave by and s mehod. sill works well, see Figure 9 (middle). When =. is considered, we observe eviden signs of insabiliy in he soluion of s mehod. The soluion will blow up rapidly when =.. u(,) 4-4 u(,) Eac PDE - Figure 9: Wih =.. Lef: =.3, propagaion of he wave by, middle: =., propagaion of he wave by s mehod, righ: dispersion relaion for λ =. Eample (Two solions soluion): We choose iniial value u(,) = 6sech, and consider periodic boundary condiions u(, ) = u(l, ), where [, L], L = 4. We se he 3

14 space sep size =. and apply he aforemenioned schemes on ime inerval [,T ] wih T =, =.. All he mehods behave sably. The profiles of s mehod and he midpoin mehod are almos indisinguishable, and he profiles for he midpoin mehod are hus no presened here. s mehod and preserve he modified energy, and accordingly he energy error of all he mehods are raher small over long ime inegraion, see Figure (lef). Afer a shor while he soluion has wo solions; one is all and he oher is shorer, see Figure (he righ wo plos). When we consider a coarser ime grid, =.37, boh mehods are sill sable, see Figure (he lef wo). However, here appear more small wiggles in he soluion by and we observe ha he soluion of will blow up soon, around =, for an even coarser ime grid =.. When we increase he ime sep size o =. and consider T =, he shape of he eac soluion is sill well preserved by s mehod, even hough here appear some small wiggles in he soluion a around =. We observe ha he soluion of s mehod will blow up when =. is considered. Similar eperimens as in his subsecion, bu for he mulisymplecic bo schemes, can be found in a paper by Ascher and McLachlan [9]. However, here we consider even coarser ime grid han here, and he numerical resuls show ha s mehod is quie sable, even hough i is linearly implici. relaive energy error H u(,) 4 u(,) 4 Figure : In his eperimen, =., =.. Lef: relaive energy errors, righ wo: propagaion of he wave by and s mehod Dispersion analysis We consider he radiional linear analysis of numerical dispersion relaions for he numerical schemes applied o he KdV equaion, geing he dispersion relaion of frequency ω and wave number ξ o be ω = ξ 3, (eac soluion) (3.6) sinω = λ( cosξ)(3cosω )sinξ, () (3.7) sinω = λ( cosξ)sinξ, () (3.8) +cosω where λ = 3. The dispersion curve is displayed in Figure (9) (righ). We observe ha s mehod is beer han a preserving he eac dispersion relaion. This coincides wih he 4

15 u(,) u(,) u(,) Figure : =.. Lef: Propagaions of he wave by, =.37, middle: propagaions of he wave by s mehod, =.37, righ: Propagaions of he wave by s mehod, =.. behaviour of he mehods applied o he nonlinear KdV equaion shown in Secion Conclusion In his paper we perform a comparaive sudy of s mehod and wha we call he polarised discree gradien (PDG) mehod. To ha end, we presen a general form encompassing a class of wo-sep mehods ha includes boh a specific case of he PDG mehod and s mehod composed wih iself. We also compare he mehods for compleely inegrable Hamilonian PDEs, he KdV equaion and he Camassa Holm equaion. Boh s mehod and he PDG mehod are linearly implici mehods, which will save he compuaional cos. A series of numerical eperimens has been performed here, for he KdV equaion wih one and wo solions, and he Camassa Holm equaion wih one and wo peakons. These eperimens show ha s mehod is more sable han he PDG mehod. They also indicae ha s mehod yields more accurae resuls, as we have winessed in he energy error and he shape and phase error when comparing o analyical soluions. Based on our resuls, we would recommend he use of s mehod if one seeks a linearly implici scheme for a Hamilonian sysem wih cubic H. Acknowledgemens The auhors would like o hank Elena Celledoni, Takayasu Masuo and Brynjulf Owren for iniiaing he work ha led o his paper, and for heir very helpful commens on he manuscrip. References [] M. E. Taylor, Parial differenial equaions I. Basic heory, vol. of Applied Mahemaical Sciences. Springer, New York, second ed.,.

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