On the necessity of negative coefficients for operator splitting schemes of order higher than two

Transcription

1 Applied Numerical Matematics 54 (2005) On te necessity of negative coefficients for operator splitting scemes of order iger tan two Sergio Blanes, Fernando Casas Departament de Matemàtiques, Universitat Jaume I, Castellón, Spain Available online 0 November 2004 Abstract In tis paper we analyse numerical integration metods applied to differential equations wic are separable in solvable parts. Tese metods are compositions of flows associated wit eac part of te system. We propose an elementary proof of te necessary existence of negative coefficients if te scemes are of order, or effective order, p and provide additional information about te distribution of tese negative coefficients. It is sown tat if te metods involve flows associated wit more general terms tis result does not necessarily apply and in some cases it is possible to build iger order scemes wit positive coefficients IMACS. Publised by Elsevier B.V. All rigts reserved. Keywords: Splitting metods; Composition metods; Effective order; Numerical integrators 1. Introduction Operator splitting scemes are numerical metods wic are particularly useful to approximate te evolution of differential equations wen tey are separable in solvable parts [17]. To be more specific, let us consider te ODE x = f(x), x 0 = x(0) R D (1) * Corresponding autor. addresses: sblanes@mat.uji.es (S. Blanes), casas@mat.uji.es (F. Casas) /$ IMACS. Publised by Elsevier B.V. All rigts reserved. doi: /j.apnum

2 24 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) 2 7 wit f : R D R D and associated vector field (or Lie operator associated wit f ) D F = f i (x). (2) x i Let us denote by ϕ te -flow of te ODE (1) for a given time step. In oter words, te exact solution is given by x() = ϕ (x 0 ). Now let us assume tat f(x) can be written as f(x)= f A (x) + f B (x), te vector field F is split accordingly as F = F A + F B and te -flows ϕ [A] and ϕ [B] corresponding to F A and F B, respectively, can be exactly computed or, equivalently, te equations x = f A (x) and x = f B (x) are solvable. Ten te composition (sometimes called Lie Trotter splitting) ψ,1 ϕ [B] () approximates ϕ wit error of order 2, i.e., ψ,1 (x 0 ) = ϕ (x 0 ) + O( 2 ), wereas te so-called Strang splitting or Störmer/Verlet/leapfrog sceme ψ,2 ϕ [A] /2 ϕ[b] /2 (4) is suc tat ψ,2 (x 0 ) = ϕ (x 0 )+O( ). Te order of approximation to te exact solution can be increased by including more maps wit fractional time steps in te composition. In general, te sceme ψ ϕ [B] b m ϕ[a] a m ϕ[b] b 1 ϕ[a] a 1, (5) (a 1,b 1,...,a m,b m ) R 2m,isoforderp if ψ = ϕ + O( p+1 ) for a proper coice of m and coefficients a i, b i. It can be assumed witout loss of generality tat in (5) none of te coefficients b 1,b 2,...,b m 1 as well as none of a 2,a,...,a m are vanising, i.e., only a 1 and/or b m can be zero, since oterwise te corresponding flows could be removed (ϕ [A] 0 = ϕ [B] 0 = id, te identity map) and te rest of te maps would be concatenated (due to te group property of te flows). Numerical scemes of order p based on te composition (5) ave been successfully applied for solving a large number of problems [10,17], including also certain partial differential equations. In fact, splitting metods are frequently used in celestial mecanics [21], quantum mecanics [6], molecular dynamics [12], accelerator pysics [7] and, in general, for numerically solving Hamiltonian dynamical systems [8,16], Poisson systems [14] and reversible differential equations [15]. It as been noticed, owever, tat some of te coefficients in (5) are negative for p wen one considers arbitrary vector fields F A and F B. In oter words, te metods always involve stepping backwards in time. Tis constitutes a problem wen te differential equation is defined in a semigroup, as arises sometimes in applications, since ten te metod can only be conditionally stable [17]. Also scemes wit negative coefficients may not be well-posed wen applied to PDEs involving unbounded operators. Te existence of backward fractional time steps in te composition metod (5) is in fact unavoidable, and can be establised as te following two teorems: Teorem 1 [20,22]. Ifp is a positive integer suc tat p, ten tere are no composition metods of te form (5) and finite m wit all te coefficients a i, b i being positive. Teorem 2 [9]. Ifp is a positive integer suc tat p, ten, for every pt-order metod (5) wit m any finite positive integer, one as min 1 i m a i < 0 and min 1 j m b j < 0.

3 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) Teorem 2 is stronger tan Teorem 1, in te sense tat it establises tat at least one of te a i and also one of te b i coefficients ave to be strictly negative, altoug a similar (and certainly non-trivial) proof strategy was used. One of te goals of tis paper is to provide an alternative, elementary proof of Teorem 2, giving in addition a more detailed analysis of ow negative coefficients are distributed in te composition. During te last few years te processing tecnique as been used to find composition metods requiring less evaluations tan conventional scemes of order p [1]. Te idea consists in enancing an integrator ψ (te kernel) wit a parametric map π : R D R D (te post-processor)as ˆψ = π ψ π 1. (6) Application of n steps of te new (and opefully better) integrator ˆψ leads to ˆψ n = π ψ n π 1, wic can be considered as a cange of coordinates in pase space. Observe tat processing is advantageous if ˆψ is a more accurate metod tan ψ and te cost of π is negligible, since it provides te accuracy of ˆψ at essentially te cost of te least accurate metod ψ. Te simplest example of a processed integrator is provided in fact by te Strang splitting (4). As a consequence of te group property of te exact flow, we ave ψ,2 = ϕ [A] /2 ϕ[b] /2 = ϕ[a] /2 ϕ[b] ϕ[a] /2 = ϕ [A] /2 ψ,1 /2 = π ψ,1 π 1 (7) wit π = ϕ [A] /2. Hence, applying te first order metod () wit processing yields a second order of approximation. Altoug initially intended for Runge Kutta metods [4], te processing tecnique as proved its usefulness mainly in te context of geometric numerical integration [10], were constant step-sizes are widely employed. We say tat te metod ψ is of effective order p if a post-processor π exists for wic ˆψ is of (conventional) order p [4], tat is, π ψ π 1 = ϕ + O ( p+1). Hence, as te previous example sows, te Lie Trotter splitting () is of effective order 2. Obviously, a metod of order p is also of effective order p (taking π = id) or iger, but te converse is not true in general. Te analysis of te order conditions of te metod ˆψ sows tat many of tem can be satisfied by π, so tat ψ must fulfill a muc reduced set of restrictions [2,]. In particular, if one takes a composition (5) for ψ, te number and complexity of te conditions to be verified by te coefficients a i, b i is notably reduced. As a consequence, by considering bot te kernel ψ and te post-processor π as compositions of basic integrators, igly efficient processed metods ave been proposed [2,2,1,17]. Neverteless, wen π is constructed as a composition (5), its computational cost is usually iger tan tat of ψ,and tus te use of te resulting processed sceme is restricted to situations were intermediate results are not frequently required. Oterwise te overall efficiency of te metod is deteriorated.

4 26 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) 2 7 To overcome tis difficulty, in [] a tecnique as been developed for obtaining approximations to te post-processor at virtually cost free and witout loss of accuracy. Te clue is to replace π by a new integrator π π obtained from te intermediate stages in te computation of ψ. As a result of te analysis carried out in [], it is generally recommended to ave a very accurate pre-processor π 1 but, on te contrary, π can safely be replaced by π, since te error introduced by te ceap approximation π is of a purely local caracter and is not propagated along te evolution (contrarily to te error in π 1 ). Here we also address te following question: do Teorems 1 and 2 also old for a composition ψ of effective order p? Observe tat, in principle, Teorem 2 applies to te wole composition ˆψ of (6), but it would noneteless be advantageous to ave te negative coefficients restricted only to te composition π. For in tat case te integration starts by computing π 1 (wic only involves positive coefficients), ten ψ (involving only positive coefficients) is evaluated once per step and finally an appropriate approximation to π may be considered wen output is required (even te crudest approximation π = id [1]). In tis way te algoritm only involves stepping forward in time and could be applied even to PDEs wit unbounded operators. Te answer to te question posed before could also be useful in te searc of efficient metods of order iger tan 2 for systems tat evolve in a semigroup, suc as te eat equation in two space dimensions [17]. Also in quantum statistical mecanics, te partition function requires computing Z = Tr(e βh ), were H is te Hamiltonian operator and β is te inverse temperature [22]. In numerical calculations a processed composition algoritm may be used to approximate e βh, and since te trace is invariant under similarity transformations, only te kernel is necessary to determine Z. If it involves only positive coefficients ten it would be possible to build up iger order convergent algoritms for tis class of problems. In Section we prove explicitly tat tis is not te case, so tat any composition metod (5) of effective order p contains necessarily some negative coefficients. 2. An elementary proof of Teorem 2 It is well known tat, for eac infinitely differentiable map g : R D R, g(ϕ (x)) admits an expansion of te form g ( ϕ (x) ) = g(x) + k k! F k [g](x) = e F [g](x), x R D, k 1 were F is te vector field (2). Similarly, for te map ψ given in (5) one as g ( ψ (x) ) = Ψ [g](x), were [10] Ψ = exp(a 1 F A ) exp(b 1 F B ) exp(a m F A ) exp(b m F B ). (8) By repeated application of te Baker Campbell Hausdorff (BCH) formula to (8) we can obtain a series of differential operators F = k 1 k F k suc tat Ψ = exp(f ), i.e., ψ is formally te exact 1-flow of te vector field F. Te sceme ψ is of order p iff 1 = F A + F B and F 2 = F = 0. In terms of te coefficients a i, b i, tis corresponds to te following order conditions:

5 order 1: order 2: order : S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) m a i = 1, m b i = 1; ( m i ) b i a j = 1 2 ; j=1 ( m 1 b i m j=i+1 a j ) 2 = 1, m ( m ) 2 a i b j = 1. (9) Te proof of Teorems 1 and 2 provided by [20,22,9] is based precisely on te fact tat a sceme of te form (5) wit m any finite positive integer and all te coefficients a i, b i being positive cannot satisfy Eqs. (9). by In te particular case of te first order metod χ = ϕ [B] j=i, te corresponding operator (8) is given exp(f A ) exp(f B ) = exp(x ) = exp ( X X 2 + X + ), (10) wit X 1 = F A + F B, X 2 = 1 2 [F A,F B ], etc., wereas for its adjoint sceme χ = χ 1 = ϕ[a] ϕ [B] one as g(χ (x)) = e X [g](x). Here[F A,F B ] stands for te Lie bracket F A F B F B F A. Since our aim is to get results valid for all pairs F A, F B of arbitrary vector fields, ten we must assume tat te only linear dependencies among nested Lie brackets of F A and F B can be derived from te skew-symmetry and te Jacobi identity of te Lie brackets. In oter words, X k, k 1, is an element of te graded free Lie algebra generated by te symbolic vector fields F A, F B, were bot ave degree one [18]. Te crucial observation tat leads to an alternative, elementary proof of Teorem 2 is te close connection existing between te splitting metod (5) ψ = ϕ [B] b m ϕ[a] a m ϕ[b] b m 1 ϕ[a] a 2 ϕ[b] b 1 ϕ[a] a 1 (11) and te composition of te first order metod χ = ϕ [B] different time steps [15]: and its adjoint χ = ϕ[a] ϕ [B] ψ = χ β 2m χ β 2m 1 χ β 2 χ β 1 χ β 0. (12) Indeed, by inserting te explicit form of χ βi and χβ i in (12) we ave ψ = ( ϕ [A] β 2m ) ( ϕ[b] β 2m ϕ [B] β 2m 1 ) ( ϕ[a] β 2m 1 ϕ [A] β 2 ) ( ϕ[b] β 2 ϕ [B] β 1 ) ( ϕ[a] β 1 ϕ [A] β 0 ) ϕ[b] β 0 = ϕ [A] β 2m ϕ[b] (β 2m +β 2m 1 ) ϕ[a] (β 2m 1 +β 2m 2 ) ϕ[b] (β 2 +β 1 ) ϕ[a] (β 1 +β 0 ) ϕ[b] β 0, were in te last equality we ave used te group property of te exact flow. If we put β 0 = β 2m = 0we recover expression (11) as soon as Ten a i = β 2i 1 + β 2i 2, b i = β 2i + β 2i 1, i = 1,...,m. (1) m a i = 2m β i = m b i. wit (14)

6 28 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) 2 7 In consequence, composition (11) can be rewritten as (12) only if (14) olds. Consistency of bot scemes require in fact tat m a i = m b i = 2m β i = 1 and it as been sown in [15] tat te order conditions for te coefficients a i, b i to get a metod of order p 1 are equivalent to te order conditions for te β i. In tat case te operator Ψ given in (8) can also be expressed as Ψ = exp( X β0 ) exp(x β1 ) exp( X β2 ) exp(x β2m 1 ) exp( X β2m ) (15) and repeated application of te BCH formula gives Ψ = exp ( f 1,1 X f 2,1 X 2 + ( f,1 X + f,2 [X 1,X 2 ] ) + O ( 4)), were te coefficients f k,j are omogeneous polynomials of degree k in te variables β i. In particular we ave f 1,1 = 2m β i, f 2,1 = 2m ( 1) i+1 β 2 i, f,1 = 2m β i. (16) Conditions f 1,1 = 1andf n,j = 0 for all n p are ten sufficient for te metod to be of order p. Proof of Teorem 2. From te preceding discussion, it is clear tat f,1 = 2m β i = 0 (17) is a necessary condition to be satisfied by any metod of order p. We suppose tat more tan two β i are different from zero because β1 + β 2 = 0 togeter wit te consistency condition β 1 + β 2 = 1aveno real solution. Now (17) can be written as m ( ) m ( ) β 2i 1 + β2i 2 + β 2m = β 2i 1 + β2i 2 = 0, for any positive integer m. In consequence β2j 1 + β 2j 2 as to be negative for some 1 j m. Butit is easy to verify tat sign(x + y ) = sign(x + y) for any x,y R, so tat a j = β 2j 1 + β 2j 2 < 0, (18) for some j suc tat 1 j m. Similarly, we can write (17) as m β0 + ( β 2i + β2i 1) m ( ) = β 2i + β2i 1 = 0 so tat β2k + β 2k 1 < 0 for some 1 k m, and again b k = β 2k + β 2k 1 < 0. (19) Distribution of te coefficients We can get more information about te distribution of te negative coefficients in te composition (5) by applying a sligtly more involved argument wic, in fact, also provides anoter demonstration of Teorem 2.

7 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) If we denote α 2i 1 = a i, α 2i = b i, i = 1,...,m, in te composition (5) ψ = ϕ [B] α 2m ϕ[a] α 2m 1 ϕ[b] α 2 ϕ[a] α 1, ten (1) implies α i = β i + β i 1, i = 1,...,2m, (20) were β j are te coefficients appearing in (12) (β 0 = β 2m = 0). Now all we need to prove Teorem 2 is to analyse ow Eqs. (17) and (20) imply tat at least one odd as well as one even α i coefficients are negative. As before, we assume tat tere are more tan two nonvanising coefficients β i and at least one of tem is negative. (a) Let us suppose first tat only one coefficient is actually negative, say β j, for some 0 <j<2m. Ten, from Eq. (17), ( ) 1/ β j = i j β i so tat β j >β i for all i j. Terefore α j = β j + β j 1 < 0 and α j+1 = β j+1 + β j < 0, i.e., two consecutive α k coefficients are negative, and tus at least one a j and one b j are negative. (b) Suppose now tat te negative coefficients are β j1,β j2,...,β jk wit j 1 <j 2 < <j k. (b.1) If β ji 1 < β ji and β ji >β ji +1 (21) for some j i I {j 1,j 2,...,j k } ten also two consecutive coefficients α k are negative, namely α ji and α ji +1. (b.2) On te oter and, wen (21) does not old for any j i I, ten te following situations are possible: (i) if β ji 1 < β ji, ten β ji <β ji +1; (ii) if β ji >β ji +1, ten β ji 1 > β ji ; (iii) finally, β ji 1 > β ji and β ji +1 > β ji. Let us suppose tat β ji +1 β ji+1 1 for all j i. Ten (β β jk = ( j1 1 + β j + ( β 1 j 1 +1) + β j2 1 + β j 2 + βj 2 +1) + + ( βj k ) β j k 1 + βj k 1 +1) + β 1/, i were contains te remaining terms (including β jk 1 and β jk +1). Since βj l 1 +β j l +βj l +1 > 0forl = 1,...,k 1, ten clearly β jk >β jk 1 and β jk >β jk +1 in contradiction wit ypotesis (b.2). Terefore β ji +1 = β ji+1 1 for some j i. Let us suppose, witout loss of generality, tat tey correspond to te first l + 1 coefficients β ji. Ten, te only

8 0 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) 2 7 possible sequence (different from tose considered before) as to be β ji 1,β ji,β ji +1,β ji +2, β ji +,...,β ji +2l 1,β ji +2l,β ji +2l+1 suc tat β ji 1 < β ji <β ji +1, β ji +2 <β ji +1, β ji +2 <β ji +,... β ji +2l 1 > β ji +2l >β ji +2l+1, were β ji +2k, k = 0, 1,...,l are te negative coefficients. Ten α ji = β ji + β ji 1 < 0, α ji +2l+1 = β ji +2l+1 + β ji +2l < 0. Also in tis case at least one a i and one b i are negative because j i and j i + 2l + 1differinan odd number. Notice tat tis is te only situation were two negative α i coefficients in a given metod do not stay in consecutive places. We ave cecked several composition metods publised in te literature aving observed tat tis occurrence is in fact quite rare: it is very muc frequent tat at least two consecutive α i coefficients are negative, and tis discussion provides an explanation of te penomenon.. Compositions of effective order p As wit te composition ψ in (5), let us consider a post-processor π in (6) formally as te exact 1-flow of te vector field P, i.e., g(π (x)) = e P [g](x) for all g, wit P = k 1 k P k. Ten one as for te processed sceme g ˆψ = Ψ [g],were Ψ = exp( F ) and te vector field F can be determined from te relation exp ( F ) = exp( P ) exp(f ) exp(p ). (22) Wit respect to te vector field P, it is natural to coose it as an element of te graded free Lie algebra generated by F A and F B.Tus,uptoordertwoin, P = (c 1 F A + c 2 F B ) + 2 c X 2 + O ( ), (2) wit c i free parameters. Notice tat π can be approximated by a composition (5), or equivalently, exp(p ) by te product (8). However, if c 1 c 2 ten i a i = c 1 c 2 = i b i and composition (12) cannot be used (as is te case, for instance, of te Strang splitting (7)). On te oter and, since c 1 F A + c 2 F B = (c 2 c 1 )F B + c 1 X 1 = (c 1 c 2 )F A + c 2 X 1, from (2) it is possible to write e P = e c 1X 1 e cf B e 2 d 1 X 2 + O ( ) (24) as well as e P = e c 2X 1 e cf A e 2 d 2 X 2 + O ( ), (25) were c = c 2 c 1 and d 1,d 2 are parameters depending on c 1,c 2,c. Since te processed sceme ˆψ is of conventional order p, ten Ψ = exp(x 1 ) + O( p+1 ) and e X 1 Ψ = Ψ e X 1 + O( p+1 ), so tat exp(c i X 1 ) in (24) and (25) can be safely removed witout loss of generality and tus we take c 1 = 0or c 2 = 0. Now we are in disposition to establis and prove te main result of tis section.

9 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) Teorem. At least one of te a i as well as one of te b i coefficients ave to be negative in te composition (5) if ψ is te kernel of a processed metod of order (or equivalently if ψ is of effective order) p. Proof. Let ψ be a composition (5) of effective order wit, say, all a i positive. Ten it is possible to construct a vector field P suc tat (24) (wit c 1 = 0) olds and terefore (22) leads to e 2 d 1 X 2 e cf B Ψ e cf B e 2 d 1 X 2 = e F = e (F A+F B ) + O ( 4), (26) or equivalently Ψ e cf B Ψ e cf B = e 2 d 1 X 2 e (F A+F B ) e 2 d 1 X 2 + O ( 4) = exp ( (F A + F B ) d 1 [X 1,X 2 ] ) + O ( 4), (27) were Ψ is given by (8). Notice tat all coefficients a i in Ψ are positive, since Ψ is associated wit te composition map ψ = ϕ [B] c ϕ[b] b m ϕ[a] a m ϕ[b] b 1 ϕ[a] a 1 ϕ[b] c, (28) wic, as previously, can be written as a composition of te first order metod χ and its adjoint χ wit coefficients β i : ψ = χ β 2k χ β 2k 1 χ β 2 χ β 1 χ β 0, (29) wit β 0 = β 2k = 0. Since te coefficient of X is zero in (27), it is clear tat f,1 := 2k β i = 0. But, as we know from te proof of Teorem 2 provided in Section 2, tis condition cannot be satisfied wit all coefficients a i positive. Similarly, if we assume tat all b i are positive ten te same argument applied to te post-processor (25) leads to te same contradiction. In fact, te explicit expression of te effective order conditions up to order can be derived in te following way. By inserting (24) in (22) and applying te BCH formula one finds ( ( (f,1 Ψ = exp X (f 2,1 + 2c)X c(f 2,1 + c) ) X + (f,2 12 ) )) c(f 2,1 + c) + d 1 [X 1,X 2 ] + O ( 4) (1) and a second order metod is obtained by taking c = 1 2 f 2,1. If we substitute tis value in (1) and take d 1 suc tat te coefficient of [X 1,X 2 ] vanises, ten ( Ψ = exp (X 1 + f,1 ) 4 f 2,1 )X 2 + O ( 4). (2) (0)

10 2 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) 2 7 Tis same result is obtained if one considers a post-processor suc tat (25) olds (wit c 2 = 0) instead of (24). In summary, it is clear tat f,1 4 f 2 2,1 = 0 () is te only condition to be satisfied by a composition to be of effective order tree. Tis condition is equivalent to te kernel condition at order tree presented in [2] using a different basis of te Lie algebra. Example. Let us consider te composition map ψ = ϕ [B] (1 b) ϕ[a] (1 a) ϕ[b] b ϕ[a] a = χ β 4 χ β χβ 2 χ β 1 χβ 0, (4) wit β 0 = 0, β 1 = a, β 2 = b a, β = 1 b, β 4 = 0, and te consistency conditions already imposed. Tis composition cannot be of order tree (tere are not enoug parameters to solve all te order conditions (9)), yet it could be of effective order tree if condition () is satisfied, wic in tis case reads 1 12ab(1 a)(1 b) = 0. But it turns out tat tis equation as no real solution if a (0, 1) as well as if b (0, 1). 4. Oter classes of composition metods Te previous results can be generalized in different contexts. For instance, let us consider a partitioned sceme built up using finite linear combinations of splitting metods of te form (5), i.e., K ψ = γ k ψ,k, (5) k=1 were ψ,k = ϕ [B] b k,n ϕ[a] a k,n ϕ[b] b k,1 ϕ[a] a k,1 (6) and it is assumed tat k γ k = 1and i a k,i = i b k,i, k = 1,...,K. Te generalization provided by te following teorem establises tat even wit partitioned scemes of te form (5) eac basic flow in a convex partition (γ k > 0forevery1 k K) must be applied for at least one backward fractional time step. On te oter and, simple polynomial extrapolation of te leapfrog metod (7) sows tat if γ k < 0 all te coefficients a k,i, b k,i may indeed be positive. Teorem 4 [9].Ifp and K are positive integers suc tat p and K 2, and γ k > 0 for k = 1,...,K, ten at least one of te coefficients a k,i as well as one of te b k,i ave to be negative in te composition (6) if ψ as order, or effective order, p. Proof. Also in tis case te proof is quite elementary. Since i a k,i = i b k,i we can write ψ,k = χβ k,2n χ β k,2n 1 χβ k,2 χ β k,1 χβ k,0 (7) and by following te same procedure as previously we find tat instead of (16) te necessary condition for (5) to be a metod of order tree or iger is now K 2n βk,i = 0. γ k k=1

11 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) 2 7 Since γ k > 0, k = 1,...,K, tere exists some j, 1 j K, suc tat 2n βj,i 0, and te previous proof can be applied. Te two-term splitting analyzed so far can be seen as a special instance of a k-term splitting of te vector field F, F = F 1 + F 2 + +F k. Suppose we ave a sceme of te form ψ = ϕ [k] a m,k ϕ[2] a m,2 ϕ[1] a m,1 ϕ[k] a 2,k ϕ[2] a 2,2 ϕ[1] a 2,1 ϕ [k] a 1,k ϕ[2] a 1,2 ϕ[1] a 1,1, (8) were ϕ [l] stands for te exact -flow of te vector field F l. If te composition (8) is of order, or effective order, p for all coices of operators F 1,...,F k, ten clearly min a i,l < 0, l = 1,...,k. 1 i m Consider now composition maps of te Strang splitting ψ,2 given by (4) (or wit te roles of te flows ϕ [A] and ϕ [B] intercanged), i.e., ψ = ψ βm,2 ψ βm 1,2 ψ β1,2 ψ β0,2. (9) Te series of differential operators S associated wit te integrator ψ,2, i.e., suc tat g ψ,2 = S [g], can be written as S = exp(x ),werex = X 1 + X + 5 X 5 +,X 1 = F and Ψ = exp(x β0 ) exp(x β1 ) exp(x βm 1 ) exp(x βm ). Now, by repeated application of te BCH formula, Ψ = exp ( f 1,1 X 1 + f,1 X + O ( 4)), were m m f 1,1 = β i, f,1 = βi. Terefore f 1,1 = 1, f,1 = 0 are necessary conditions for ψ to be of order p. In fact, since f 2,1 = 0in tis case, tey are also te conditions to be satisfied by ψ to ave effective order p = and te following teorem can be establised. Teorem 5. If p is any positive integer suc tat p and ψ,2 is te Strang splitting (or Störmer/Verlet sceme) (4), ten at least two consecutive coefficients a i, b i ave to be negative in te composition (9) (wen it is expressed in terms of te basic flows ϕ [A], ϕ[b] ) if ψ is of order, or effective order, p. Even more, at least two coefficients a i1, a i2 ave to be negative. Proof. By substituting in (9) te expression of te basic metod ψ βi,2 = ϕ [A] a composition of te type (5) wit β i /2 ϕ[b] β i ϕ[a] β i /2, we obtain b i = β i, a i = 1 2 (β i + β i 1 ), i = 1,...,m, (40)

12 4 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) 2 7 if β 0 = β m = 0. It is immediate to ceck tat if tere exists one negative coefficient, say β j < 0, 1 j m 1, and β j >β j 1, (41) ten a j < 0, b j < 0, wereas if β j >β j+1, (42) ten b j < 0, a j+1 < 0. In oter words, as soon as one of te β i is negative and its absolute value is iger tan te previous one or te next one ten te corresponding composition (5) as, at least, two consecutive coefficients wic are negative. Let us analyse te different possibilities arising from te order condition f,1 = 0. (i) First, suppose tere exists only one negative coefficient β j < 0, 1 j m 1. Ten ( ) 1/ β j = i j β i and bot conditions (41) and (42) are satisfied so tat, according to te previous discussion, a j < 0, b j < 0anda j+1 < 0. (ii) Suppose now tat tere are k 2 negative coefficients, β j1,β j2,...,β jk < 0 suc tat tey do not satisfy conditions (41) and (42). Observe tat tey cannot be consecutive, oterwise eiter (41) or (42) are satisfied. Ten we can write condition f,1 = 0as (β β jk = ( j1 1 + ) ( β j β jk ) ) β j k 1 + β 1/, i were contains te remaining terms, including β jk 1 and β jk +1. Since βj i 1 + β j i > 0, i = 1,...,k 1, ten β jk 1 < β jk, β jk +1 < β jk, conditions (41) and (42) are in fact satisfied by β jk and terefore b j1,...,b jk 1,a jk,b jk,a jk +1 < 0. (iii) Finally, consider te case in wic β j1,β j2,...,β jk < 0(k 2) and only one of te coefficients β ji, i = 1,...,k, satisfies eiter (41) or (42). For instance, suppose tat β j1 is suc tat β j1 >β j1 1 (and terefore a j1 < 0). Ten, condition f,1 = 0 can be expressed as ( β j1 + βj 1 +1) + βi = 0, i j 1,j 1 +1 but βj 1 + βj 1 +1 > 0, since (42) is not satisfied by β j 1, so tat ( ) ( ) β j2 + βj β jk + βj k +1 + β i < 0, were, as before, contains te remaining (positive) terms. In consequence, tere must exist some 2 i k suc tat βj i + βj i +1 < 0. Terefore we ave at least b j 1,...,b jk < 0anda j1,a ji +1 < 0. If β j1 satisfies (42) instead, a similar strategy applies and te same conclusion follows. Tis result, togeter wit te discussion of Section 2 justifies wy it is so frequent tat at least two consecutive coefficients are negative.

13 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) Composition metods wit all coefficients being positive Let us consider now te second order differential equation y = g(y), (4) wic can be written in te form (1) by taking x (x 1,x 2 ) = (y, y ) and f A (x) = (x 2, 0), f B (x) = (0,g(x 1 )), or equivalently F A x 2, F B g(x 1 ). x 1 x 2 Tis equation frequently appears in relevant problems arising in classical and quantum mecanics: tere te operator F A is related to te kinetic energy (quadratic in momenta) and F B is associated wit te potential energy. Now te flow corresponding to F C [F B, [F A,F B ]] is explicitly and exactly computable and, in addition, [F B,F C ]=0, so tat it makes sense to compute te 1-flow ϕ [B,C] associated wit te b,c vector field bf B + c F C and include it into te composition (5): ψ = ϕ [B,C] b m,c m a m ϕ[b,c] b 1,c 1 a 1. (44) In tis case ψ cannot always be written as te composition of a first order sceme and its adjoint, and Teorem 2 does not necessarily apply. For instance ψ = ϕ [B] /6 ϕ[a] /2 ϕ[b,c] 2/, /72 ϕ[a] /2 ϕ[b] /6 (45) is a metod of order four [11] and ψ = ϕ [A] /2 ϕ[b,c], /24 ϕ[a] /2 (46) is a metod of effective order four [19]. In te last case we can write ψ = χ/2 χ /2 wit χ ϕ [B,C], /6 ϕ [A]. However, if we analyse te corresponding operator exp(x ) = exp(x X 2 + X + ) associated wit χ,wefindtatx =[X 1,X 2 ]/6. Ten X is not an independent element and its contribution can be cancelled wit a proper coice of te map π, tus giving a fourt-order metod. Numerical experiments suggest tat tis is te igest order one can get wit te composition (44) wit positive coefficients and a rigorous proof is at present under investigation. However, metods of effective order six as well as of order six are known to exist wit all coefficients b i being positive. On te oter and, if we consider a Hamiltonian system of te form H = T(p)+ V(q), wit T quadratic in p and V(q) a polynomial function up to degree four in q (or, in general, if g(y) is a polynomial function up to degree tree), ten F E [F A, [F A, [F A, [F A,F B ]]]] vanis or depends only on te momenta, i.e., [F A,F E ]=0, and its flow can be computed exactly. In addition F D [F B, [F B, [F A, [F A,F B ]]]] depends only on te coordinates and tus [F B,F D ]=0. Tus one may consider composition maps involving te 1-flows ϕ [A,E] a,e 5, ϕ [B,C,D] b,c,d 5 corresponding to te vector fields af A + e 5 F E and bf B + c F C + d 5 F D, respectively. In particular, te generalised leapfrog splitting sceme ψ = ϕ [A,E] /2,e 5 ϕ [B,C,D],c,d 5 ϕ [A,E] /2,e 5, (47)

14 6 S. Blanes, F. Casas / Applied Numerical Matematics 54 (2005) 2 7 wit c = 1 24, d = , e = 1 is a metod of effective order six, since tese coefficients satisfy te 2880 kernel conditions collected in [2] up to tis order. We sould recall tat metods (45) (47) are particular examples of composition scemes involving only positive coefficients. Te possible existence of oter families of composition metods of order p wit positive coefficients is, at te time being, an open question of great interest, for instance, in te numerical integration of nonreversible systems. Acknowledgements Tis work as been partially supported by te Generalitat Valenciana troug te project GRU- POS0/002 and Fundació Bancaixa (project P1.1B ). SB is also supported by te Ministerio de Educación y Ciencia (MEC) troug a contract in te programme Ramón y Cajal 2001 and te project BFM Finally, te work of FC was performed at te Department of Applied Matematics and Teoretical Pysics, University of Cambridge, wile on sabbatical leave from Universitat Jaume I, tanks to a mobility grant from te MEC (Spain). We tank te tree referees for teir comments wic elped greatly to improve te presentation of te paper. Since te completion of tis work, S.A. Cin as publised a different proof of Teorem ; see [5]. References [1] S. Blanes, Hig order numerical integrators for differential equations using composition and processing of low order metods, Appl. Numer. Mat. 7 (2001) [2] S. Blanes, F. Casas, J. Ros, Symplectic integration wit processing: A general study, SIAM J. Sci. Comput. 21 (1999) [] S. Blanes, F. Casas, A. Murua, On te numerical integration of ordinary differential equations by processed metods, SIAM J. Numer. Anal. 42 (2004) [4] J. Butcer, Te effective order of Runge Kutta metods, in: J.L. Morris (Ed.), Conference on te Numerical Solution of Differential Equations, Lecture Notes in Matematics, vol. 109, Springer, Berlin, 1969, pp [5] S.A. Cin, Quantum statistical calculations and symplectic corrector algoritms, Pys. Rev. E 69 (2004) [6] H. De Raedt, Product formula algoritms for solving te time dependent Scrödinger equation, Comput. Pys. Rep. 7 (1987) [7] A.J. Dragt, D.T. Abell, Symplectic maps and computation of orbits in particle accelerators, in: J.E. Marsden, G.W. Patrick, W.F. Sadwick (Eds.), Integration Algoritms and Classical Mecanics, American Matematical Society, Providence, RI, 1996, pp [8] E. Forest, R. Rut, Fourt-order symplectic integration, Pysica D 4 (1990) [9] D. Goldman, T.J. Kaper, Nt-order operator splitting scemes and nonreversible systems, SIAM J. Numer. Anal. (1996) [10] E. Hairer, C. Lubic, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algoritms for Ordinary Differential Equations, Springer Series in Computational Matematics, vol. 1, Springer, Berlin, [11] P.-V. Koseleff, Formal Calculus for Lie Metods in Hamiltonian Mecanics, P.D. Tesis, Lawrence Berkeley Laboratory LBID-200, Tec. Report UC-405, Berkeley, CA, [12] B.J. Leimkuler, S. Reic, R.D. Skeel, Integration metods for molecular dynamics, in: J.P. Mesirov, K. Sculten, D.W. Sumners (Eds.), Matematical Approaces to Biomolecular Structure and Dynamics, IMA Volumes in Matematics and Its Applications, vol. 82, Springer, New York, 1996, pp [1] M.A. López-Marcos, J.M. Sanz-Serna, R.D. Skeel, Explicit symplectic integrators using Hessian-vector products, SIAM J. Sci. Comput. 18 (1997)

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