Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems

Transcription

1 Celestial Mecanics and Dynamical Astronomy manuscript No. (will be inserted by te editor) Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems Sergio Blanes Arie Iserles Received: date / Accepted: date Abstract We consider Sundman and Poincaré transformations for te long-time numerical integration of Hamiltonian systems wose evolution occurs at different time scales. Te transformed systems are numerically integrated using explicit symplectic metods. Te scemes we consider are explicit symplectic metods wit adaptive time steps and tey generalise oter metods from te literature, wile exibiting a ig performance. Te Sundman transformation can also be used on non-hamiltonian systems wile te Poincaré transformation can be used, in some cases, wit more efficient symplectic integrators. Te performance of bot transformations wit different symplectic metods is analysed on several numerical examples. Keywords Symplectic Integrators Adaptive time step Hamiltonian systems 1 Introduction We consider te long-time numerical integration of dynamical systems wose evolution is governed by te Hamiltonian function H = 1 2 p p + V (q), (1) SB acknowledges te support of te Ministerio de Ciencia e Innovación (Spain) under te coordinated project MTM C3 (co-financed by te ERDF of te European Union). Sergio Blanes Instituto de Matemática Multidisciplinar, Universitat Politècnica de Valencia, E-4622 Valencia, Spain. serblaza@imm.upv.es Arie Iserles DAMTP, Centre for Matematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 WA, United Kingdom A.Iserles@damtp.cam.ac.uk

2 2 Sergio Blanes, Arie Iserles wit q, p R d, and suc tat te system evolves at different time scales. Effective long-time numerical integration requires to use adaptive time steps, consequently we wis to implement symplectic metods wit adaptive time steps. To tis purpose, we consider bot te Sundman and te Poincaré transformations. Te scemes proposed generalise oter metods from te literature and allow us to build new explicit adaptive symplectic scemes exibiting a ig performance. Numerical solution of Hamiltonian systems is frequently carried out by symplectic integrators (SIs) due to teir good performance in moderate and longtime integration Hairer et al (26); Iserles (28); Leimkuler and Reic (24); McLaclan and Quispel (22); Sanz-Serna and Calvo (1994). SIs belong to te family of Geometric Numerical Integrators (metods wic preserve important qualitative and geometric properties of te underlying differential system) and are arguably te most popular metods in tis class. Certain qualitative properties of te evolution, like symplecticity, are preserved and, in general, tey exibit smaller error growt along te numerical trajectory. Splitting metods applied to separable Hamiltonian systems are te most frequently used symplectic integrators. Denoting by Φ H t te t-flow of te system, z(t) = Φ H t (z()), wit z = (q, p), te exact flow for one time step,, can be approximated by a composition of te flows associated to te kinetic energy Φ [T ] and potential energy, Φ [V ]. Te second order V T V leap-frog/störmer/verlet metod is given by te composition Ψ Φ [V ] /2 Φ[T ] Φ[V ] /2 : p n+1/2 = p n qv (qn) 2 q n+1 = q n + p n+1/2 p n+1 = p n+1/2 2 qv (q n+1). (2) (Analogously, one can also take te T V T version.) More accurate results can be obtained by using iger order metods. Splitting symplectic metods exist for a range of problems wit different structure, leading to ig-performance numerical scemes. For instance, splitting Runge Kutta Nyström (RKN) metods are tailored for Hamiltonian systems wit quadratic kinetic energy, and tere are also splitting metods tailored for perturbed systems were te Hamiltonian takes te form ( ) 1 H = H (p, q) + εh I (q) = 2 p p + V (q) + εv I (q), (3) were H is exactly solvable and < ε 1, an important suc case being perturbed Kepler problems. We can ten adapt te Verlet metod (2) to te composition Ψ [2] Φ[H I ] /2 Φ[H ] Φ [H I ] /2 = ΦH + O(ε 3 ) (4) to gain a factor ε in te accuracy Kinosita et al (1991); Wisdom and Holman (1991). A systematic procedure to build iger-order metods was introduced in Creutz and Gocksc (1989); Suzuki (199); Yosida (199), and since ten considerable effort as been expanded in order to construct new symplectic integrators wit smaller local errors at a given computational cost. A significant number of new split symplectic integrators ave been publised for a wide number of problems

3 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 3 wit different structures like RKN metods or metods for near-integrable systems (see Blanes et al (28); Hairer et al (26); Laskar and Robutel (21); Leimkuler and Reic (24); McLaclan (1995a); McLaclan and Quispel (22, 26); Sanz-Serna and Calvo (1994) and references terein). Unfortunately, te ig performance of symplectic integrators is typically associated wit te use of constant time steps. In contrast, adaptive variable time-step metods are often superior to fixed time-step metods wen applied to problems wit varying evolutionary time scales, since tey lead to more regular problems wit reduced local errors. Once standard tecniques for canging te time step are included in a symplectic integrator, several of te favourable properties of symplectic metods are lost Calvo and Sanz-Serna (1993); Gladman et al (1991); Skeel (1993). Tis procedure to cange te time step does not destroy te symplectic structure but, rougly speaking, it can be considered as te exact solution of a perturbed time-dependent Hamiltonian, H(q, p) + δ H(q, p, t). (In te constant-step scenario H is independent of t.) Unless te time step is canged properly, secular error terms appear in te perturbed Hamiltonian and te errors in energy and position grow similarly to standard non-symplectic integrators: te advantages of symplecticity are tus lost. Integrators wit variable time step and non-secular error terms ave been proposed in te literature. Tey consist of a regularisation of te time dt dτ = g(q, p) (5) wit g a positive-definite function, wereby one integrates wit a constant time step in te fictive time, τ Blanes and Budd (25); Bond and Leimkuler (1998); Calvo et al (1998); Hairer (1997); Hairer and Söderlind (25); Holder et al (21); Huang and Leimkuler (1997); Mikkola (1997); Mikkola and Aarset (22); Mikkola and Tanikawa (1999); Preto and Tremaine (1999). Te Levi Civita or Kustaaneimo Stiefel regularizations for solving te Kepler problem consider of a time transformation of tis kind, wic is ten combined wit a canonical transformation, and tey ave been successfully used for many years in celestial mecanics Blanes and Budd (24); Stiefel and Sceifel (1971). We consider te Sundman and te Poincaré transformations in a general form wic include most of te existing metods as particular cases and improve teir performance for most problems. Tat performance teir accuracy and computational cost alike depends on te coice of te regularisation function, g, as well as in te way te system is split for its numerical integration. In section 2 we review te Sundman transformation and present new tecniques to construct efficient explicit metods. In section 3 we review te Poincaré transformation and present new splitting metods to solve efficiently te evolution for perturbed Hamiltonian systems or wit quadratic kinetic energy. In section 4 we present several numerical examples to illustrate ow te new metods apply as well as to illustrate teir relative performance.

4 4 Sergio Blanes, Arie Iserles 2 Explicit adaptive SIs using te Sundman transformation Te Sundman transformation takes te real time as a new coordinate and introduces a fictitious time, τ in te following manner, q g(q, p)p d p dτ = g(q, p) V (q). (6) t g(q, p) Tis equation loses te symplectic structure and, in addition, most explicit metods available in te literature for te numerical integration of te Hamiltonian (1) cannot be used or are igly inefficient in tis setting. It as been sown, owever, tat if a reversible time-symmetric metod is used, many good properties for long-time integrations are retained. Note tat if te regularisation function g is frozen at eac step (or at eac stage inside a step) ten we can use an explicit symplectic integrator and te coordinates (q, p) will evolve troug a symplectic transformation. Tus, it is only necessary to look for an appropriate procedure to freeze te function g in a manner wic renders te wole numerical integration reversible and time-symmetric, leading to an adaptive symplectic and time-reversible integrator. We introduce an auxiliary scalar variable, z, and a positive-definite and invertible function, G(z), suc tat G(z) = g(z). To avoid some instabilities, we differentiate tis equation so, te system to be solved is d dτ q G(z)p p G(z) V (q) = t G(z) z ( G(g(q, p)) 1 ) ( ) G(z) g(q, p) f were f J H(q, p) and J denotes te standard (2d) (2d) symplectic matrix. Numerical integration of tis problem as been considered in te literature for different coices of te function G(z), te monitor function, g(q, p) and by splitting te system in different ways. Since G(z)/g(q, p) = 1 is a first integral of te system, it as been usual to substitute G(z) by g(q, p) in different places on te rigt-and side of te equation, and in particular in te equation for dz/dτ. For example, Huang and Leimkuler (1997); Leimkuler and Reic (24) employs te coice G(z) = z wile Hairer and Söderlind (25); Mikkola and Aarset (22) (see also (Hairer et al, 26, Sec. VIII.3.2)) use G(z) = 1/z. Tese are two particular cases of te more general function G(z) = z α, were te coice α = corresponds to no regularisation. As we will see, te coice of G(z) plays an important role in te performance of te algoritm. Te monitor function g is also very important in constructing efficient algoritms Blanes and Budd (25). For te system (1), fast evolutions usually occur close to te singularities of te potential. Tus, it makes sense to consider a function depending only on te coordinates, say, g(q). However, oter coices are also valid and te main differences impacts on te complexity wen computing te equation for z (were te prime denotes te derivative wit respect to τ). Finally, one as to decide ow to split te system in order to use a timesymmetric splitting metod wic preserves symplecticity for te coordinates q, p. For example, in Hairer and Söderlind (25) (see also (Hairer et al, 26, Sec. (7)

5 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 5 VIII.3.2)), were G(z) = 1/z, g(q) = (q q) γ (for problems were te singularity of te potential is at te origin), te function G(z) is replaced by g(q) in te equation of z, and te system is split as follows 1 q z p d p 1 V (q) = z +. (8) dτ t 1 z z γ q p q q }{{}}{{} f B f A Te system is ten solved using te composition Ψ = Φ (B) /2 Φ(A) Φ (B) (9) /2 wit δτ. Here, Φ (B) is used only to cange te time step for te real time. /2 Consequently, one can replace Φ (A) by te desired symplectic integrator to advance q, p and t. Tis is a second-order approximation in te auxiliary variable z, wic is not relevant to te accuracy of te metod. Te practical order of accuracy follows from te order of te metod used to approximate Φ (A). Tis is a simple time-reversible explicit symplectic integrator and it is peraps te most efficient algoritm for low to moderate accuracies. However, if accurate results are desired, ig-order metods are necessary. Hig-order metods usually require many evaluations per step and demonstrate teir superior performance wen relatively large time steps are used in lieu of te extra per-step cost. However, tis sceme canges te time step only wen te flow Φ (B) is computed, ence to use a relatively large time step can be dangerous wen te system moves close to te singularities because te sceme as no opportunity to adapt te time step. We ten propose to split te system as follows, q G(z)p d p G(z) V = + +. (1) dτ t G(z) ( z G(g) 1 ) G(z)( g f) }{{}}{{}}{{} f A f B f C Te rigt and side of te equation for dt/dτ could be put into any of te tree vector fields and eac part would be still exactly solvable but it is advantageous to keep it in eiter f A or f C. We propose to use te following symmetric second order composition S [2] = Φ(A) /2 Φ(B) /2 Φ(C) Φ (B) /2 Φ(A) /2. (11) Oter sequences of te maps are also valid. Next, to reac ig accuracy one can use tis metod as te basic metod to build iger order metods by composition, i.e. S [2p] = S [2] α m S[2] α 1

6 6 Sergio Blanes, Arie Iserles were te coefficients α i are cosen appropriately to build efficient metods of order 2p in te time step Hairer et al (26); McLaclan (1995b); McLaclan and Quispel (22); Soproniou and Spaletta (25). Notice tat in tis approac te time step is adjusted along te internal stages of te metod instead of adjusting te time step by te end of te step. Constructing metods of order four or six, it is usually more efficient to consider a composition of a first order metod, say, Φ = Φ (A) Φ = Φ(C) Φ (B) Φ (B) Φ (C) and its adjoint Φ (A) in an appropriate sequence Blanes and Moan (22). Te coice of te monitor function We ave to remark tat te coice of te monitor function g is essential in constructing efficient metods Blanes and Budd (25); Budd et al (21); Calvo et al (1998); Hairer (1997); Hairer et al (26). Te numerical accuracy of a given sceme can be similar for different coices of g wile its computational cost can cange drastically. Ten, one as to coose te most appropriate monitor function wic must also satisfy te required conditions (e.g. preserving scaling invariance of te system or any reversing symmetry) and tis as to be done in tandem wit te coice of te function G(z) in order to obtain simple and easy to compute expression for z. For example, if te potential function is given by V (r) wit r = (q q) 1/2 ten it seems appropriate to coose g = g(r) because ten, replacing G(z) by g(r), te equation for z becomes z = F (r)(q p), wit F (r) a given scalar function. Anoter important property to be analysed is ow te coice of te function G(z) affects te performance of te numerical metods. To sum up, te Sundman transformation allows us to build explicit time reversible adaptive integrators to ig order. However, RKN metods or metods tailored for near integrable systems can not be used, unless te system is split as in (8), wic as some limitations as previously mentioned. To circumvent tis problem, we will consider te Poincaré transformation. 2.1 Some oter particular cases Te sceme tat we ave proposed generalises oter scemes used to numerically solve te system (6) wic ave appeared in te literature. In Leimkuler and Reic (24) an algoritm (previously obtained in Huang and Leimkuler (1997)) is presented wic corresponds to G(z) = z, considering z = g in te equation of z, and employing te splitting q zp d p z V = + dτ t z z g( g f) }{{}}{{} f A f B (12) were we let g(q, p) = f. Wit tis splitting and te coice of g, te final algoritm as to be necessarily implicit. In Leimkuler and Reic (24) te - flow Φ B is approximated by te first order explicit Euler metod, ΦB, and its

7 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 7 implicit adjoint metod, Φ B, and constructs te following implicit symmetric second order metod S = ΦB /2 Φ A Φ B /2. (13) Tis metod requires te computation of te Hessian of te potential and for tis reason it can be computationally costly for many problems. In Mikkola and Aarset (22), te autors consider a monitor function depending only on te coordinates, g(q), let G(z) = 1/z (altoug te metod is presented in terms of te function Ω(q) = 1/g(q)) so tat z = ( qg p)/g and te algoritm proposed is basically equivalent to considering te splitting 1 q z p d p g V = dτ t 1 +. (14) z z 1 ( qg p) g }{{}}{{} f f B A Note tat te equation for f B is exactly solvable because te equation for z is linear in p. Wit tis splitting, in Mikkola and Aarset (22) te autors construct te sceme S = Φ A /2 ΦB Φ A /2 (15) wic is ten taken as te basic metod to be used wit te Gragg Bulirsc Stoer extrapolation metod. However, an extrapolation metod based on a symmetric second order symplectic integrator is not appropriate because it loses symplecticity and its benefits for long-time integration. One can obtain, owever, pseudo-symplectic integrators by extrapolation if te basic metod is of iger order Blanes et al (1999); Can and Murua (2) and qualitative properties are preserved up to iger order tan te order of te metod. We recommend is procedure only wen ig accuracy is desired, say, to nearly round-off level and relatively sort time integrations. Alternatively, one can solve te system (14) by standard splitting metods for separable systems. Here, bot q and p evolve troug symplectic maps only if g V is te gradient of a potential, and tis deserves furter investigation. 3 Te Poincaré transformation Given a Hamiltonian H(q, p), te Poincaré transformation introduces te following new Hamiltonian in te extended pase space, H(q, p, q t, p t ) = g(q, p)(h(q, p) + p t ) (16) were q t t and p t is its conjugate momentum. Note tat H is autonomous, terefore H does not depend on q t, and p t is a constant of motion wose value is usually taken as p t = H wit H H(q(), q()), consequently H = along te solution. Te structure of te Hamiltonian H differs in general from te structure of H, i.e. if H is separable in solvable parts, in general, H will not be separable in solvable parts.

8 8 Sergio Blanes, Arie Iserles Te Poincaré transformation applied to (1) corresponds to solving te evolution for te Hamiltonian (16), wic we split ( ) H(q, p, q t, p t 1 ) = g(q, p) 2 p p + p t + g(q, p)v (q). (17) Taking te monitor function ( ) γ 1 1 g(q, p) = 2 p p + p t and considering 1 2 p p + p t = V (q), te Hamiltonian being solved becomes separable, H(q, p, q t, p t ) = ( 1 2 p p + p t ) γ ( V (q)) γ. (18) Tis is equivalent to te Hamiltonian used in Preto and Tremaine (1999), were it was taken wit te monitor function g = f( 1 2 p p + p t ) f( V (q)) H(q, p) + p t (19) and it was assumed tat f(u) = u γ for some γ. Te particular case f(u) = log u was previously considered in Mikkola (1997); Mikkola and Tanikawa (1999). f(u) must be an analytic function, were te apparent singularity at H = p t is removable: take te Taylor series of f about te point V (q) and use 1 2 p p+p t = H(q, p)+p t V (q). Since numerically 1 2 p p+p t V (q), we ave g f ( V (q)), were f(u) must be a function suc tat f (u) >, wit no singularities in te domain of interest. Te system is still separable, but it is no longer quadratic in te kinetic energy and RKN metods cannot be used. In practice, we ave observed in some numerical examples tat 1 2 p p + p t (or V (q)) can be negative along te trajectory and te algoritm breaks down. We can circumvent tis problem if we consider instead te splitting ( γ H(q, p, q t, p t 1 ( ) = p) 2 p V (q) p t) γ. (2) Separable Hamiltonian wit kinetic energy quadratic in momenta If we consider te Hamiltonian (1) and a monitor function depending only on te coordinates, g(q), te Poincaré transformation leads to te following enlarged Hamiltonian system wic we split as follows H = K 1 (q, p) + K 2 (q, p t ) = 1 2 g(q)p p + g(q)(v (q) + p t ). (21) Now, K 2 (q, p t ) is exactly solvable and K 1 (q, p) is, in general, not solvable but it is a product of solvable parts. On te oter and, it is important to notice tat K 1 is quadratic in momenta. If {K 1, K 2 } denotes te Poisson bracket of te functions K 1, K 2, it is easy to see tat {K 2, {K 2, {K 2, K 1 }}} =, displaying an algebraic structure similar to Nyström problems. Tus, symplectic splitting Nyström metods can be used, and tis is usually more accurate and stable for tese problems tan general symplectic integrators.

9 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 9 For tis purpose, we need an algoritm to approximate efficiently and accurately te evolution for te Hamiltonian K 1 = 1 2 g(q)p p. In Blanes and Budd (25) it is sown tat in some cases one can find a canonical transformation, (q, p) (Q, P) were q = Φ(Q), Φ (Q) T p = q, suc tat K 1 depends only on te new momenta. Unfortunately, tis occurs rarely and we look for an alternative procedure. Tis can be done in different ways, and te most appropriate one will depend on te particular problem. If g(q) is ceap to compute function e.g. te singularity of te potential involves only few contributions (i.e. if it involves only two bodies in a N-body problem) we can numerically solve efficiently te equations for K 1 to roundoff accuracy at eac stage. For instance, we can use a logaritmic metod. Notice tat te evolution for one time step τ of K 1 = 1 2 g(q)p p is equivalent to te evolution for one time step δ = E 1 τ wit E 1 = K 1 of te Hamiltonian K L = K L,1 + K L,2 = log(g(q)) + log(p p), (22) (during tis evolution K 1 is kept constant) as is immediate to see from te Hamiltonian equations. At near collisions, 1 2 p p and g take large and small values, respectively, but logaritmic functions reduce tese large values and maintain te effectiveness of te splitting. Te Hamilton equations for K L,1 and K L,2 are: dq dτ =, dp dτ = 1 g qg, dq dτ = 2 p p p, (23) dp dτ =. In general, we can take g in a very simple form, just by considering te singularities of te potential. If te system is scaling invariant (or close to it) as appens for most practical potentials wit singularities or strong interactions, te optimal regularisation function g is closely related wit te coice wic preserves tis scaling invariance Blanes and Budd (24, 25); Budd et al (21). Ten, if we take g = (q q) γ/2 (24) for an appropriate coice of γ, we ave 1 g qg = γ q q q rendering bot parts of (23) trivial to compute wit simple and ceap aritmetic operations, so te evolution for K 1 can be easily computed to roundoff accuracy wit marginal extra cost. It is important to bear in mind tat, solving K in (22), te value of E 1 canges from one stage to te next for te entire metod. As we ave already mentioned, to cange te Hamiltonian at eac step from information originating in te previous step introduces, in general, secular errors. To diminis tese errors to nearly roundoff, te system as to be numerically solved to a very ig accuracy, so tese errors can be sufficiently diminised for practical purposes. Tis can be acieved, e.g. using a ig order composition.

10 1 Sergio Blanes, Arie Iserles A nearly-integrable Hamiltonian Let us now consider te perturbed Hamiltonian H(q, p) = H (q, p) + εh 1 (q), (25) were H is exactly solvable and < ε 1, but occasionally te perturbation can approac a singularity and it can take large values. For tis problem it is convenient to split te Hamiltonian into te dominant part and te perturbation because te error of numerical metods reduces considerably, but we need to adapt te time step wen te perturbation is more relevant. Tis is te case, for instance, wit te N-body problem in te solar system, were te Hamiltonian can be split into pure Kepler problems and weak interactions between planets. In some cases, owever, close approaces between te planets or asteroids can occur and, in spite of ε being a small parameter, te Hamiltonian H 1 (q) can be nearly singular. In tis case it is convenient to introduce adaptivity. Tere are two natural coices for te regularisation function: g(h + p t ) and g(q). In te first case, te Hamiltonian to solve is (considering tat H +p t = εh 1 ) H = g(h + p t )(H + p t ) + εg( εh 1 )H 1. (26) Te Hamiltonian g(h + p t )(H + p t ) exibits te same difficulty as H, and it is also exactly solvable. However, for te coice g(u)u = u γ we ave found in several problems tat H +p t takes negative values. In tat case, one can integrate, backward in time, te Hamiltonian, H, in order to keep bot parts positive. Te evolution of a Hamiltonian system is unaltered by adding an arbitrary constant so, we can introduce a constant δ suc tat εh 1 + δ <, consequently H + p t >, and te new Hamiltonian to consider is H = g(h + p t )(H + p t ) g( εh 1 δ)( εh 1 δ), (27) wic as lost te structure of a perturbed system: te performance of te splitting metods tailored for perturbed systems is ten seriously degraded. However, if we take e.g. g(q) = (q q) γ, te Hamiltonian becomes H = (q q) γ (H + p t ) + ε(q q) γ H 1 (q), (28) wic retains te near integrable structure, but we need to solve accurately and efficiently te Hamiltonian K 1 = (q q) γ (H + p t ). (29) We can use te logaritmic metod as previously, te main trouble being tat H + p t must be positive definite (if not, we can take K 1 = (q q) γ ( H p t ) and integrate backward in time te Hamiltonian (q q) γ ( H p t )) and eac stage would need to evaluate te Hamiltonian H several times. If tis part is not computationally costly and te perturbation H 1 is te most costly part of te algoritm, tis splitting migt be useful because it benefits from te perturbed structure. We conclude wit an interesting open problem, efficient time-reversible SIs for Hamiltonians of te form H = H 1 (q, p)h 2 (q, p), were H 1 and H 2 are exactly solvable Hamiltonian functions.

11 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 11 4 Numerical Examples We consider te integration of several examples by considering bot te Sundman and te Poincaré transformations jointly wit appropriate splitting and composition metods. Given a symmetric second order (implicit or explicit) metod, S [2], we consider m-stage symmetric compositions of order n as follows S mn S [2] α 1 S[2] α 2 S[2] α k S[2] α k+1 S[2] α k S[2] α 2 S[2] α 1 wit m = 2k + 1. Te following composition metods are cosen for tat purpose S 5 4: te 5-stage fourt-order composition, S 9 6: te 9-stage sixt-order composition, S 17 8: te 17-stage eigt-order composition, and wose coefficients can be found in Hairer et al (26). In Soproniou and Spaletta (25) more elaborated compositions wit additional extra stages are obtained wit sligtly improved performance. Te following splitting metods are considered for problems were RKN metods or metods for near-integrable systems can be used RKN 6 4: te 6-stage fourt-order BAB metod from Blanes and Moan (22), RKN 11 6: te 11-stage sixt-order BAB metod from Blanes and Moan (22), M84: te 5-stage (8,4) BAB metod from McLaclan (1995a) for perturbed systems. 4.1 Te one-dimensional Kepler problem Let us consider te Kepler problem, wic possesses periodic orbits wit close approaces. Suc an orbit as a radial coordinate q wit associated momentum p wic evolves according to te Hamiltonian H = 1 2 p2 1 q + ε q 2, (3) were ε is te angular momentum. In numerical tests we take te initial conditions (q, p ) = (1, ) and ε =.1 (wic allows for very close approaces). We first study te performance of different algoritms wen using te Sundman transformation. We take G(z) = 1/z, integrate until t = 1 and measure te average relative error in energy wen considering te splitting sown in (8) (9) or wen considering (1) (11). In te first case, Te compositions are used to approximate Φ (A) in (9) as composition of te leapfrog sceme as te basic metod, or tey are used as compositions of te second-order sceme (11). Te time step is adjusted in all cases suc tat te final time is reaced using approximately 5 evaluations of te potential. Fig. 1 displays te results obtained wen considering te regularisation function g = q γ for different values of γ. It is clear from te results tat adjusting te time step troug te variable z along te internal stages of te metod allows us to greatly improve te performance of te metods once ig-order metods are used to reac ig accuracy. We next repeat te experiment for G(z) = z α and different values of α. We measure te accuracy of S 17 8 wen te monitor function g = q γ is used wit its

12 12 Sergio Blanes, Arie Iserles S (REL. ERROR) 5 7 S 5 4 S 9 6 (REL. ERROR) 5 7 S γ γ Fig. 1 Average error in energy for te numerical integration of te Hamiltonian (3) using te Sundman transformation wit te splitting from (1)-(11) (left panel) or wen considering (8)-(9) (rigt panel). Te following metods are compared: te leap-frog S 1 2 (dotted lines), S 5 4 (dased lines), S 9 6 (dotted-dased lines) and S 17 8 (solid lines). optimal value γ = 1.6 and te most accurate splitting (1) (11). Figure 2 sows te results obtained wen solving te equation for z in two different ways: ( z γ ) = α qγ/α 1 p z α (31) were q, p are taken constants along its integration wic as exact solution (broken lines) or, once we replace z α by q γ, z = γ α qγ+γ/α 1 p (32) (solid lines). We observe tat te two coices lead to considerably different results and it is clear tat te coice of te function G(z) can be relevant to te performance of te metods. Te usual coice G = 1/z is very close to optimal if one considers z = 1/q γ (it corresponds to taking α = 1 in (32)) as it as been frequently done in te literature, and it is clearly better tan te coice z = q γ (i.e. to take α = 1 in (32)). We ave repeated te numerical experiments for te more general one-dimensional potential, V = 1/r β ε/r 2β, for different coices of β and ε and te coice G = 1/z was laways very close to optimal one.

13 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 13.5 (REL. ERROR) α Fig. 2 Same as Fig. 1 for g = q γ and γ = 1.6 wen considering te splitting sown in (1)-(11) and te eigt-order metod S We take G(z) = z α for different values of α. Solid lines sow te results wen te equation (31) is used and broken lines sow te results wen te equation (32) is used. Let us now integrate te system using te Poincaré transformation and te regularisation function g = q γ. In tat case (21) takes te form K = T (q, p) + V (q, q t ) = 1 ( 2 qγ p 2 + q γ 1 q + ε ) q 2 + pt. (33) We take te same initial conditions (so, p t = ( 1 2 p2 q 1 + ε ) = 1 ε) and q 2 γ = 1.35, wic is rougly te optimal value for tis type of metods Blanes and Budd (25), and integrate until t = 1. We integrate tis system using te metod RKN 11 6 as follows Φ [6] = 12 i=1 Φ [T,n] a i Φ[V ] b i (34) were a 12 = and te last map in one step is reused in te following step (in practice, te cost corresponds to 11 stages), and were q n+1 = q n, Φ [V ] b i : p n+1 = p n b i V q, t n+1 = t n + b i q γ. We compare its performance versus te S 9 6 and S 17 8 scemes wic use te following generalization of te leapfrog metod, (35) Φ [2] = Φ[V ] /2 Φ[T,n] Φ [V ] /2 (36)

14 14 Sergio Blanes, Arie Iserles 3 (ERROR) RKN 11 6 S 9 6 S (t) (ERROR) RKN 11 6 S 9 6 S (t) Fig. 3 Error in energy (averaged every 1 steps) for different metods and using all of tem approximately te same number of evaluations. Te logaritmic Hamiltonian is solved using te S 9 6 metod (top figure) and te S 17 8 metod (bottom figure). as te basic metod to build te iger order ones. In bot cases, Φ [T,n] denotes a nt-order approximation to te evolution associated to T (q, p). Here, T (q, p) is a product of a small and a large term near te collision. Tis evolution is approximated using te splitting of symmetric and symplectic integrators of order n for te logaritmic Hamiltonian H = γ log(q) + 2 log(p) (37) wit a time step E 1. Te exact solution is equivalent to te evolution associated to T (q, p). We use S 9 6 and S 17 8 wit te leapfrog as te basic metod to approximate Φ [T,n]. Fig. 3 exibits te error in energy (te average error every 1 steps because te error oscillates). Eac metod is used wit te fictive time step δτ = m, were m is te number of stages of eac metod, ence all metods need approximately te same number of evaluations to reac te final real time. In te top figure we sow te results wen te logaritmic Hamiltonian is solved using te S 9 6 metod. After some time, a linear error growt in energy is observed. We repeated te experiments but solving te logaritmic Hamiltonian wit te S 17 8 metod (bottom figure) and te error growt as disappeared in te interval of interest. Tis sceme can be considered as a pseudo-symplectic integrator (a metod

15 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 15 wic preserves simplecticity at a iger order tan te order of te metod) and it is clear ow important it is to solve tis part accurately. We ave observed a good beaviour in te numerical experiments for long-time integration as well as te good performance of te RKN metods for problems wit quadratic kinetic energy, and for tis reason it makes good sense to consider tis tecnique for oter families of problems. 4.2 A perturbed Kepler problem We consider in tis subsection te two-dimensional perturbed Kepler problem H = 1 2 (p2 1 + p 2 2) 1 r + ε r 3, (38) q1 2 + q2 2, wic inter alia describes in first approximation te dynamics of a r = satellite moving into te gravitational field produced by a sligtly oblate planet. We take as initial conditions q 1 = 1 e, q 2 =, p 1 =, p 2 = (1 + e)/(1 e) wic, for te unperturbed problem, would correspond to an orbit of period 2π, eccentricity e and energy 1 2. We integrate until te final real time t = 1 for e =.8, ε = 1 3 and measure te average relative error in energy versus te number of force evaluations for different coices of fictitious time steps. We compare te performance of te metods wen using te Sundman and te Poincaré transformation. For tis problem we take te monitor function g = r γ for γ = 3 2, wic is close to optimal in bot cases. In te Sundman transformation we take G(z) = 1/z and te splitting 1 q z p d p 1 V (q) = + + z dτ t 1 z γ q p z }{{} q q }{{}}{{} f A f B f C and we consider te S 9 6 and te S 17 8 metods as a composition of te symmetric second order (11). Next, we consider te Poincaré transformation H = r γ 1 2 (p2 1 + p 2 2) + r γ ( 1 r + + ε r 3 + pt (39) ). (4) We integrate te Hamiltonian r γ 1 2 (p2 1 + p 2 2) troug te integration of te logaritmic Hamiltonian K = γ ( ) 2 log(q2 1 + q2) log 2 (p2 1 + p 2 2) (41) using a scaled time. Numerical integration of tis part requires very simple and fast evaluations and we neglect its cost in te results we present (we ave solved tis part using te S 17 8 metod, but for tis problem similar results were obtained using te S 9 6 metod). We solve te separable system (4) using te RKN 6 4 and te RKN 11 6 metods.

16 16 Sergio Blanes, Arie Iserles 3 4 (REL. ERROR) (EVALUATIONS) Fig. 4 Average error in energy in te numerical integration of (38) (for ε = 1 3, e =.8 and integrated up to t = 1). Te results for te following scemes are sown: S 17 8 wit te Sundman transformation (das-dotted line) and for te logaritmic Hamiltonian (42) (line wit circles), te Poincaré transformation for te Hamiltonian (41) using te RKN metods of order four (dased line) and of order six (solid line), and te sixt-order RKN metod wen no regulariozation is considered (γ = ) (line wit squares). Finally, we consider te logaritmic metod Mikkola and Tanikawa (1999); Preto and Tremaine (1999) corresponding to te integration of te Hamiltonian ( ) ( 1 K = log 2 (p2 1 + p 2 2) + p t 1 log r ε ) r 3. (42) Tis is an efficient metod for perturbed Kepler problems because it exactly solves te pure Kepler problem Preto and Tremaine (1999). Since te Hamiltonian as no kinetic energy quadratic in momenta, we integrate te system using S 9 6 and S 17 8 (we carried out numerical experiments of tis metod in te previous onedimensional problem, wic resulted in very low performance). Fig. 4 sows te most relevant results: S 17 8 wit te Sundman transformation (das-dotted line) and for te logaritmic Hamiltonian (42) (line wit circles), te Poincaré transformation for te Hamiltonian (41) using te RKN metods of order four (dased line) and of order six (solid line). As an illustration, we sow te results of te sixt-order RKN metod wen no regulariozation is considered (γ = ) (line wit squares). We ave observed in numerical examples not reported ere tat bot te Sundman and te Poincaré transformations exibit very similar accuracies wen used wit te appropriate regularization function (and function G(z) in te first case) and wen tey are integrated using te same splitting metod. Te main difference occurs wen different splitting metods can be used wit eac formulation. We ave not considered te computational cost of bot coices since migt be problem dependent. Te Sundman transformation is more general and can be used

17 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 17 for solving oter non Hamiltonian problems wile te Poincaré transformation allows in some cases to use more efficient splitting metods, but is constrained to Hamiltonian systems. Te logaritmic Hamiltonian (42) can provide efficient algoritms for perturbed Kepler problems, but its performance wit oter problems is not so clear Adaptive metods for near-integrable systems Finally, we analyse te performance of te Poincaré transformation for perturbed Hamiltonians wile employing te monitor function given in (27). In particular, we consider ( 1 H = 2 (p2 1 + p 2 2) 1 ) γ ( r + pt ε ) γ. (43) r 3 Here, 1 2 (p2 1 + p 2 2) 1 r + pt is a positive function only if ε r is a positive function. 3 If ε >, te Hamiltonian to be integrated would be ( H = 1 2 (p2 1 + p 2 2) + 1 ) γ ( ε ) γ r pt +. (44) r 3 In te following we will only consider tis problem wit negative values of ε. Te Hamiltonian is separable into solvable parts. Te case γ = 1 corresponds to te nonregurlarized case and as exactly te same complexity and computational cost as for different values of γ if one considers tat te evolution of a Hamiltonian H 1 is te same as te evolution of H 2 = F (H 1 ) wit a scaled time F (H 1 ). Tis splitting for nonregularised problems (γ = 1) is convenient for a small value of ε and low eccentricities, and tis can be furter improved by using splitting metods tailored for perturbed problems Laskar and Robutel (21); McLaclan (1995a). We consider ε = 1 3 and different values for te eccentricities (for te unperturbed problem), e =.2, e =.5 and e =.8. We use te following fourtorder splitting metods: S 5 4 and M84. Figure 5 sows te results obtained wen no regularization is considered (γ = 1) (left figure). Te convenience in using a splitting metod tailored for perturbed problems is clear. We ave repeated te experiment taking te regularization wit γ =.4 (tis is close to te optimal value we ave observed for tis problem and ig eccentricities) (rigt figure). We observe tat for e >.2 it is always more efficient to use regularization. Now, S 5 4 sows similar or even superior performance tan M84. Tis is due to te fact tat te regularization makes bot parts relatively small and of similar size, and it is not advantageous to use metods tailored for perturbed problems. In te following, wen tis regularization is considered, we will only use S m n metods. Finally, we compare te performance of different S m n splitting metods applied to (43) versus te RKN mn metods applied to (4). Te cost is very muc problem dependent, and we measure ere te number of evaluations in lieu of te cost (tis value for te cost sould be adjusted depending on te problem). Fig. 6 sows in te left panel te results obtained for te coices e =.8 and ε = 1 3 : solid lines correspond to te RKN 6 4 and RKN 11 6 metods for te Hamiltonian (4) and dased lines correspond to te S 5 4, S 9 6 and S 17 8 metods applied to (43) wit γ =.4. In tis case te perturbation contributes strongly and te RKN metods exibit te best performance. We ave repeated te numerical experiment for te coices e =.5 and ε = 1 5, wic correspond to a moderate eccentricity wit a

18 18 Sergio Blanes, Arie Iserles 2 γ=1 2 γ= e= (REL. ERROR) (REL. ERROR) e= e=.5 e= (EVALUATIONS) 9 1 e=.2 e= (EVALUATIONS) Fig. 5 Average error in energy in te numerical integration of (43) for ε = 1 3, for different values of te eccentricity, e, and up to t = 1. Two fourt-order splitting metods are used: S 5 4 for general separable systems (dased lines) and te (8,4) 5-stage BAB given in McLaclan (1995a) tailored for perturbed problems. Te left figure corresponds to no regularization (γ = 1) and te rigt figure corresponds to te regularization wit γ =.4. very small perturbation (rigt panel). In tis case, te metods for te perturbed problem wit regularisation sow te best performance. For illustration, we include te results for S 17 8 wen no regularisation is considered (γ = 1) (dotted line) (te results for te M84 metod remains very close to te RKN 11 6 metod, and for clarity in te presentation it as not been sown in te figure). From te numerical experiments we observe tat bot procedures for te Poincaré transformation exibit good performance. Te best coice is bound to depend on te problem. Wile te RKN metods applied to (4) need an efficient algoritm to solve te part wic mixes coordinates and momenta, te splitting for te Hamiltonian (43) needs bot parts to be positive definite. Were tis not te case, one could ave considered te Hamiltonian (28) wic requires an efficient numerical integration of te Hamiltonian (29). Tis calls for furter researc.

19 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems 19 5 e=.8, ε= e=.5, ε= (REL. ERROR) (REL. ERROR) (EVALUATIONS) (EVALUATIONS) Fig. 6 Average error in energy in te numerical integration of (38) integrated using te S 5 4, S 9 6 and S 17 8 metods applied to (43) wit γ =.4 (dased lines), and te RKN 6 4 and RKN 11 6 metods applied to (4) wit γ = 3/2 (solid lines). Tese metods can be distinguised by te slope of te curves. Left panel: corresponds to te coice e =.8 and ε = 1 3. Rigt panel: corresponds to te coice e =.5 and ε = 1 5. Te results for S 17 8 wen no regularisation is considered (γ = 1) is also sown (dotted line). 4.3 Te two-fixed-centres problem We finally consider te two-dimensional two-fixed-centres problem H = 1 2 (p2 1 + p 2 2) 2µ 2(1 µ), (45) r 1 r 2 wit r 1 = (q 1 c) 2 + q2 2, r 2 = (q 1 + c) 2 + q2 2. We take µ =.4, c = 1 and initial conditions q 1 = 1/2, q 2 =, p 1 =, p 2 = 3 wic as close approaces to one of te singularities (see Figure 7).

20 2 Sergio Blanes, Arie Iserles q q 1 Fig. 7 Solution for te two-fixed-centres problem (45) for t [, 1] and parameters given in te text. For tis problem we take te monitor function g = r γ 1 rγ 2 for γ = 3 2, and we take te Sundman transformation wit G(z) = 1/z and te splitting d dτ 1 q z p p = + t z γ }{{} f A g V (q) r1 2 + R ) z r2 2 }{{} f C ( R1 } {{ } f B (46) wit R 1 = (q 1 c)p 1 + q 2 p 2, R 2 = (q 1 + c)p 1 + q 2 p 2. Tis corresponds, basically, to te split (14), as proposed in Mikkola and Aarset (22), but separated in tree parts for simplicity. We consider te S 17 8 metod as a composition of te symmetric second order (11) and we take a fictive time step suc tat te final real time, t = 1, is reaced wit approximately 5 evaluations of te potential, and measure te error in energy along te time integration. We repeated te numerical experiment replacing te term g V (q) in f C, wic is not te gradient of a potential, by ( V (q))/z, wic preserves symplecticity. Te results are sown in Fig. 8 were te igest performance of te symplectic sceme is clear. As previously mentioned, tis deserves furter investigation. If one considers te Poincaré transformation ten RKN symplectic metods can be used if te Hamiltonian K 1 = r γ 1 rγ 2 (p2 1 + p 2 2)/2 is exactly solved, or numerically solved up to ig accuracy. For tis problem it is clear tat we can not use a

21 Explicit Adaptive Symplectic Integrators for solving Hamiltonian Systems g grad(v) 1.5 (1/z) grad(v) (ERROR ENERGY) (ERROR ENERGY) (t) (t) Fig. 8 Error in energy along te time integration for te two-fixed-centres problem (45) for te split (46) (left panel) and wen te term gv (q) in f C is replaced by ( V (q))/z (rigt panel). canonical transformation to solve it, but te logaritmic metod can be used very easily to ig accuracy and low computational cost. 5 Conclusions We ave considered te Sundman and te Poincaré transformation for te numerical integration of separable Hamiltonian systems evolving at different time scales. We ave considered bot transformation in teir general form. Tis allowed us to obtain, in a simple form, most algoritms proposed in te recent literature as particular cases. We ave also constructed new igly efficient metods, displaying teir best performance wen a ig accuracy is desired. Te numerical experiments suggest tat, in general, bot te Sundman and te Poincaré transformations provide similar accuracy if an appropriate regularization function is cosen (wic can differ in eac case) and te same splitting metod is used in bot cases. Te main difference remains in te fact tat te Poincaré transformation allows, in some cases, to retain te structure of te original Hamiltonian (e.g. quadratic in momenta or near-integrable) and one can use splitting metods tailored for tese problems. Tis requires te numerical integration of

22 22 Sergio Blanes, Arie Iserles new Hamiltonian functions wit a relatively simple and very particular structure wose efficient integration allows to end up wit igly efficient algoritms. Te Sundman transformation is, owever, more general and can be easily used on non Hamiltonian systems. Te results presented in tis work easily extend to time-dependent potential functions V (q, t) by considering te time as a new coordinate as well as to oter separable Hamiltonian functions. References Blanes S, Budd CJ (24) Hig order symplectic integrators for perturbed amiltonian systems. Celestial Mecanics and Dynamical Astronomy 89: Blanes S, Budd CJ (25) Adaptive geometric integrators for amiltonian problems wit approximate scale invariance. SIAM J Sci Comput 26: Blanes S, Moan PC (22) Practical symplectic partitioned runge kutta and runge kutta nyström metods. J Comp Appl Mat 142: Blanes S, Casas F, Ros J (1999) Extrapolation of symplectic integrators. Celestial Mecanics and Dynamical Astronomy 75: Blanes S, Casas F, Murua A (28) Splitting and composition metods in te numerical integration of differential equations. Bol Soc Esp Mat Apl 45: Bond SD, Leimkuler B (1998) Time-transformations for reversible variable stepsize integration. Numerical Algoritms 19:55 71 Budd CJ, Leimkuler B, Piggott MD (21) Scaling invariance and adaptivity. Appl Numer Mat 39: Calvo MP, Sanz-Serna JM (1993) Te development of variable-step symplectic integrators, wit applications to te two-body problem. SIAM J Sci Comput 14: Calvo MP, Sanz-Serna JM, López-Marcos MA (1998) Variable step implementations of geometric integrators. Appl Numer Mat 28:1 16 Can RPK, Murua A (2) Extrapolation of symplectic metods for amiltonian problems. Appl Numer Mat 34: Creutz M, Gocksc A (1989) Higer-order ybrid monte carlo algoritms. Pys Rev Lett 63:9 12 Gladman B, Duncan M, Candy J (1991) Symplectic integrators for long-term integrations in celestial mecanics. Celestial Mecanics 52: Hairer E (1997) Variable time step integration wit symplectic metods. Appl Numer Mat 25: Hairer E, Söderlind G (25) Explicit, time reversible, adaptive stepsize control. SIAM J Sci Comput 26: Hairer E, Lubic C, Wanner G (26) Geometric Numerical Integration. Structure- Preserving Algoritms for Ordinary Differential Equations, vol 31, Second edn. Springer-Verlag Holder T, Leimkuler B, Reic S (21) Explicit variable step-size and timereversible integration. Appl Numer Mat 39: Huang W, Leimkuler B (1997) Te adaptive verlet metod. SIAM J Sci Comput 18: Iserles A (28) A First Course in te Numerical Analysis of Differential Equations, Second edn. Cambridge University Press

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