Energy-preserving variant of collocation methods 1

Transcription

1 European Society of Computational Metods in Sciences and Engineering (ESCMSE) Journal of Numerical Analysis, Industrial and Applied Matematics (JNAIAM) vol.?, no.?, 21, pp ISSN Energy-preserving variant of collocation metods 1 E. Hairer 2 Université de Genève, Section de Matématiques, 2-4 rue du Lièvre, CH-1211 Genève 4, Switzerland Received 15 October, 29 Abstract: We propose a modification of collocation metods extending te averaged vector field metod to ig order. Tese new integrators exactly preserve energy for Hamiltonian systems, are of arbitrarily ig order, and fall into te class of B-series integrators. We discuss teir symmetry and conjugate-symplecticity, and we compare tem to energypreserving composition metods. c 21 European Society of Computational Metods in Sciences and Engineering Keywords: Hamiltonian systems, energy-preserving integrators, B-series, Runge Kutta metods, collocation, conjugate-symplecticity, symmetry, Gaussian quadrature. Matematics Subject Classification: 65P1, 65L6 PACS: 45.2.Jj, 2.3.Hq, 2.6.Lj 1 Introduction We consider Hamiltonian differential equations, written in te form ẏ = J 1 H(y), (1) were J is a skew-symmetric constant matrix, and te Hamiltonian H(y) is assumed to be sufficiently differentiable. For its numerical integration one would ideally like to ave energypreservation and symplecticity of te discrete flow. Bot properties cannot be satisfied at te same time unless te integrator produces te exact solution (see [6, page 379]). Tere is a uge literature on symplectic integrators, and backward error analysis permits to prove tat tey nearly preserve te energy (tey exactly preserve a modified Hamiltonian). Metods tat exactly preserve energy (energy-momentum metods, discrete gradient metods) ave been considered since several decades. More recently, te existence of energy-preserving B- series metods as been sown in [5], and a practical integrator as been proposed in [1]. It is called averaged vector field metod and is defined by y n+1 = y n + ) J 1 H ((1 τ)y n + τ y n+1 dτ. (2) 1 Tis work is supported by te Fonds National Suisse, project No Corresponding autor. Ernst.Hairer@unige.c

2 2 E. Hairer Tis metod exactly preserves te energy for an arbitrary Hamiltonian, and in contrast to projection-type integrators, only requires evaluations of te vector field. It is symmetric and its Taylor series as te structure of a B-series. For polynomial Hamiltonians, te integral can be evaluated exactly, and te implementation is comparable to tat of te implicit mid-point rule. Tere is muc activity applying tis metod and various modifications to polynomial Hamiltonian systems (see [2, 3]) and also to Hamiltonian partial differential equations. Anoter interesting class of integrators are extended collocation metods and Hamiltonian Boundary Value Metods, wic exactly preserve energy of polynomial Hamiltonian systems (see [9, 1]). In tis article we combine te ideas of bot approaces. At te one and we extend te metod (2) to iger order, similar to te way ow te implicit mid-point rule can be extended to collocation Runge Kutta metods. On te oter and, we consider a limit of te metod proposed in [9] to cover also non-polynomial Hamiltonians. In Section 2 we give te precise definition of te class of considered metods, we interpret tem as Runge-Kutta metods wit a continuum of stages, we study teir order, we prove teir exact energy conservation, and we investigate teir symmetry. Section 3 is devoted to te study ow close te metods are to symplectic integrators. We interpret te numerical solution as a B-series. We sow tat many of te algebraic conditions on te B-series coefficients, wic caracterize symplecticity, are satisfied. We ten prove tat te metods of maximal order 2s are conjugate-symplectic up to order at least 2s + 2, and we prove tat composition metods of order 4 based on (2) cannot be conjugate-symplectic up to an order iger tan 4. We terminate tis article wit a discussion of te new class of integrators (Section 4). 2 Te new class of integrators Altoug our interest is mainly in Hamiltonian systems (1), we define te metods for general initial value problems ẏ = f(y), y(t ) = y. In te following we use te notation l i (τ) = s j=1,j i 2.1 Definition τ c j c i c j, b i = for te Lagrange basis polynomials in interpolation. l i (τ)dτ Energy-preserving collocation metods. Let c 1,...,c s be distinct real numbers (usually c i 1) for wic b i for all i. We consider a polynomial u(t) of degree s satisfying u(t ) = y (3) u(t + c i ) = 1 l i (τ)f ( u(t + τ) ) dτ. (4) b i Te numerical solution after one step is ten defined by y 1 = u(t + ). Approximating te integral wit te interpolatory quadrature formula corresponding to te nodes c 1,..., c s, we obtain u(t + c i ) = f ( u(t + c i ) ) in place of (4), and te metod reduces to a classical collocation metod. Because of tis connection we call te new metods (wit abuse of notation) energy-preserving collocation metods. c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

3 Energy-preserving variant of collocation metods 3 Denoting k i := u(t + c i ), te derivative of u(t) and te polynomial u(t) itself become s u(t + τ) = l i (τ)k i (5) i=1 u(t + τ) = y + s i=1 τ l i (σ)dσ k i. (6) Inserting (6) into (4) yields a nonlinear equation k = G(k) for te finite dimensional vector k = (k 1,..., k s ). Tis equation can be solved by fixed-point iteration or by Newton tecniques. Every evaluation of G(k) requires te computation of integrals wic can be computed exactly for polynomial vector fields or approximated numerically by an accurate quadrature formula. Instead of writing te polynomial u(t) in terms of y and u(t + c i ), we can also express it in terms of y and u = ( u(t + ĉ i ), i = 1,...,s ), were te ĉ i are distinct numbers tat need not coincide wit te collocation points c i. Wit k i from (4) inserted into (6) we obtain a nonlinear equation u = Ĝ(u) wic can again be solved iteratively. If te appearing integrals are replaced by a fixed ig-order quadrature, te resulting metod is closely related to te approac presented in [1]. 2.2 Interpretation as Runge Kutta metod and order For teoretical investigations of te metod, it is more convenient to interpret it as a Runge Kutta metod wit a continuum of stages τ [, 1]: Y τ = y + A τ,σ f(y σ )dσ, y 1 = y + B σ f(y σ )dσ. (7) Here, Y τ y(t + C τ ), were C τ = A τ,σ dσ. In our situation, te internal stages Y τ are te values of te polynomial u(t + τ), and te coefficients are given by s 1 τ C τ = τ, A τ,σ = l i (α)dα l i (σ), B σ = 1. (8) b i=1 i Teorem 1. Let r be te order of te interpolatory quadrature formula based on te nodes c 1,..., c s. Ten, te energy-preserving collocation metod as { 2s for r 2s 1 order = 2r 2s + 2 for r 2s 2. Proof. Using te facts tat g(σ)dσ = s i=1 b ig(c i ) for polynomials of degree r 1 and tat s i=1 l i(τ)g(c i ) = g(τ) for polynomials of degree s 1, one sees tat our metod satisfies te simplifying assumptions (partial integration is elpful for te verification of D(ζ)) B(ρ) : C(η) : D(ζ) : B τ Cτ k 1 dτ = 1, k = 1,...,ρ k A τ,σ Cσ k 1 dσ = 1 k Ck τ, k = 1,...,η B τ Cτ k 1 A τ,σ dτ = B σ k (1 Ck σ), k = 1,...,ζ wit ρ =, η = min(s, r s + 1) and ζ = min(s 1, r s). Te statement now follows from a classical result by Butcer (see for example [7, p. 28]), wic states tat te order of a Runge Kutta metod satisfying te simplifying assumptions B(ρ), C(η) and D(ζ) is at least min(ρ, 2η + 2, η + ζ + 1). Tis proves te statement. c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

4 4 E. Hairer It is surprising tat te order of our new metod is not te same as for te corresponding classical collocation metod (wic is equal to te order r of te quadrature formula). For r = 2s 1 it is iger, wereas for r 2s 3 it is lower. We notice tat te order based on Newton Côtes quadrature drops to 2 wen s is even. We also see tat te order of te new metods is always even. 2.3 Examples Case s = 1. Writing te polynomial u(t) as u(t +τ) = y +τ(y 1 y ), we recover te averaged vector field metod (2), independently of te coice of c 1. Case s = 2. We consider nodes c 1, c 2 satisfying (c 1 + c 2 ) + c 1 c 2 =, so tat te corresponding quadrature formula as at least order r = 3, e.g., Gaussian nodes c 1,2 = or Radau nodes c 1 = 1 3, c 2 = 1. Te Runge Kutta coefficients become C τ = τ, A τ,σ = τ ( (4 3τ) 6 (1 τ)σ ), B σ = 1, and we notice wit surprise tat te metod is independent of te coice of c 1. If we write te polynomial u(t) (wic is of degree 2) as a linear combination of y = u(t ), y 1 = u(t + ) and Y 1/2 := u(t + /2), ten tis metod becomes Y 1/2 = y + y 1 = y + ( ) 2 σ f ( u(t + σ) ) dσ f ( u(t + σ) ) dσ. Tis metod is of order 4, it is symmetric and exactly preserves te energy for Hamiltonian systems as will be sown in Sections 2.4 and 2.5 below. Tere is an interesting connection to te metods studied in [9]. If we approximate te integrals in (9) by te Lobatto quadrature of order eigt, we obtain precisely te extended Lobatto IIIA metod of order four [9]. Case s = 3. We can eiter take any quadrature formula of order r 2s 1 = 5 and proceed as for te case s = 2. Anoter possibility is to exploit te fact tat A τ,σ is known by (8) to be a polynomial of degree 2 in σ. Te simplifying assumption C(s) tus uniquely determines its coefficients and yields A τ,σ = τ ( (9 18τ + 1τ 2 ) 12 (3 8τ + 5τ 2 )σ + 3 (1 3τ + 2τ 2 )σ 2) togeter wit C τ = τ and B σ = 1. Representing te polynomial u(t) of degree 3 as linear combination of y, Y 1/3, Y 2/3, and y 1, te metod reads Y 1/3 Y 2/3 y 1 = y + = y + = y + ( σ σ2) f ( u(t + σ) ) dσ ( σ 2 9 σ2) f ( u(t + σ) ) dσ f ( u(t + σ) ) dσ and is of order 6. If we represent te polynomial u(t) by different values, say y, Y α, Y β, y 1, we get anoter representation of te same metod. (9) (1) c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

5 Energy-preserving variant of collocation metods Exact energy-preservation Te proof of energy conservation for te new class of metods is as simple as for metod (2). From te fundamental teorem of calculus we ave H ( u(t + ) ) H ( u(t ) ) = u(t + τ) T H ( u(t + τ) ) dτ. Substituting (5) for u(t + τ) and replacing k i = u(t + c i ) wit (4), were f(y) = J 1 H(y), yields for tis expression s 1 b i=1 i l i (τ) H ( u(t + τ) ) T dτ J T l i (τ) H ( u(t + τ) ) dτ wic vanises by te skew-symmetry of te matrix J. As a consequence our metod satisfies H(y 1 ) = H(y ), so tat H(y n ) = H(y ) is exactly preserved along te numerical solution. 2.5 Symmetry A numerical one-step metod y n+1 = Φ (y n ) is symmetric, if Φ 1 = Φ. For a standard s-stage Runge Kutta metod tis is te case, if teir coefficients satisfy a s+1 i,s+1 j + a i,j = b j. For a continuous Runge Kutta metod (7) tis condition can be written as A 1 τ,1 σ + A τ,σ = B σ. (11) For te energy-preserving collocation metod wit coefficients given by (8), tis condition is equivalent to (after differentiation wit respect to τ) s i=1 1 ( ) l i (τ)l i (σ) l i (1 τ)l i (1 σ) =. (12) b i Lemma 2. If te nodes c 1,...,c s are symmetric, i.e., c s+1 i = 1 c i for all i, ten te energypreserving collocation metod is symmetric. Proof. Tis follows from (12), because for symmetric nodes c 1,...,c s we ave b i = b s+1 i and l i (1 τ) = l s+1 i (τ). As a consequence, all metods presented in Section 2.3 are symmetric. Notice tat, in contrast to standard collocation metods, te symmetry of te nodes is a sufficient but not necessary condition for te symmetry of te metod (in fact, a metod based on te non-symmetric Radau nodes is identical to te metod based on te symmetric Gaussian nodes). 3 On teir conjugate-symplecticity Numerical metods in te class of B-series integrators, wic exactly conserve te energy for Hamiltonian systems, cannot be symplectic. It is of interest to investigate weter suc metods are conjugate to a symplectic integrator (up to a certain order). Te proof of our results is based on a representation of te numerical solution in terms of trees and elementary differentials. Our notation closely follows te monograp [6, Cap. III] (see also [4] for a recent survey on tis approac). c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

6 6 E. Hairer 3.1 Trees and symplecticity Let T = {,,,...} be te set of rooted trees. We denote u = [u 1,..., u m ] te tree tat is obtained by grafting te roots of te trees u 1,..., u m to a new vertex wic becomes te root of u. Te order u is te number of vertices of te tree u, te symmetry coefficient is defined recursively by σ( ) = 1, σ(u) = σ(u 1 ) σ(u m )µ 1!µ 2!, (13) were te integers µ 1, µ 2,... count equal trees among u 1,..., u m, and elementary differentials F(u) are given by F( )(y) = f(y), F(u)(y) = f (m) (y) ( F(u 1 )(y),..., F(u m )(y) ). From Runge Kutta teory it is known tat te Taylor series of te exact solution, of te numerical solution, and of te internal stages of te metod (7) can be written as B-series, y(t +) = B(e, y ), y 1 = B(φ, y ) and Y τ = B(φ τ, y ), were B(a, y) = y + u T Te coefficients for te exact solution are recursively defined by u a(u)f(u)(y). (14) σ(u) e( ) = 1, e(u) = 1 u e(u 1)... e(u m ) for u = [u 1,...,u m ]. (15) Tose for te numerical solution and te internal stages are defined by φ τ ( ) = C τ, φ( ) = 1 and φ τ (u) = A τ,σ φ σ (u 1 )... φ σ (u m )dσ, φ(u) = B τ φ τ (u 1 )... φ τ (u m )dτ. For te study of symplecticity of a numerical B-series metod, te expressions a(u, v) := a(u v) + a(v u) a(u)a(v) for u, v T (16) play an important role. Here, te so-called Butcer product u v of two trees u = [u 1,..., u m ] and v is defined by u v = [u 1,..., u m, v]. Te mapping Φ (y) = B(a, y) is a symplectic transformation for Hamiltonian vector fields (i.e., its Jacobian matrix satisfies Φ (y)t JΦ (y) = J), if and only if a(u, v) = for all u, v T. Tis caracterization, due to Calvo and Sanz-Serna, is explained and discussed in [6, Cap. VI.7]. Teorem 3. Assume tat a Runge Kutta metod (wit finitely many or a continuum of stages) is of order r and satisfies te simplifying conditions C(η) and D(ζ) wit r η ζ. Ten, we ave φ(u, v) = for min( u, v ) ζ. (17) Proof. In a first step we prove te statement for u = τ k wit k ζ, were τ k = [,..., ] is te busy tree of order k. For a tree v = [v 1,..., v m ] we multiply te simplifying assumption D(ζ) wit φ σ (v 1 )... φ σ (v m ), and integrate over σ. Tis yields for k ζ (for a continuum of stages) B τ C k 1 τ A τ,σ dτ φ σ (v 1 )... φ σ (v m )dσ = 1 k B σ (1 C k σ)φ σ (v 1 )... φ σ (v m )dσ. c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

7 Energy-preserving variant of collocation metods 7 On te oter and te condition C(η) implies for k η tat B τ A τ,σ Cσ k 1 dσ φ τ (v 1 )... φ τ (v m )dτ = 1 k B τ C k τ φ τ(v 1 )... φ τ (v m )dτ. Adding bot equations and using te fact tat φ(τ k ) = 1/k for k r (te metod is of order r), we obtain φ(τ k v) + φ(v τ k ) = φ(τ k )φ(v) wic proves φ(τ k, v) = for k ζ and for v T. We next notice tat a recursive application of te simplifying assumption C(η) implies tat φ τ (w) = e(w)c τ w for all trees w wit w η, were e(w) is te coefficient defined in (15). We now fix a tree u = [u 1,...,u m ] satisfying u = k η, and we apply tis property to u j and u. Tis implies φ(u) = k e(u)φ(τ k ), φ(u v) = e(u 1 )... e(u m )φ(τ k v), and φ(v u) = k e(u)φ(v τ k ) for all trees v. Consequently, we ave φ(u, v) = ke(u)φ(τ k, v). Since φ(τ k, v) = for k ζ and ζ η, tis completes te proof of te teorem. For te energy-preserving collocation metod of maximal order 2s, te simplifying conditions C(s) and D(s 1) old. Terefore te symplecticity condition φ(u, v) = is satisfied wen at least one tree among u and v as s 1 vertices. For te leading error term, were trees of order 2s + 1 are involved, only pairs (u, v) wit u = s and v = s + 1 give tus rise to non-zero φ(u, v), and due to te simplifying assumption C(s) tey are related by φ(u, v) = e(u)e(v) e(τ s )e(τ s+1 ) φ(τ s, τ s+1 ) for u = s, v = s + 1. (18) 3.2 Conditional conjugate-symplecticity We consider te energy-preserving collocation metod of order 2s. It cannot be symplectic (see also Section 3.1). It is terefore interesting to study weter it can be conjugate to a symplectic metod: does tere exist a cange of coordinates χ (y) = y + O( 2s ), suc tat χ 1 Φ χ is a symplectic integrator? If tis appens, its long-time beavior is te same as tat of a symplectic metod. Unfortunately, it turns out tat tis aim is too ambitious. We terefore study weter our metods satisfy some weaker condition. We call te metod y n+1 = Φ (y n ) of order r conjugate-symplectic up to order q (q > r), if tere exists a cange of coordinates χ (y) = y + O( r ), suc tat te Jacobian matrix of Ψ = χ 1 Φ χ satisfies Ψ (y)t JΨ (y) = J + O(q+1 ). In suc a situation te modified differential equation (in te sense of backward error analysis) is Hamiltonian up to terms of size O( q ). For nearly integrable systems te global error is tus bounded by O(t r + t 2 q ), so tat te metod beaves like a symplectic integrator on intervals of lengt O( r q ), see [6, page 436]. Lemma 4. If a symmetric integrator Ψ satisfies Ψ (y)t JΨ (y) = J + O(2m ), ten it automatically also satisfies Ψ (y)t JΨ (y) = J + O(2m+1 ). ( Proof. An integrator is symmetric if Ψ Ψ (y) ) = y. Differentiating tis relation wit respect to y yields Ψ ( Ψ (y) ) Ψ (y) = I = identity. Te assumption implies tat Ψ (ŷ) T JΨ (ŷ) = J + C(ŷ) 2m + O( 2m+1 ) wit ŷ = Ψ (y). We multiply tis relation from te rigt wit Ψ (y) and from te left wit Ψ (y)t. Since ŷ = y+o() and Ψ (y) = I+O(), tis yields J = Ψ (y)t JΨ (y)+c(y)2m +O( 2m+1 ). However, te assumption written wit in place of gives Ψ (y)t JΨ (y) = J +C(y)2m + O( 2m+1 ). Bot relations togeter can be true only wen C(y) =. Tis proves te statement. c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

8 8 E. Hairer Te main result of tis section is te following property concerning conjugate-symplecticity. It sows tat besides exact energy-preservation, te new class of metods as also excellent symplecticity features. Teorem 5. Te energy-preserving collocation metod of order 2s is conjugate-symplectic up to order 2s + 2. Proof. We ave seen in Section 3.1 tat te numerical metod is a B-series Φ (y) = B(φ, y). We terefore searc for a cange of coordinates tat can also be written as a B-series χ (y) = B(c, y). To satisfy χ (y) = y + O( 2s ) and because we are only interested in te leading error term, we require c(u) = for trees wit u 2s. Te study of conjugate-symplecticity eavily relies on te fact tat te composition of two B-series is again a B-series, B(a, B(b, y)) = B(b a, y), were te product on te coefficients is given by (b a)(u) = b(u) + b(u \ σ)a(σ u ). (19) σ S(u) Here, S(u) denotes te set of ordered subtrees of u (an ordered subtree is a connected subset of te vertices containing te root), u \ σ is te collection of trees tat remains wen σ is removed, and σ u is te rooted tree given by te vertices of σ wit edges induced from te tree u (we use te notation of [4]). Te proof of te teorem is in several steps. a) As composition of B-series, te conjugated metod Ψ = χ 1 Φ χ is also a B-series Ψ (y) = B(ψ, y), and a transcription of te relation χ Ψ = Φ χ into coefficients of B-series yields (ψ c)(w) = (c φ)(w) for trees w T. (2) Te statement of te teorem, written in terms of te coefficients of B-series, is as follows: find coefficients c(w) for trees w satisfying w = 2s (we assume c(w) = for trees wit w 2s), suc tat te coefficients ψ(w), defined by (2), satisfy ψ(u, v) = for pairs u, v wit u + v = 2s + 1. (21) For trees wit u + v 2s, te relation (21) is satisfied, because te metod Ψ (y) is of order 2s, and trees wit u + v = 2s + 2 need not be considered by Lemma 4, because χ (y) = χ (y) guarantees te symmetry of Ψ (y). b) Exploiting te fact tat c(w) only for trees wit w = 2s and using te composition law (19), te relation (2) becomes for te trees u v and v u (wit u + v = 2s + 1) ψ(u v) + ψ(v u) + σ S 2s(u v) σ S 2s(v u) { c(v) if u = c(σ u v ) = φ(u v) + else, c(σ v u ) = φ(v u) + { c(u) if v = else, were S 2s (w) denotes te set of subtrees of w wit exactly 2s vertices. We add bot relations and ten we subtract te product ψ(u)ψ(v) = φ(u)φ(v) from bot sides. Te term c(v) (resp. c(u)) on te rigt-and side, wic is present only for u = (resp. v = ), cancels wit te subtree σ S 2s (v u), were u = is removed (resp. σ S 2s (u v), were v = is removed). Since all oter σ S 2s (v u) can be written eiter as σ = û v wit û S u 1 (u) or as σ = u ˆv wit ˆv S v 1 (v), we see tat te symplecticity condition (21) becomes equivalent to φ(u, v) = c(û, v) + c(u, ˆv) for pairs u, v wit u + v = 2s + 1 (22) û S u 1 (u) ˆv S v 1 (v) c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

9 Energy-preserving variant of collocation metods 9 (notice tat te set S u 1 (u) is empty wen u = ). Te proof of te teorem is complete if we can find coefficients c(w) tat satisfy tis relation. c) In tis part of te proof we define te coefficients c(w) of te conjugacy mapping χ (y). We sall sow in part (d) tat tey are solution of te linear system (22). We start wit assuming c(τ s, τ s ) = 1 s φ(τ s, τ s+1 ) (23) c(u, v) = e(u)e(v) e(τ s )e(τ s ) c(τ s, τ s ) if u = v = s (24) c(u, v) = else, (25) were e(u) are te coefficients of te exact solution, defined in (15). Let us first explain te meaning of defining values for te expressions c(u, v) = c(u v) + c(v u) (notice tat c(u) = for u < 2s). We let be te equivalence relation defined by u v v u, so tat equivalence classes are free (or un-rooted) trees. Tis means tat te root can be moved arbitrarily in te tree. If te value c(w) is known for one rooted tree in an equivalence class, ten te knowledge of c(u, v) uniquely defines te coefficients c(w) for all oter trees in tis equivalence class. For a tree of te form u u, te value of c(u u) is determined by (23)-(25), because c(u, u) = 2c(u u). Terefore, wenever an equivalence class contains a trees of te form u u, te values of c(w) are uniquely defined for all trees w in tis equivalence class. For equivalence classes tat do not contain a tree of te form u u, we can arbitrarily fix te value c(w) for one tree in te equivalence class, wic ten uniquely defines te values for te remaining trees. Tis freedom in te coice of te coefficients c(w) reflects te fact tat wenever a cange of coordinates χ transforms te metod Φ (y) to a symplectic mapping up to a certain order, te same will be true for χ ϕ, if ϕ is a symplectic transformation. d) We prove tat te coefficients defined in part (c) are a solution of te linear system (22). By te symmetry of φ(u, v) we can assume u < v, so tat te first sum in (22) vanises by (25), because û s 1. If u < s, and ence v > s + 1, also te second sum in (22) vanises, and te equation is satisfied by Teorem 3, because φ(u, v) = in tis case. It remains to consider te situation, were u = s and v = s + 1. Using (24), ten Lemma 6 below, ten e(τ s ) = 1/s and e(τ s+1 ) = 1/(s + 1), and finally (23), yields for te rigt-and side of (22) ˆv S s(v) c(u, ˆv) = ˆv S s(v) e(u) e(ˆv) e(τ s )e(τ s ) c(τ s, τ s ) = (s + 1)e(u)e(v) e(τ s )e(τ s ) c(τ s, τ s ) = e(u)e(v) e(τ s )e(τ s+1 ) φ(τ s, τ s+1 ). Comparing tis relation wit (18) proves tat every set of coefficients c(w) satisfying (23)-(25) solves te linear system (22). Lemma 6. Te coefficients e(v) of te B-series for te exact solution (cf. definition (15)) satisfy (s + 1)e(v) = ˆv S s(v) e(ˆv) for trees v wit v = s + 1. (26) Proof. For te busy tree v = τ s+1 = [,..., ] of order s + 1, te sum in (26) is over s copies of ˆv = τ s. Since e(τ s ) = 1/s, tis proves (26) for v = τ s+1. For an arbitrary tree v, te proof is by induction on its order. Let v = [v 1,..., v m ] and assume te formula (26) be true for v 1,..., v m. We split te sum over ˆv S s (v) as ˆv S s(v) e(ˆv) = 1 s ( ˆv 1 S s1 (v 1) e(ˆv 1 )e(v 2 )...e(v m ) ˆv m S sm (v m) ) e(v 1 )e(v 2 )...e(ˆv m ), c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

10 1 E. Hairer were s j + 1 = v j. By induction ypotesis te rigt-and expression becomes 1 ( v v m ) e(v 1 )e(v 2 )...e(v m ) = e(v 1 )e(v 2 )...e(v m ) = (s + 1)e(v) s wic completes te proof of te lemma. Te obvious question is now weter te energy-preserving collocation metods can be conjugate-symplectic up to an order iger tan 2s + 2. Te answer is negative for s = 1: te recent publication [2] sows tat te metod (2) is not conjugate-symplectic up to an order iger tan 4. Te question is still open for s 2. Since many symplecticity conditions are satisfied by te new class of metods (see Teorem 3), we expect te energy-preserving collocation metod of order 2s to be conjugate-symplectic up to an order iger tan 2s + 2 (it could even by 4s). We do not pursue tis question in te present article. Taking up te discussion of te beginning of Section 3.2, we see tat te metods of order 2s proposed in tis article not only preserve exactly (up to round-off errors) te total energy, but also beave like symplectic metods on long time intervals. For nearly integrable Hamiltonian systems, Teorem 5 implies tat te global error is bounded by O(t 2s + t 2 2s+2 ) and, if our conjecture is true, by O(t 2s ) as long a t 2s const. 3.3 Comparison wit composition metods An alternative way of designing ig order energy-preserving integrators is by composition. Let Φ (y) denote te averaged vector field metod (2), wic is symmetric and of order two. Te symmetric composition metod is of order four if (see [6, Section V.3.2]) Φ [4] = Φ γ m... Φ γ2 Φ γ1 Φ γ2... Φ γm (27) γ ( γ γ m ) = 1, γ ( γ γ3 m) =. (28) It is not of order six for linear differential equations (e.g., te armonic oscillator) if γ ( γ γ5 m). (29) Teorem 7. A symmetric composition (27) based on te averaged vector field metod (2) and satisfying (28)-(29) cannot be conjugate-symplectic up to an order iger tan 4. Proof. Te proof is muc simplified if we work wit modified differential equations in te sense of backward error analysis. It as been sown in [3] tat te modified differential equation of te averaged vector field metod (2) is were ẏ = f(y) + 2 f 3 (y) + 4 f 5 (y) +... (3) f 3 (y) = 1 12 (f f f)(y), f 5 (y) = u =5 a(u) σ(u) F(u)(y) wit coefficients a(u) given in Table 1. It follows from Lemma 5.9 of [6, page 94] tat te modified differential equation of te composition metod Φ [4] is ẏ = f(y) + 4( α f 5 (y) + β [ f, [f, f 3 ] ] (y) ) + O( 6 ), (31) c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

11 Energy-preserving variant of collocation metods 11 u 72 a(u) b(u) Table 1: B-series coefficients for te modified differential equation (31) were α = γ ( γ γm) 5 and β is some constant depending on te coefficients γi of te metod Φ [4]. Recall tat [f, f 3] = f 3f f f 3. Applying tis operation a second time sows tat te double commutator as te structure of a B-series [ f, [f, f3 ] ] (y) = b(u) σ(u) F(u)(y) u =5 wit coefficients b(u) of Table 1. Tis implies tat te leading term of te local truncation error is 5( α f 5 (y) + β [ f, [f, f 3 ] ] (y) ), so tat te B-series coefficients of te metod Φ [4] are φ(u) = e(u) + α a(u) + β b(u) for trees u wit u = 5. Teorem 8.3 of [6, page 224] gives necessary (and sufficient) conditions for a fourt order metod to be conjugate-symplectic up to order 5. Tese conditions are fulfilled if and only if α =. Tis proves te statement of te teorem. 4 Conclusion and discussion For Hamiltonian differential equations tere is a long-standing dispute on te question weter in a numerical simulation it is more important to preserve energy or symplecticity. Many people give more weigt on symplecticity, because it is known (by backward error analysis arguments) tat symplectic integrators conserve a modified Hamiltonian. Te present article proposes a class of integrators tat guarantees energy conservation up to macine precision. Te metods are not symplectic, but tose of maximal order 2s are conjugate-symplectic up to at least order 2s + 2 (possibly to a muc iger order). Terefore, we expect tese metods to beave like symplectic metods wen tey are applied to an integration over very long time intervals. It is not common to use numerical metods, were integrals over te vector field ave to be evaluated. However, for polynomial vector fields tis work is reduced to te computation of integrals over polynomials, and tis can be done once for all in te beginning of te integration. Terefore, te complexity of te metods is comparable to tat of implicit Runge Kutta metods wit te same number of stages. In a parallel computing environment, vector field evaluations can be done in parallel, wen te integrals are approximated by a ig order quadrature formula. Te proposed metods of maximal order 2s are symmetric. Tis implies tat for nearly integrable reversible differential equations teir long-time beavior is comparable to tat of symplectic integrators (see [6, Capter XI]). Te use of variable step sizes retains energy conservation, but destroys symplecticity (and conjugate-symplecticity). If te step sizes are selected in a reversible way (for example, using te tecniques of [8]), te excellent long-time beavior of symmetric integrators can be conserved. c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)

12 12 E. Hairer Acknowledgment Tis work emerged during te international conference ICNAAM 29, Retymno, Crete, Greece, September 29. We are in particular grateful to Luigi Brugnano wose presentation was te stimulus of te present work, and to Brynjulf Owren for pointing out te pseudo-conjugatesymplecticity of te second order averaged vector field metod. References [1] L. Brugnano, F. Iavernaro, and D. Trigiante. Analysis of Hamiltonian boundary value metods (HBVMs) for te numerical solution of polynomial Hamiltonian dynamical systems. Submitted for publication, 29. [2] E. Celledoni, R. I. McLaclan, D. I. McLaren, B. Owren, G. R. W. Quispel, and W. M. Wrigt. Energy-preserving Runge Kutta metods. M2AN Mat. Model. Numer. Anal., 43(4): , 29. [3] E. Celledoni, R. I. McLaclan, B. Owren, and G. R. W. Quispel. Energy-preserving integrators and te structure of B-series. NTNU Report No. 5, 29. [4] P. Cartier, E. Hairer, and G. Vilmart. Algebraic structures of B-series. To appear in Foundations of Comput. Mat., 21. [5] E. Faou, E. Hairer, and T.-L. Pam. Energy conservation wit non-symplectic metods: examples and counter-examples. BIT, 44:699 79, 24. [6] E. Hairer, C. Lubic, and G. Wanner. Geometric Numerical Integration. Structure-Preserving Algoritms for Ordinary Differential Equations. Springer Series in Computational Matematics 31. Springer-Verlag, Berlin, 2nd edition, 26. [7] E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer Series in Computational Matematics 8. Springer, Berlin, 2nd edition, [8] E. Hairer and G. Söderlind. Explicit, time reversible, adaptive step size control. SIAM J. Sci. Comput., 26: , 25. [9] F. Iavernaro and D. Trigiante. Hig-order symmetric scemes for te energy conservation of polynomial Hamiltonian problems. JNAIAM J. Numer. Anal. Ind. Appl. Mat., 4(1-2):87 11, 29. [1] G. R. W. Quispel and D. I. McLaren. A new class of energy-preserving numerical integration metods. J. Pys. A, 41(4):4526, 7, 28. c 21 European Society of Computational Metods in Sciences and Engineering (ESCMSE)