Hamiltonian Boundary Value Methods (Energy Preserving Discrete Line Integral Methods) 1 2

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1 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE) Journal of Numerical Analysis, Indusrial and Applied Mahemaics (JNAIAM) vol. 5, no. -2, 2, pp ISSN Hamilonian Boundary Value Mehods (Energy Preserving Discree Line Inegral Mehods) 2 Luigi Brugnano 3 Diparimeno di Maemaica U. Dini, Universià di Firenze Viale Morgagni 67/A, I-534 Firenze, Ialy Felice Iavernaro 4 Diparimeno di Maemaica, Universià di Bari Via Orabona 4, I-725 Bari, Ialy Donao Trigiane 5 Diparimeno di Energeica S.Secco, Universià di Firenze Via Lombroso 6/7, I-534 Firenze, Ialy Received Ocober 25, 29; acceped in revised form April 5, 2. Absrac: Recenly, a new family of inegraors (Hamilonian Boundary Value Mehods) has been inroduced, which is able o precisely conserve he energy funcion of polynomial Hamilonian sysems and o provide a pracical conservaion of he energy in he non-polynomial case. We sele he definiion and he heory of such mehods in a more general framework. Our aim is on he one hand o give accoun of heir good behavior when applied o general Hamilonian sysems and, on he oher hand, o find ou wha are he opimal formulae, in relaion o he choice of he polynomial basis and of he disribuion of he nodes. Such analysis is based upon he noion of exended collocaion condiions and he definiion of discree line inegral, and is carried ou by looking a he limi of such family of mehods as he number of he so called silen sages ends o infiniy. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering Keywords: Hamilonian problems, exac conservaion of he Hamilonian, energy conservaion, Hamilonian Boundary Value Mehods, HBVMs, discree line inegral. Mahemaics Subjec Classificaion: 65P, 65L5. Inroducion We consider canonical Hamilonian problems in he form ẏ = J H(y), y( ) = y R 2m, () Published elecronically Ocober 5, 2 2 Work developed wihin he projec Numerical mehods and sofware for differenial equaions. 3 luigi.brugnano@unifi.i 4 felix@dm.uniba.i 5 rigian@unifi.i

2 8 L. Brugnano, F. Iavernaro, and D. Trigiane where J is a skew-symmeric consan marix, and he Hamilonian H(y) is assumed o be sufficienly differeniable. For is numerical inegraion, he problem is o find numerical mehods which preserve H(y) along he discree soluion {y n }, since his propery holds for he coninuous soluion y(). So far, many aemps have been made inside he class of Runge-Kua mehods, he mos successful of hem being ha of imposing he sympleciciy of he discree map, considering ha, for he coninuous flow, sympleciciy implies he conservaion of H(y). Concerning symplecic inegraors, a backward error analysis permis o prove ha hey exacly conserve a modified Hamilonian, even hough his fac clearly does no always guaranee a proper qualiaive behavior of he discree orbis. On he oher hand, i is possible o follow differen approaches o derive geomeric inegraors which are energy-preserving. This has been done, for example, in he pioneering work [6], and laer in [4], where discree gradien mehods are inroduced and sudied. An addiional example of energy-preserving mehod is he Averaged Vecor Field (AVF) mehod defined in [5] (see also [4]). By he way, he laer mehod can be rerieved by he mehods here sudied. More recenly, in [2] a new family of one-sep mehods has been inroduced, capable of providing a numerical soluion {y n } of (), along which he energy funcion H(y) is precisely conserved, in he case where his funcion is a polynomial (see also [,, ]). These mehods, named Hamilonian Boundary Value Mehods (HBVMs hereafer), may be also hough of as Runge-Kua mehods where he inernal sages are spli ino wo caegories: - he fundamenal sages, whose number, say s, is relaed o he order of he mehod; - he silen sages, whose number, say r, has o be suiably seleced in order o assure he energy conservaion propery for a polynomial H(y) of given degree ν; he higher is ν, he higher mus be r. The resuling mehod is denoed by HBVM(k,s), 6 where k = s + r is he oal number of unknown sages. In [2, ] i has also been shown ha hese new mehods provide a pracical conservaion of he energy even in he non-polynomial case: he erm pracical means ha, in many general siuaions, when he number of silen sages is high enough, he mehod makes no disincion beween he funcion H(y) and is polynomial approximaion, being he laer in a neighborhood of size ε of he former, where ε denoes he machine precision. Anoher relevan issue o be menioned is ha he compuaional cos for he soluion of he associaed nonlinear sysem is essenially independen of he number of silen sages, and only depends on s (see [2, ]). This comes from he fac ha he silen sages are acually linear combinaions of he fundamenal sages. These wo aspecs moivae he following quesion: wha is, if any, he limi mehod when he number of silen sages grows o infiniy? This quesion was firs posed by Erns Hairer, 7 who also provided a parial answer by saing formulae (2), which he called Energy Preserving varian of Collocaion Mehods (EPCMs, hereafer) [7]. We provide a proof of his saemen by clarifying he connecion beween he limi formulae and HBVMs: we show ha acually one can define several differen limi mehods, 8 each one associaed o he specific polynomial basis, as well as o he choice of he abscissae disribuion, used o consruc he sequence of HBVMs. For example, EPCMs are based upon he use of Lagrange polynomials, while, working wih he shifed Legendre basis, yields o differen limi mehods, ha we have called Infiniy Hamilonian Boundary Value Mehods (in shor, -HBVMs or HBVM(,s), being s he number of he unknown fundamenal sages). Our aim in his paper is hreefold: 6 The denominaion HBVM wih k seps and degree s was used in [2]. 7 During he inernaional conference ICNAAM 29, Rehymno, Cree, Greece, 8-22 Sepember 29, afer he alks, by he firs wo auhors, where HBVMs were presened. 8 In he sense ha hey generae differen discree problems. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

3 Hamilonian Boundary Value Mehods 9. We sele he definiion of HBVMs in a more general framework, also deriving he general formulaion of he limi formulae lim k HBVM(k,s). In paricular, we show ha such limi coincides wih EPCMs if he Lagrange polynomial basis is used (Secion 2). 2. In Secion 3, we inroduce he new class of -HBVMs, which are he limi formulae corresponding o he HBVMs based upon he shifed Legendre polynomial basis. We prove ha he order of such formulae is he same as he Gauss-Legendre mehods, ha is 2s (where s is he number of he unknown fundamenal sages). 3. We menion he case where H(y) belongs o vecor spaces differen from ha of polynomials, hus providing a naural (and rivial) generalizaion of he original formulae (see Secion 4). Moreover, in he polynomial case, we deermine he opimal disribuion of he nodes (Secion 5). We sress ha any finie approximaion of EPCMs or -HBVMs based on quadraures leads back o HBVM(k,s) mehods, for k high enough. We address all he poins lised above, by slighly modifying he approach followed o define he class of HBVMs in [2]. 2 Reformulaion of Hamilonian BVMs The key formula which HBVMs rely on, is he line inegral and he relaed propery of conservaive vecor fields: H(y ) H(y ) = h σ( + τh) T H(σ( + τh))dτ, for any y R 2m, (2) where σ is any smooh funcion such ha σ( ) = y, σ( + h) = y. (3) Here we consider he case where σ() is a polynomial (of degree a mos s), yielding an approximaion o he rue soluion y() in he ime inerval [, + h]. The numerical approximaion for he subsequen ime-sep, y, is hen defined by (3). Afer inroducing a se of s disinc abscissae c,...,c s, ( < c i ), 9 we se Y i = σ( + c i h), i =,...,s, (4) so ha σ() may be hough of as an inerpolaion polynomial, Y i, i =,...,s, being he inernal sages. Le us consider he following expansions of σ() and σ() for [, + h]: σ( + τh) = γ j P j (τ), τ σ( + τh) = y + h γ j P j (x)dx, (5) where {P j ()} is any suiable basis of he vecor space of polynomials of degree a mos s and he (vecor) coefficiens {γ j } are o be deermined. Before proceeding, one imporan remark is in order. 9 As a convenion, when c = is o be considered, as in he case of he Lobao abscissae in [, ], hen c = is formally added o he abscissae c,..., c s, and he subsequen formulae are modified accordingly. More general funcion spaces will be considered in he sequel. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

4 2 L. Brugnano, F. Iavernaro, and D. Trigiane Remark. As will be clear in a while, we observe ha he numerical mehod which he following procedure will define is basis-dependen, in ha o differen choices of he basis {P j ()} here will, in general, correspond differen numerical mehods. In his secion, in order o le he heory be presened as general as possible, we leave he basis no beer specified. This will allow us o achieve he resuls lised a poin. in he inroducion. The quesion abou how o choose he basis properly is faced in Secion 3, where -HBVMs will be inroduced. Therefore, jus in he presen secion, o avoid confusion, we will always specify wha is he basis we are working wih. This will be no necessary anymore saring from Secion 3, afer deermining he opimal basis. In his secion we assume ha H(y) is a polynomial, which implies ha he inegrand in (2) is also a polynomial so ha he line inegral can be exacly compued by means of a suiable quadraure formula. I is easy o observe ha in general, due o he high degree of he inegrand funcion, such quadraure formula canno be solely based upon he available abscissae {c i }: one needs o inroduce an addiional se of abscissae, ĉ,...,ĉ r, disinc from he nodes {c i }, in order o make he quadraure formula exac: σ( + τh) T H(σ( + τh))dτ = (6) β i σ( + c i h) T H(σ( + c i h)) + i= r ˆβ i σ( + ĉ i h) T H(σ( + ĉ i h)), where β i, i =,...,s, and ˆβ i, i =,...,r, denoe he weighs of he quadraure formula corresponding o he abscissae {c i } and {ĉ i }, respecively, i.e., s β i = c j r ĉ j d, i =,...,s, c i c j c i ĉ j ˆβ i =,j i s c j ĉ i c j r,j i i= ĉ j d, ĉ i ĉ j i =,...,r. According o [], he righ-hand side of (6) is called discree line inegral, while he vecors Ŷ i = σ( + ĉ i h), i =,...,r, (8) are called silen sages: hey jus serve o increase, as much as one likes, he degree of precision of he quadraure formula, bu hey are no o be regarded as unknowns since, from (5), hey can be expressed in erms of linear combinaions of he fundamenal sages (4). In [2], he mehod HBVM(k,s), wih k = s + r is hen defined by subsiuing he quaniies in (5) ino he righ-hand side of (6) and by choosing he unknowns {γ j } in order ha he resuling expression vanishes. Insead of carrying ou our compuaion on he righ-hand side of (6), as was done in [2], we apply he procedure direcly o he original line inegral appearing in he lef-hand side. Of course, since hese wo expressions are equal, he final formula will exacly mach he HBVM(k,s) mehod, wrien in a differen guise. Wih his premise, by considering he firs expansion in (5), he conservaion propery reads (7) γ T j P j (τ) H(σ( + τh))dτ =, (9) c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

5 Hamilonian Boundary Value Mehods 2 which, as is easily checked, is cerainly saisfied if we impose he following se of orhogonaliy condiions γ j = η j P j (τ)j H(σ( + τh))dτ, j =,...,s, () wih {η j } suiably nonzero scaling facors ha will be defined in a while. Then, from he second relaion of (5) we obain, by inroducing he operaor L(f;h)σ( + ch) = () c σ( ) + h η j P j (x)dx P j (τ)f(σ( + τh))dτ, c [,], ha σ is he eigenfuncion of L(J H;h) relaive o he eigenvalue λ = : σ = L(J H;h)σ. (2) Definiion. Equaion (2) will be called he Maser Funcional Equaion defining σ. Remark 2. We also observe ha, from () and he firs relaion in (5), one obains he equaions σ( + c i h) = η j P j (c i ) P j (τ)j H(σ( + τh))dτ, i =,...,s, (3) which may be viewed as exended collocaion condiions according o [, Secion 2], where he inegrals are (exacly) replaced by discree sums (see, e.g., (6) (7)). where To pracically compue σ, we se (see (4) and (5)) Y i = σ( + c i h) = y + h a ij = ci a ij γ j, i =,...,s, (4) P j (x)dx, i,j =,...,s. (5) Insering () ino (4) yields he final formulae which define he HBVMs class based upon he basis {P j }: Y i = y + h η j a ij P j (τ) J H(σ( + τh))dτ, i =,...,s. (6) The consans {η j } have o be chosen in order o make he formula consisen. Problem (4) (6) can be acually solved, provided ha all he {η j } are differen from zero, and he marix c P cs (x)dx... P (x)dx.. c P cs s(x)dx... P s(x)dx is nonsingular (which we shall obviously assume hereafer). Indeed, such a marix allows us, by using (5), o reformulae equaion (6) in erms of he (unknown) fundamenal sages (4). Le us now formally se and repor a few examples for possible choices of he basis {P j (x)}. f(y) = J H(y), (7) For example, he choice P j (x) = x j, j =,..., s, would lead o η =, and η j =, j = 2,..., s. This implies ha, wih his choice of he basis, σ can only be a line (i.e., s = ). c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

6 22 L. Brugnano, F. Iavernaro, and D. Trigiane. In [] we have chosen {P (x),...,p s (x)} as he Newon basis. This has allowed us he consrucion of a family of mehods of order 2 and In [2], he abscissae {c = } {c i } {ĉ i } are disposed according o a Lobao disribuion wih k + poins in [,] and {P (x),...,p s (x)} is he shifed Legendre basis in he inerval [,]. 2 Consequenly, choosing in (6) =,h =, and f(y(τ)) = P j (τ), he consisency condiion yields ( η j = Pj (x)dx) 2 = 2j, j =,...,s, (8) which is exacly he value found in [2]. In such a case, i has been shown ha he resuling mehod has order 2s, jus he same as he generaing Lobao IIIA mehod (obained for k = s). 3. In a similar way, when using he Lagrange basis {l j (x)}, by seing f(y(τ)), one obains η j = /b j wih s x c i b j = l j (x)dx, l j (x) =. (9) c j c i i=, i j Consequenly, formulae (6) become Y i = y + h a ij l j (τ) J H(σ( + τh))dτ, i =,...,s. (2) b j Moreover, by inroducing he new variables K i = σ( + c i h), which are herefore relaed o he Y i as Y i = y + h a ij K j, i =,...,s, sysem (2) can be recas in he equivalen form provided by he exended collocaion condiions (3): K i = l i (τ)j H(σ( + τh))dτ, i =,...,s. (2) b i Formulae (2) are he ones E. Hairer proposed in he general case, ha is for any kind of Hamilonian funcion. They were called Energy Preserving varian of Collocaion Mehods (EPCMs) [7]. The above discussion hen proves ha if he inegral can be subsiued by a finie sum, as in he case where H(y) is a polynomial, formulae (6), and consequenly (2), become a HBVM(k,s), wih a suiable value of k. 3 For sake of compleeness, we repor he nonlinear sysem associaed wih he HBVM(k, s) mehod, in erms of he fundamenal sages {Y i } and he silen sages {Ŷi} (see (8)), by using he noaion (7). In his conex, hey represen he discree counerpar of (6), and may be direcly rerieved by evaluaing, for example, he inegrals in (6) by means of he (exac) quadraure formula inroduced in (6): r Y i = y + h η j a ij P j (c l ) f(y l ) + ˆβ l η j a ij P j (ĉ l ) f(ŷl) = y + h l= β l η j a ij ( l= β l P j (c l )f(y l ) + l= ) r ˆβ l P j (ĉ l )f(ŷl), i =,...,s. 2 More precisely, P j (x) is here he shifed Legendre polynomial of degree j, j =,..., s. 3 The same clearly happens when he inegral is only approximaed by a finie sum. l= (22) c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

7 Hamilonian Boundary Value Mehods 23 From he above discussion i is clear ha, in he non-polynomial case, supposing o choose he abscissae {ĉ i } so ha he sums in (22) converge o an inegral as r = k s, he resuling formula is (6). 4 Consequenly, EPCMs may be viewed as he limi of HBVMs family, when he Lagrange basis is considered, as he number of silen sages grows o infiniy. The above argumens also imply ha HBVMs may be as well applied in he non-polynomial case since, in finie precision arihmeic, HBVMs are indisinguishable from heir limi formulae (6), when a sufficien number of silen sages is inroduced. The aspec of having a pracical exac inegral, for k large enough, was already sressed in [2,, ]. 3 Infiniy Hamilonian Boundary Value Mehods As is easily argued (and emphasized in Remark ), he choice of he basis along which σ( + τh) is expanded (see (5)), somehow influences he shape of he final formulae (6), ha is, o wo differen polynomial bases here may correspond wo differen families of formulae. 5 The quesion hen naurally arises abou he bes possible choice of he basis o consider. This issue has been a crucial poin in devising he class of HBVMs in [2] and deserves a paricular aenion. 6 Indeed, we recall ha our final goal is o devise mehods ha make he sum (9), represening he line inegral, vanish. This is accomplished by he orhogonaliy condiions (), whose effec is o make null each erm of he sum in (9). I follows ha such condiions are in general oo demanding, in ha hey are sufficien bu no necessary o ge conservaiveness. In fac, he sum could in principle vanish even in he case when wo ore more of is erms are differen from zero. This exra consrain may affec he general properies of he conservaive mehods we are ineresed in, and in paricular heir order. This was a problem already encounered in [] where he auhors realized ha he use of he Newon basis didn assure he expeced growh of he order of he resuling mehod when he degree of he polynomial σ( + τh) was increased. This barrier has been definiively overcome in [2], where i was undersood ha he proper polynomial basis o be used by defaul was ha of he shifed Legendre polynomials in he inerval [, ]. We emphasize ha, conrary o wha happens for he Lagrange and Newon bases, he Legendre polynomials are orhogonal and symmeric in he inerval [,] and in addiion hey are abscissae-free, ha is hey by no means depend on he specific disribuion of he abscissae {c i } adoped. This, in urn, implies ha he Maser Funcional Equaion (2) is independen of he choice of boh he abscissae {c i } and {ĉ i }: he only requiremen being ha (6) holds rue. 7 From he above argumens, i is clear ha he orhogonaliy condiions (), i.e., he fulfillmen of he Maser Funcional Equaion (2), is only a sufficien condiion for he conservaion propery (9) o hold. Such a condiion becomes also necessary, when he basis {P j } is orhogonal. Theorem. Le {P j } be an orhogonal basis on he inerval [,]. Then, assuming H(y) o be suiably differeniable, (9) implies ha each erm in he sum has o vanish. Proof Le us consider, for simpliciy, he case of an orhonormal basis, and he expansion g(τ) H(σ( + τh)) = l ρ l P l (τ), ρ l = (P l,g), l, where, in general, (f,g) = f(τ)g(τ)dτ. 4 This obvious requiremen for he abscissae will be always assumed in he sequel. 5 For example, see he mehod presened in subsecion The argumen presened here is he analog of he one appearing in [, Remark 3.]. 7 We emphasize ha his is no he case when using, for example, he Lagrange basis. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

8 24 L. Brugnano, F. Iavernaro, and D. Trigiane Subsiuing ino (9), yields γj T (P j,g) = γ T j P j, ρ l P l = l γj T ρ j =. Since his has o hold whaever he choice of he funcion H(y), one concludes ha γ T j ρ j =, j =,...,s. (23) Remark 3. In he case where {P j } is he shifed Legendre basis, from (23) one derives ha γ j = Sρ j, i =,...,s, where S is any nonsingular skew-symmeric marix. The naural choice S = J hen leads o (), wih η j = (P j,p j ) = 2j. The use of he Legendre basis allows he resuling mehods o have he bes order and sabiliy properies ha one can expec. This aspec is elucidaed in he wo heorems and he corollary below, which represen he main resul of he presen work. Alhough, up o now, we have mainained he reamen of HBVMs a a general level, i is clear ha, in view of he resul presened in Theorem 2, when he curve σ( + τh) is assumed of polynomial ype, we will implicily adop he Legendre basis. 8 This imporan assumpion will be incorporaed in he HBVM mehods from now on: if needed, he use of any oher kind of basis will be explicily saed, in order no o creae confusion. Taking ino accoun he consisency condiions (8), formula (6) (5) akes he form: Y i = y + h (2j )a ij P j (τ) J H(σ( + τh))dτ, i =,...,s. (24) If he Hamilonian H(y) is a polynomial, he inegral appearing a he righ-hand side is exacly compued by a quadraure formula, hus resuling ino a HBVM(k,s) mehod wih a sufficien number of silen sages. As already sressed in he previous secion, in he non-polynomial case such formulae represen he limi of he sequence HBVM(k,s), as k. Definiion 2. We call he new limi formula (24) an Infiniy Hamilonian Boundary Value Mehod (in shor, -HBVM or HBVM(,s)). We emphasize ha, in he non-polynomial case, (24) becomes an operaive mehod, only afer ha a suiable sraegy o approximae he inegral is aken ino accoun (see he nex secion for addiional examples). In he presen case, if one discreizes he Maser Funcional Equaion () (2), HBVM(k, s) are hen obained, essenially by exending he discree problem (22) also o he silen sages (8). In order o simplify he exposiion, we shall use (7) and inroduce he following noaion: { i } = {c i } {ĉ i }, {ω i } = {β i } {ˆβ i }, y i = σ( + i h), f i = f(σ( + i h)). (25) The discree problem defining he HBVM(k,s) hen becomes, wih η j = 2j, y i = y + h i η j P j (x)dx k ω l P j ( l )f l, i =,...,k. (26) 8 Acually, he erm Hamilonian Boundary Value Mehod has been coined in [2], afer inroducing he Legendre basis. l= c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

9 Hamilonian Boundary Value Mehods 25 We can cas he se of equaions in vecor form, by inroducing he vecors y = (y T,...,y T k )T, e = (,...,) T R k, and he marices I, P R k s, wih as I ij = i P j (x)dx, P ij = P j ( i ), and Λ = diag(η,...,η s ), Ω = diag(ω,...,ω k ), (27) y = e y + h(iλp T Ω) I f(y), (28) wih an obvious meaning of f(y). Consequenly, he mehod can be seen as a Runge-Kua mehod wih he following Bucher ableau:. IΛP T Ω (29) k ω... ω k Remark 4. We observe ha, provided ha he marix Λ is independen of he basic abscissae {c i } (as in he case of he Legendre basis), he role of such abscissae and of he silen abscissae {ĉ i } is inerchangeable. This is no rue, for example, for he Newon and Lagrange bases. The following resul hen holds rue. Theorem 2. Provided ha he quadraure has order a leas 2s (i.e., i is exac for polynomials of degree a leas 2s ), HBVM(k,s) has order p = 2s 2deg(σ), whaever he choice of he abscissae c,...,c s. Proof From he classical resul of Bucher (see, e.g., [9, Theorem 7.4]), he hesis follows if he simplifying assumpions C(s), B(p), p 2s, and D(s ) are saisfied. By looking a he mehod (28) (29), one has ha he firs wo (i.e., C(s) and B(p), p 2s) are obviously fulfilled: he former by he definiion of he mehod, he second by hypohesis. The proof is hen compleed, if we prove D(s ). Such condiion can be cas in marix form, by inroducing he vecor ē = (,...,) T R s, and he marices Q = diag(,...,s ), D = diag(,..., k ), V = ( j i ) R k s, (see also (27)) as i.e., QV T Ω ( IΛP T Ω ) = ( ē e T V T D ) Ω, Since he quadraure is exac for polynomial of degree 2s. one has ( I T ΩV Q ) ij = = ( k l ω l l= ( δ i PΛI T ΩV Q = ( eē T DV ). (3) P i (x)dx(j j l ) ) ( = ) P i (x)dx(j j )d ) P i (x)x j dx, i =,...,s, j =,...,s, where he las equaliy is obained by inegraing by pars, wih δ i he Kronecker symbol. Consequenly, ( PΛI T ΩV Q ) ( ) = η ij l P l ( i ) P l (x)x j dx = ( j i ), i =,...,k, j =,...,s, l= c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

10 26 L. Brugnano, F. Iavernaro, and D. Trigiane ha is, (3), where he las equaliy follows from he fac ha η l P l () l= P l (x)x j dx = j, j =,...,s. Concerning he sabiliy, he following resul holds rue. Theorem 3. For all k such ha he quadraure formula has order a leas 2s 2deg(σ), HBVM(k,s) is perfecly A-sable, whaever he choice of he abscissae c,...,c s. Proof As i has been previously observed, a HBVM(k,s) is fully characerized by he corresponding polynomial σ which, for k sufficienly large (i.e., assuming ha (6) holds rue), saisfies he Maser Funcional Equaion () (2), which is independen of he choice of he nodes c,...,c s (since we consider he Legendre basis). When, in place of f(y) = J H(y) we pu he es equaion f(y) = λy, we have ha he collocaion polynomial of he Gauss-Legendre mehod of order 2s, say σ s, saisfies he Maser Funcional Equaion, since he inegrands appearing in i are polynomials of degree a mos 2s, so ha σ = σ s. The proof complees by considering ha Gauss-Legendre mehods are perfecly A-sable. A worhwhile consequence of Theorems 2 and 3 is ha one can ransfer o HBVM(,s) all hose properies of HBVM(k,s) which are saisfied saring from a given k k on: for example, he order and sabiliy properies. Corollary. Whaever he choice of he abscissae c,...,c s, HBVM(,s) (24) has order 2s and is perfecly A-sable. Remark 5. From he resul of Corollary, i follows ha HBVM(,s) has order 2s and is perfecly A- sable for any choice for he abscissae c,...,c s. Since such abscissae can be arbirarily chosen, we can formally place hem a he roos of he Gauss-Legendre polynomial of degree s. On he oher hand, by considering ha, a such abscissae, by seing {l i (c)} and {b i } he corresponding Lagrange polynomials and quadraure weighs, respecively (see (9)), b i P j (x)l i (x)dx = b i b r P j (c r )l i (c r ) = P j (c i ), r= one obains (wih η j = 2j and by using he noaion (7)): σ ( + c i h) = = η j P j (c i ) P j (τ)f(σ( + τh))dτ j =,...,s, ( ) η j P j (x)l i (x)dx P j (τ)f(σ( + τh))dτ b i = η j P j (τ) b i = l i (τ)f(σ( + τh))dτ, b i P j (x)l i (x)dx f(σ( + τh))dτ i =,...,s. Consequenly, for any choice of he abscissae {c i }, HBVM(,s) provide he same polynomial σ as he opimal EPCMs (2) of order 2s [7]. 9 Conversely, an EPCM is opimal (i.e., i has order 2s) only when 9 In his sense, hey are equivalen, even hough hey generae differen discree problems. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

11 Hamilonian Boundary Value Mehods 27 he abscissae c,...,c s define a quadraure formula of order a leas 2s, whereas differen choices resul in mehods of lower order [7, Theorem ]. Remark 6. We also observe ha, due o he choice of he shifed Legendre polynomial basis (see (24)) HBVM(,s) = lim k HBVM(k,s), whaever is he choice of he fundamenal abscissae {c i }. Consequenly, for all k large enough, so ha he Maser Funcional Equaion (2) holds rue (e.g., in he case of a polynomial Hamilonian H(y)), all HBVM(k,s) provide he same polynomial σ of degree s, independenly of he choice of he abscissae {c i }. Hence, hey are equivalen o each oher. This resul doesn change in he case where H(y) is no a polynomial, provided ha H(y) is sufficienly differeniable. In his case, in fac, one formally obains, in place of he Maser Funcional Equaion (2), an equaion of he form σ k = L(J H;h)σ k + ψ k (h), (3) where ψ k (h) = O(h q k s+2 ), q k being he degree of precision of he quadraure a he righ-hand side in (6), so ha q k as k. From (2) and (3), one hen obains ha as h, assuming ha f is Lipschizian wih consan µ, and for a suiable consan M independen of h: i.e., σ k σ hµm σ k σ + ψ k (h), σ k σ ( hµm) ψ k (h) = O(h q k s+2 ), k. One hen concludes ha, when using finie precision arihmeic, σ k is indisinguishable from σ, for all k large enough. Example. As previously menioned, for he mehods sudied in [2], based on a Lobao disribuion of he nodes {c =,c,...,c s } {ĉ,...,ĉ k s }, one has ha deg(σ) = s, so ha he order of HBVM(k,s) urns ou o be 2s, wih a quadraure saisfying B(2k). Example 2. For he same reason, when one considers a Gauss disribuion for he abscissae {c,...,c s } {ĉ,...,ĉ k s }, one also obains a mehod of order 2s wih a quadraure saisfying B(2k). This case will be furher sudied in Secion 5. Remark 7. Finally, we also menion ha, from Remark 4, HBVM(k,s) are symmeric mehods, 2 provided ha he abscissae { i } (see (25)) are symmerically disribued (see also [2]). 4 Generalizaion of Hamilonian BVMs The approach ha has allowed he consrucion of mehods ha conserve energy funcions of polynomial ype is quie general: ha is, by no means i depends on he paricular vecor space generaing he curve σ() nor on he quadraure echnique used. As was emphasized in [, Secion 2], i solely relies on he following wo ingrediens: he definiion of discree line inegral and he exended collocaion condiions (3), which zero he line inegral (6). Therefore, in a more general conex, his procedure can be formalized as follows. One firs picks a curve σ( + τh), τ [,], joining wo poins of he phase space y = σ( ) and y = σ( + h). Such a curve is assumed o lie in a proper finie dimensional vecor space W = span{p (x),...,p s (x)}, where now P j (x), j =,...,s, are any linearly independen funcions. Therefore he curves σ() and σ() will admi an expansion in he form (5). 2 According o he ime reversal symmery condiion defined in [3, p. 28]. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

12 28 L. Brugnano, F. Iavernaro, and D. Trigiane The fundamenal hypohesis, for his approach o work, is ha he choice of W mus guaranee ha he funcions P j (τ) H(σ( + τh)) appearing in (9) (and hence σ() T H(σ())) be elemenary inegrable, ha is hey are required o admi a primiive ha can be expressed in erms of elemenary funcions. If his is he case, all he seps performed o obain (6) may be repeaed wih he inegral subsiued by he primiive. This represens a generalizaion of wha done for polynomial Hamilonian funcions no only because he vecor space W may be generaed by non-polynomial funcions bu also because he analyic soluion of he line inegral may be carried ou by any available echnique. Hereafer, we repor a couple of examples in he class (24) A mehod of order wo We consider a separable Hamilonian funcion (for simpliciy we assume m = ) Le σ() be he segmen joining y = (q,p ) T o y = (q,p ) T : H(q,p) = V (p) U(q). (32) σ( + τh) = y + τ(y y ). We have c =, c =, and he corresponding mehod (24) becomes: q q h V (p + τ(p p ))dτ = p p = h U (q + τ(q q ))dτ V (p ) V (p ) p p U(q ) U(q ) q q. (33) Formula (33) is one of he simples discree gradien mehods due o Ioh and Abe [2], whose general form, for non-separable Hamilonian funcions wih one degree of freedom, reads q q H(q,p ) H(q,p ) h = J q q p p H(q,p ) H(q,p ). (34) h p p The vecor appearing a he righ-hand side of (34) is obained by replacing he parial derivaives of H(q, p) wih incremens along he q and p axes. Mehod (34) is in general firs order and nonsymmeric. However, when confined o separable Hamilonian sysems, i urns ou o be second order and symmeric A mehod of order four To consruc a mehod of order four in he form (24) applied o (32), we pick a curve σ() of degree wo, based upon he abscissae c =, c = /2, and c 2 =. Such a mehod has been already described in [] for polynomial Hamilonian funcions: here we consider is generalizaion o he non-polynomial case. Seing Y = (q /2,p /2 ) T and, observing ha Y 2 = (q,p ) T, he wo componens of he curve σ( +τh) are ( ) ( σ ( + τh) 2(q 2q /2 + q )τ 2 ) (3q 4q /2 + q )τ + q =. (35) σ 2 ( + τh) 2(p 2p /2 + p )τ 2 (3p 4p /2 + p )τ + p 2 While he mehod in Secion 4. is equivalenly obainable by applying eiher (24) or (6), he same is no rue for he fourh-order mehod derived in Secion 4.2 where, he use of he Lagrange basis, would produce a coefficien b 2 = (appearing as a denominaor in he resuling formulae (2)). 22 A generalizaion of (34) inroduced in [3] also becomes mehod (33) when applied o Hamilonian funcions in he form (32). c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

13 Hamilonian Boundary Value Mehods 29 Consequenly, (24) becomes Y Y 2 q /2 p /2 q p = = q p q p + h + h ( 3 2 τ )V (σ 2 ( + τh))dτ ( 3 2 τ )U (σ ( + τh))dτ V (σ 2 ( + τh))dτ U (σ ( + τh))dτ Subsiuing (35) ino (36) we obain a sysem in he unknowns q /2, p /2, q, p. Looking a (36), we realize ha even in he simpler case of a sysem deriving from a Hamilonian funcion in he form (32), he elemenary inegrabiliy of he inegrals in (24) is no a priori guaraneed. This means ha, in his case, we canno arrive a a general formula analogous o (33), in erms of U(q) and V (p). On he oher hand, in several cases of ineres, such primiive can be explicily compued: hereafer we repor a significan example, which we shall use laer in he numerical ess in Secion 5. Example 3. The role of his example is also o show ha, when finie precision arihmeic is used, i may be no convenien o use he infinie version of he mehods, even if he inegrals can be analyically evaluaed. This will be eviden from he numerical resuls in Secion 5.4. The sysem we consider is he one defined by he Hamilonian funcion H(q,p) = a(log q q) + b(log p p), (37) where a and b are posiive consans. The associaed sysem () reads ( ) ( ) q = b p, ṗ = a q. (38)., (36) This sysem is sricly relaed o he Loka-Volerra model { q = bq ( p), ṗ = ap( q), (39) in ha sysem (39) may be recas as he Poisson sysem ẏ = η(q,p) J H(y), where η(q,p) = qp is called inegraing facor. Sysems (39) and (38) share he same Hamilonian funcion (37) as firs inegral and, consequenly, hey share he same curves as rajecories in he phase plane. Mehod (36) applied o (39) reads q /2 q h/2 p /2 p h/2 = b b log( p /p ) + b p 8p /2 +7p p 2p /2 +p 2 C p 2p /2 +p ( arcanh( 3p +4p /2 p ) arcanh( p ) 4p /2 +3p ) C C a 3 4 a log( q /q ) a q 8q /2 +7q q 2q /2 +q 2 C 2 q 2q /2 +q ( arcanh( 3q +4q /2 q ) arcanh( q ) 4q /2 +3q ) C 2 C 2, (4) q q h p p h = [ b 2b arcanh( p 4p /2 +3p ) + arcanh( 3p ] 4p /2 +p ) C C C [ a + 2a arcanh( q 4q /2 +3q ) + arcanh( 3q ] 4q /2 +q ) C 2 C 2 C 2, (4) c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

14 3 L. Brugnano, F. Iavernaro, and D. Trigiane where C = (p 2 + 6p 2 /2 + p2 8p p /2 2p p 8p /2 p ) /2, C 2 = (q 2 + 6q 2 /2 + q2 8q q /2 2q q 8q /2 q ) /2. 5 HBVMs based upon Gauss quadraure As anicipaed in Example 2, we now sudy he properies of he HBVM(k,s) which is defined over he se of k disinc abscissae, {,..., k } {c,...,c s } {ĉ,...,ĉ k s }, coinciding wih he Gauss-Legendre nodes in [, ], i.e., he roos of he shifed Legendre polynomial of degree k. The corresponding polynomial σ has hen degree s. By virue of Theorems 2 and 3 (see also Remark 7), such mehods are symmeric, perfecly A-sable, and of order 2s. They reduce o Gauss- Legendre collocaion mehods, when k = s, and are exac for polynomial Hamilonian funcions of degree ν, provided ha k νs 2. (42) By recalling wha saed in Remark 6, for all k sufficienly large so ha (6) holds, HBVM(k,s) based on he k Gauss-Legendre abscissae in [,] are equivalen o HBVM(k,s) based on k + Lobao abscissae in [,] (see [2]), since boh mehods define he same polynomial σ of degree s. 23 As maer of fac, we have run HBVM(k,s) based on Gauss-Legendre nodes, and HBVM(k,s) based on he Lobao nodes, obaining he same resuls on he polynomial es problems repored in [2], which are briefly recalled in he sequel. 5. Tes problem Le us consider he problem characerized by he polynomial Hamilonian (4.) in [5], H(p,q) = p3 3 p 2 + q6 3 + q4 4 q , (43) having degree ν = 6, saring a he iniial poin y (q(),p()) T = (,) T, so ha H(y ) =. For such a problem, in [5] i has been experienced a numerical drif in he discree Hamilonian, when using he fourh-order Lobao IIIA mehod wih sepsize h =.6, as confirmed by he plo in Figure. When using he fourh-order Gauss-Legendre mehod he drif disappears, even hough he Hamilonian is no exacly preserved along he discree soluion, as is shown by he plo in Figure 2. On he oher hand, by using he fourh-order HBVM(6,2) wih he same sepsize, he Hamilonian urns ou o be preserved up o machine precision, as shown in Figure 3, since such mehod exacly preserves polynomial Hamilonians of degree up o 6. In such a case, he numerical soluions obained by using he Lobao nodes {c =,c,...,c 6 = } or he Gauss-Legendre nodes {c,...,c 6 } are he same. 5.2 Tes problem 2 The second es problem, having a highly oscillaing soluion, is he Fermi-Pasa-Ulam problem (see [8, Secion I.5.]), defined by he Hamilonian H(p,q) = 2 m ( p 2 2i + p 2 ω 2i) 2 + i= 4 m m (q 2i q 2i ) 2 + (q 2i+ q 2i ) 4, (44) i= i= 23 In he non-polynomial case, hey converge o he same HBVM(, s), as k. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

15 Hamilonian Boundary Value Mehods 3 x 6 H Figure : Fourh-order Lobao IIIA mehod, h =.6, problem (43): drif in he Hamilonian. 3.5 x H Figure 2: Fourh-order Gauss-Legendre mehod, h =.6, problem (43): H 6. 2 x 6 H Figure 3: Fourh-order HBVM(6,2) mehod, h =.6, problem (43): H 6. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

16 32 L. Brugnano, F. Iavernaro, and D. Trigiane Table : Maximum difference beween he numerical soluions obained hrough he fourh-order HBVM(k, 2) mehods based on Lobao abscissae and Gauss-Legendre abscissae for increasing values of k, problem (45), 3 seps wih sepsize h =.. k h = wih q = q 2m+ =, m = 3, ω = 5, and saring vecor p i =, q i = (i )/, i =,...,6. In such a case, he Hamilonian funcion is a polynomial of degree 4, so ha he fourh-order HBVM(4,2) mehod, eiher when using he Lobao nodes or he Gauss-Legendre nodes, is able o exacly preserve he Hamilonian, as confirmed by he plo in Figure 6, obained wih sepsize h =.5. Conversely, by using he same sepsize, boh he fourh-order Lobao IIIA and Gauss-Legendre mehods provide only an approximae conservaion of he Hamilonian, as shown in he plos in Figures 4 and 5, respecively. 5.3 Tes problem 3 (non-polynomial Hamilonian) In he previous examples, he Hamilonian funcion was a polynomial. Neverheless, as observed above, also in his case HBVM(k,s) are expeced o produce a pracical conservaion of he energy when applied o sysems defined by a non-polynomial Hamilonian funcion ha can be locally well approximaed by a polynomial. As an example, we consider he moion of a charged paricle in a magneic field wih Bio- Savar poenial. 24 I is defined by he Hamilonian [2] H(x,y,z,ẋ,ẏ,ż) = (45) [ ( ẋ α x ) 2 ( ) ] 2 + ẏ α y 2 + (ż + α log( )) 2, 2m 2 wih = x 2 + y 2, α = eb, m is he paricle mass, e is is charge, and B is he magneic field inensiy. We have used he values m =, e =, B =, wih saring poin x =.5, y =, z =, ẋ =., ẏ =.3, ż =. By using he fourh-order Lobao IIIA mehod, wih sepsize h =., a drif is again experienced in he numerical soluion, as is shown in Figure 7. By using he fourh-order Gauss-Legendre mehod wih he same sepsize, he drif disappears even hough, as shown in Figure 8, he value of he Hamilonian is preserved wihin an error of he order of 3. On he oher hand, when using he HBVM(6,2) mehod wih he same sepsize, he error in he Hamilonian decreases o an order of 5 (see Figure 9), hus giving a pracical conservaion. Finally, in Table we lis he maximum absolue difference beween he numerical soluions over 3 inegraion seps, compued by he HBVM(k,2) mehods based on Lobao abscissae and on Gauss-Legendre abscissae, as k grows, wih sepsize h =.. As expeced, he difference ends o, as k increases, since he wo sequences of mehods end o he same limi, given by he HBVM(,2) (see (24) wih s = 2). 24 This kind of moion causes he well known phenomenon of aurora borealis. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

17 Hamilonian Boundary Value Mehods 33 2 x H H Figure 4: Fourh-order Lobao IIIA mehod, h =.5, problem (44): H H 3. 4 x 4 2 H H Figure 5: Fourh-order Gauss-Legendre mehod, h =.5, problem (44): H H 3..5 x 4.5 H H Figure 6: Fourh-order HBVM(4,2) mehod, h =.5, problem (44): H H 4. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

18 34 L. Brugnano, F. Iavernaro, and D. Trigiane 6 x H H Figure 7: Fourh-order Lobao IIIA mehod, h =., problem (45): drif in he Hamilonian. 5 x 3 H H Figure 8: Fourh-order Gauss-Legendre mehod, h =., problem (45): H H 3. 2 x 5 H H Figure 9: Fourh-order HBVM(6,2) mehod, h =., problem (45): H H 5. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

19 Hamilonian Boundary Value Mehods Tes problem 4 (non-polynomial Hamilonian) We finally solve he Hamilonian sysem (38) by using he Iho-Abe mehod (33), he fourh-order formula (4) (4), and he HBVM(,2), which has order four and degree of precision (ha is, according o (42), i precisely conserves he energy of polynomial Hamilonians of degree up o ). We have se a = b = in formula (37), and inegraed over a ime inerval [,5] wih sepsize h =.5 and (q,p ) = (.5,.5) as iniial condiion. Figure repors he numerical Hamilonian funcion associaed wih he hree mehods. The occurrence of jumps in he firs wo graphs (lef picure) is due o he fac ha boh formulae (33) and (4) (4) may become ill-condiioned for cerain values of he sae vecor. For example (see Figure ), a he wo consecuive imes = and = 283, he sae vecors associaed wih he Ioh-Abe mehod (33) are, respecively, [q,p ] ( ,.42656) T, [q 2,p 2 ] ( ,.67353) T, which shows ha q may be very close o q 2 even for large values of h. This causes some cancellaion in he subracion a he righ-hand side of (33) and, hence, a jump of he subsequen branch of he numerical rajecory on a differen level curve. However, since, in general, he numerical rajecory densely fills he level curve H(y) = H(y ), i may be argued ha he occurrence of such jumps are sysemaic and frequen when he dynamics is raced over a long ime. The use of finie arihmeic evenually desroys he heoreical conservaion propery. A similar argumen may be applied o discuss he behavior of he fourh-order mehod (4) (4). Alhough he HBVM mehod does no provide a heoreical conservaion of he energy, as is he case for he above cied mehods, is behavior in finie arihmeic would sugges he opposie (see he righ picure in Figure ), as already emphasized a he beginning of Example 3. 6 Conclusions In his paper, he newly inroduced Hamilonian Boundary Value Mehods (HBVMs), a class of numerical mehods able o exacly preserve polynomial Hamilonians of any degree, have been re-derived in a unifying framework. Such framework relies on he use of line inegrals, which are approximaed by suiable discree counerpars (acually, hey are exac, when he Hamilonian is a polynomial). In his conex, he limi of he mehods, as he number of he so called silen sages ends o infiniy, is easily obained. When he underlying polynomial basis upon which he HBVM is consruced is he Lagrange basis, such limi formulae coincide wih he recenly inroduced Energy Preserving varian of Collocaion Mehods; if insead one uses he shifed Legendre polynomial basis, he corresponding HBVMs have he highes possible order and so do heir limi formulae, he Infiniy Hamilonian Boundary Value Mehods, independenly of he considered abscissae. Any limi formula, when discreized, fall ino he HBVMs class. Possible exensions of he approach have been also skeched, along wih a number of numerical ess. Such ess confirm ha, in he limi of he silen sages ending o infiniy, all HBVMs wih s (unknown) fundamenal sages end o he same limi mehod, which is characerized by he eigenfuncion (relaive o he uni eigenvalue) of a cerain operaor, which is independen of he choice of he abscissae. 7 Acknowledgemens This work emerged from fruiful discussions wih Erns Hairer during he inernaional conference IC- NAAM 29, Rehymno, Cree, Greece, 8 22 Sepember 29. I has also benefied from his many subsequen commens. We wish also o hank an anonymous referee for he useful commens. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

20 36 L. Brugnano, F. Iavernaro, and D. Trigiane H H =283 H H 4 2 = Figure : Lef picure: absolue error of he Hamilonian funcion (37) evaluaed along he numerical soluions compued by he Ioh-Abe mehod (33) (lower curve) and formula (4) (4) (upper curve). The jumps are sympomaic of ill-condiioning of he formulae for cerain values of he soluion. Righ picure: he same kind of plo produced by he HBVM formula of order 4 and k = Gaussian abscissae ( H H 2 ) =283.5 p = q Figure : Trajecory in he phase plane compued by he Ioh-Abe mehod (33). The small circles locae he soluion a he wo consecuive imes = and = 283. The very close values of he variable q for such wo poins causes loss of significan digis in he subsequen branch of he rajecory. c 2 European Sociey of Compuaional Mehods in Sciences and Engineering (ESCMSE)

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