A SIXTH ORDER AVERAGED VECTOR FIELD METHOD *

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1 Journal of Computational Mathematics Vol.34, No.5, 6, doi:.48/jcm.6-m5-65 A SIXTH ORDER AVERAGED VECTOR FIELD METHOD * Haochen Li Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Jiangsu, 3, China lihaochen bjut@sina.com Yushun Wang ) Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Jiangsu, 3, China wangyushun@njnu.edu.cn Mengzhao Qin Lsec, Academy of Mathematics and System Sciences, Chinese Academy of Science, PoBox 79, Beijing, China qmz@lsec.cc.ac.cn Abstract In this paper, based on the theory of rooted trees and B-series, we propose the concrete formulas of the substitution law for the trees of order 5. With the help of the new substitution law, we derive a B-series integrator extending the averaged vector field (AVF) methods for general Hamiltonian system to higher order. The new integrator turns out to be order of six and exactly preserves energy for Hamiltonian systems. Numerical experiments are presented to demonstrate the accuracy and the energy-preserving property of the sixth order AVF method. Mathematics subject classification: 65D5, 65L5, 65L7, 65P. Key words: Hamiltonian systems, B-series, Energy-preserving method, Sixth order AVF method, Substitution law.. Introduction With the fast development of computer, geometric methods become more and more powerful in scientific research. A numerical method which can conserve the geometric properties of a system is called geometric method [, 9]. Geometric methods, such as symplectic methods, symmetric methods, volume-preserving methods, energy-preserving methods and so on, have been successfully used in many application areas [4,, 4, 7, 8,, 4, 7, 3 33]. The conservation of the energy function is one of the most relevant features characterizing a Hamiltonian system. Methods that exactly preserve energy have been considered for several decades and many energy-preserving methods have been proposed [5 7,, 6]. Here we list some examples. The discrete gradient method is one of the most popular methods for designing integral preserving schemes for ordinary differential equations, which was perhaps first discussed by Gonzalez [5]. Matsuo proposed discrete variational method for nonlinear wave equation [5]. L. Brugnano and F. Iavernaro proposed Hamiltonian boundary value methods [, ]. More recently, the existence of energy-preserving B-series methods has been shown in [3], and a * Received April, 5 / Revised version received October 3, 5 / Accepted January 9, 6 / Published online September 4, 6 / ) Corresponding author

2 48 H.C. LI, Y.S. WANG AND M.Z. QIN practical integrator which is the averaged vector field (AVF) method of order two has been proposed [5, 6, 6, 8]. This method exactly preserves the energy of Hamiltonian systems, and in contrast to projection-type integrators, only requires evaluations of the vector field. It is symmetric and its Taylor series has the structure of a B-series. For polynomial Hamiltonians, the integral can be evaluated exactly, and the implementation is comparable to that of the implicit mid-point rule [6]. In recent years, there has been growing interest in high-order AVF methods, and the second, third and fourth order AVF methods have been proposed in succession [8]. It is shown that the theory of B-series and the substitution law obtained by substituting a B-series into the vector field appearing in another B-series play an important role in constructing high order methods [3, 4]. The substitution law for the trees of order 4 has been shown in [8, 9, 4]. The fourth order AVF method is obtained by the concrete formulas of the substitution law for the trees of order 4. To construct a B-series method which not only has high order accuracy but also preserves the Hamiltonian is an important and interesting topic. However, the concrete formulas of the substitution law for the trees of order 5 have not been proposed, as the corresponding calculations are sufficiently complicated. There are two aims in this paper. The first aim is to propose the concrete formulas of the substitution law for the trees of order 5. As we know, a low order B-series integrator can be extend to high order by the substitution law. Using the new obtained concrete formulas of the substitution law for the trees of order 5, one can extend a low order geometric B-series integrator to sixth order naturally, easily and automatically, such as the symplectic integrator, the energy-preserving integrator, the momentum-preserving integrator and so on. Using the new obtained substitution law, we also easily obtained the same sixth order symplectic integrator in [9]. The second aim is to derive a sixth order AVF method for Hamiltonian systems. By expanding the second order AVF method into a B-series and considering the substitution law for the trees of order 5, a new method can be constructed. The new method is derived by the concrete formulas of the substitution law for the trees of order 5. We prove that the new method is of order six and can also preserve the energy of Hamiltonian systems exactly. The paper is organized as follows: In Sect., we introduce the AVF method. In Sect. 3, we recall a few definitions and properties related to trees and B-series. The substitution law for the trees of order 4 is shown and we obtain that for the trees of order 5. In Sect. 4, we derive the sixth order AVF method, and prove that the new method is of order six and it can preserve the Hamiltonian. A few numerical experiments are given in Sect. 5 to confirm the theoretical results. We finish the paper with conclusions in Sect. 6.. The AVF Method and Its Energy-Preserving Property Here we briefly discuss the AVF method and its energy-preserving property. We consider a Hamiltonian differential equation, written in the form ż f(z) S H(z), z() z, z R n, (.) where f(z) : R n R, S is a skew-symmetric constant matrix, n is an even number, and the Hamiltonian H(z) is assumed to be sufficiently differentiable. From system (.), we can get dh(z(t)) dt H(z) T f(z) H(z) T S H(z). (.)

3 A Sixth Order Averaged Vector Field Method 48 Therefore, the flow of the system (.) preserves the Hamiltonian H(z) exactly. The so-called AVF method [5, 8] is defined by z z h f(ξz + ( ξ)z )dξ, (.3) where h is the time step. The AVF method (.3) can be rewritten as We can obtain z z h S H(ξz + ( ξ)z )dξ. h (H(z ) H(z )) d h dξ H(ξz + ( ξ)z )dξ ( ) T ( ) z z H(ξz + ( ξ)z )dξ h ( T H(ξz + ( ξ)z )dξ) S H(ξz + ( ξ)z )dξ. It follows that the Hamiltonian H is conserved at every time step. McLachlan and Quispel also proposed a higher order energy-preserving method [8] z i z i h ( ) δj i + αh fk(ẑ)f i j k (ẑ) f j (ξz + ( ξ)z )dξ, (.4) where α is an arbitrary constant, δ j i is the Kronecker delta, and we can take e.g. ẑ z or ẑ (z +z )/. For α, we recover the second order method (.3). For α and ẑ z, the method is of order three. For α and ẑ (z + z )/, the method is of order four. Up to now, the highest order of existing AVF methods is four, and here we derive a AVF method of order six. 3. Preliminaries In this section, we briefly recall a few definitions and properties related to rooted trees and B-series [3, 8, 3, 4, 7 9]. 3.. Trees and B-series Let denote the empty tree. Definition 3.. (Unordered trees [4]) The set T of (rooted) unordered trees is recursively defined by T, [τ,..., τ m ] T, τ,..., τ m T, (3.) where is the tree with only one vertex, and τ [τ,..., τ m ] represents the rooted tree obtained by grafting the roots of τ,..., τ m T to a new vertex. Trees τ i are called the branches of τ. We note that τ does not depend on the ordering of τ,..., τ m. For instance, we do not distinguish between [, [ ]] and [[ ], ].

4 48 H.C. LI, Y.S. WANG AND M.Z. QIN Definition 3.. (Coefficients [8]) The following coefficients are defined recursively for all trees τ [τ,..., τ m ] T : τ + m τ i (the order, i.e. the number of vertices), i ( τ )! α(τ) τ!... τ m! α(τ )... α(τ m ) µ!µ!... σ(τ) α(τ )... α(τ m ) µ!µ!... γ(τ) τ γ(τ )... γ(τ m ) where the integers µ, µ,... count equal trees among τ,..., τ m. (Connes M oscovici weights), (symmetry), (density), Definition 3.3. (Elementary differentials [7]) For a vector field f : R d R d, and for an unordered tree τ [τ,..., τ m ] T, the so-called elementary differential is a mapping F f (τ) : R d R d, recursively defined by F f ( )(z) f(z), F f (τ)(z) f m (z)(f f (τ )(z),..., F f (τ m )(z)). Definition 3.4. (B-series [7]) For a mapping a : T { } R, a formal series of the form is called a B-series. B f (a, z) a( )z + τ T h τ σ(τ) a(τ)f f (τ)(z) Theorem 3.. (Exact solution [8]) If z(t) denotes the exact solution of (.), it holds for all j, j! z(j) () σ(τ)γ(τ) F f (τ)(z ). τ T, τ j Therefore, letting γ( ), the exact solution of (.) is (formally) given by z(h) B f ( γ, z ). 3.. Basic tools for trees 3... Partitions and skeletons In order to manipulate trees more conveniently, it is useful to consider the set OT of ordered trees defined below. Definition 3.5. (Ordered trees [8]) The set OT of ordered trees is recursively defined by OT, (ω,..., ω m ) OT, ω,..., ω m OT. In contrast to T, the ordered tree ω depends on the ordering ω,..., ω m. Neglecting the ordering, a tree τ T can be considered as an equivalent class of ordered trees, denoted τ ω. Therefore, any function ψ defined on T (such as order, symmetry, density,... ) can be extended to OT by putting ψ(ω) ψ(ω) for all ω OT. Moreover, for all τ T, we can choose a tree ω(τ) OT such as τ ω(τ) [8].

5 A Sixth Order Averaged Vector Field Method 483 Definition 3.6. (Partitions of a tree [8]) A partition p θ of an ordered tree θ OT is the (ordered) tree obtained from θ by replacing some of its edges by dashed ones. We denote P (p θ ) {s,..., s k } the list of subtrees s i T obtained from p θ by removing dashed edges and by neglecting the ordering of each subtree. We denote #(p θ ) k the number of s i s. We observe that precisely one of the s i s contains the root of θ. We denote this distinguished tree by R(p θ ) T. We denote P (p θ ) P (p θ )\{R(p θ )} the list of s i s that do not contain the root of θ. Eventually, the set of all partitions p θ of θ is denoted P(θ). Finally, for τ T, we put P(τ) P(ω(τ)) where ω(τ) OT is given in definition 3.5. Definition 3.7. (Skeleton of a partition []) The skeleton χ(p τ ) T of a partition p τ P(τ) of a tree τ T is the tree obtained by replacing in p τ each tree of P (p τ ) by a single vertex and then dashed edges by solid ones. We can notice that χ(p τ ) #(p τ ). Table 3.: The concrete formulas of the substitution law defined in (3.) for trees of order 4 [8,4]. b a( ) a( ), b a( ) a( )b( ), b a( ) a( )b( ) + a( )b( ), b a( ) a( )b( ) + a( )b( )b( ) + a( )b( ) 3, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( ) 3, b a( ) a( )b( ) + 3a( )b( )b( ) + 3a( )b( ) b( ) + a( )b( ) 4, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( )b( ) + a( )b( ) 4, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( ) b( ) + a( )b( ) 4, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( ) + 3a( )b( ) b( ) + a( )b( ) Substitution law Theorem 3.. (Substitution law [8,9,4]) Let a, b : T { } R be two mappings with b( ). Given a field f : R d R d, consider the (h-dependent) field g : R d R d defined by hg(z) B f (b, z), Then, there exists a mapping b a : T { } R satisfying B g (a, z) B f (b a, z),

6 484 H.C. LI, Y.S. WANG AND M.Z. QIN and b a is defined by b a( ) a( ), τ T, b a(τ) p τ P(τ) a(χ(p τ )) δ P (p τ ) b(δ). (3.) In [8, 4], the concrete formulas of the substitution law for the trees of order 4 was proposed (see table 3.). In this paper, we obtain the concrete formulas of the substitution law for the trees of order 5 (see table 3.). Table 3.: The concrete formulas of the substitution law defined in (3.) for trees of order 5. b a( ) a( )b( ) + 4a( )b( )b( ) + 6a( )b( ) b( ) + 4a( )b( ) 3 b( ) + a( )b( ) 5, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( ) b( ) + a( )b( ) b( ) + a( )b( )b( ) + a( )b( ) 3 b( ) + a( )b( ) 3 b( ) + a( )b( ) 5, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( ) b( ) + 3a( )b( )b( ) + 4a( )b( ) 3 b( ) + a( )b( ) 5, b a( ) a( )b( ) + a( )b( )b( ) + 3a( )b( )b( ) + 3a( )b( ) b( ) + 3a( )b( ) b( ) + a( )b( ) 3 b( ) + 3a( )b( ) 3 b( ) + a( )b( ) 5, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( ) b( ) + a( )b( ) 3 b( ) + a( )b( ) 3 b( ) + a( )b( ) 3 b( ) + a( )b( ) 5, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( ) 3 b( ) + 3a( )b( ) 3 b( ) + a( )b( ) 5,

7 A Sixth Order Averaged Vector Field Method 485 b a( ) a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( ) b( ) + a( )b( ) b( ) + a( )b( ) 3 b( ) + a( )b( ) 3 b( ) + a( )b( ) 5, b a( ) a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + 3a( )b( ) b( ) + 3a( )b( )b( ) + 4a( )b( ) 3 b( ) + a( )b( ) 5 b a( ) a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) b( ) + a( )b( ) b( ) + a( )b( )b( ) + a( )b( ) 3 b( ) + a( )b( ) 3 b( ) + a( )b( ) 3 b( ) + a( )b( ) Sixth Order AVF Method 4.. The second order AVF method and its B-series Consider an ordinary differential equation and the second order AVF method [8] ż f(z), z R n, z(t ) z, (4.) Φ f h (z ) z z + h Theorem 4.. The AVF method (4.) can be expanded into a B-series Φ f h (z ) B f (a, z ) a( )z + τ T where a( ) a( ), and for all τ [τ,..., τ m ] T, a(τ) m + a(τ ) a(τ m ). f(ξz + ( ξ)z )dξ. (4.) h τ σ(τ) a(τ)f f (τ)(z ), (4.3) Proof. We develop the derivatives of (4.), by Leibniz s rule, and obtain [ z (q) h ] (q) f(ξz + ( ξ)z )dξ [ (q) [ q h f(ξz + ( ξ)z )dξ] + q f(ξz + ( ξ)z )dξ].

8 486 H.C. LI, Y.S. WANG AND M.Z. QIN This gives, for h, z (q) : z (q) h q[ f(ξz + ( ξ)z )dξ] q h, q, and considering z h z, we can obtain ż z f(ξz + ( ξ)z )dξ f(z ), (4.4a) ξf (ξz + ( ξ)z )żdξ f(z ) f (z )ż, (4.4b) z (3) 3 ( 3 f (z )(ż, ż) + f (z ) z), (4.4c) and so on. We now insert in (4.4a) recursively the computed derivatives ż, z,... into the right side of the subsequent formulas. Letting f (q) : f (q) (z ) and F (τ) F f (τ)(z ), we can obtain ż f γ( )α( )a( )F ( ), z f f γ( )α( )a( )F ( ), (4.5) z (3) 3 3 f (f, f) + ( 3) ( )f f f γ( )α( )a( )F ( )+γ( )α( )a( )F ( ), and so on. For all τ [τ,..., τ m ] T, letting α µ, µ,... count equal trees among τ,..., τ m, we can obtain τ γ(τ)α(τ)a(τ)f (τ) ( τ )! τ!... τ m! µ!µ!..., where the integers αξ m f m (z )(γ(τ )α(τ )a(τ )F (τ ),..., γ(τ m )α(τ m )a(τ m )F (τ m ))dξ [ τ γ(τ )... γ(τ m )] [αα(τ )... α(τ m )] [ ] m+ a(τ ) a(τ m ) f m (z )(F (τ ),..., F (τ m )) [ ] γ(τ)α(τ) m + a(τ ) a(τ m ) F (τ). So we have z (q) γ(τ)α(τ)a(τ)f (τ), τ T, τ q where a( ), and for all τ [τ,..., τ m ] T, a(τ) m + a(τ ) a(τ m ). Letting a( ), and considering σ(τ) Φ f h (z ) z a( )z + τ T τ! α(τ)γ(τ), we obtain h τ σ(τ) a(τ)f f (τ) B f (a, z ). The proof is completed.

9 A Sixth Order Averaged Vector Field Method 487 The B-series (4.3) can be rewritten as Φ f h (z ) z + hf ( ) + h F ( ) + h 3 ( 3 F ( ) + 4 F ( )) + h4 ( 4 6 F ( ) + 6 F ( ) + 6 F ( ) + 8 F ( )) + h5 ( 5 4 F ( ) + 8 F ( ) + F ( ) F ( ) + F ( ) + F ( ) + F ( ) + 6 F ( ) + 9 F ( )) Sixth order AVF method Let a : T { } R be a mapping satisfying a( ), a( ), and let f : R d R d be a field. We consider here the numerical flow Φ g h, where g : Rd R d denotes the modified field of f, whose B-series expansion is Φ g h (z) B g(a, z). The fundamental idea of obtaining sixth order AVF method consists in interpreting the numerical solution z Φ g h of the initial value problem z() z, ż g(z) as the exact solution of a modified differential equation ż f(z) [3, 4]. Theorem 4.. If z Φ f h (z ), which is a numerical solution of (4.), can be expanded into a B-series Φ f h (z ) z B f (a, z ), a( ), and b : T { } R be a mapping with b( ) satisfying τ T, b a(τ) γ(τ), (4.6) then we can obtain a new numerical solution Φ g h (z ) B g (a, z ), satisfying Φ g h (z ) φ f h (z ), where φ f h (z ) denotes the exact solution of (4.), and g : R n R n is a (h-dependent) field defined by hg(y) B f (b, y). Proof. From Theorem 3., φ f h (z ) B f ( γ, z ). And from Theorem 3., we obtain Φ g h (z ) B g (a, z ) B f (b a, z ) B f ( γ, z ) φ f h (z ). The proof is completed. Letting e(τ) γ(τ), we calculate b(τ) defined in (4.6) for trees of order as follows: τ, a( )b( ) e( ), a( ), e( ), b( ), τ, a( )b( ) + a( )b( ) e( ), a( ), e( ), b( ). In the same way, we can obtain b(τ) for trees of order 5 (see table 4.) and b(τ) for trees of order 6.

10 488 H.C. LI, Y.S. WANG AND M.Z. QIN Then we obtain the exact modified field g(z ) h B f (b, z ) and the sixth order modified field ğ(z ) f(z ) h F ( ) + h4 [6F ( ) + 4F ( ) + F ( ) + 4F ( ) + F ( ) 7 + F ( ) + 3F ( ) + 8F ( )], satisfying g(z ) ğ(z ) + O(h 6 ) and Φ g h (z ) φ f h (z ). z + h ǧ(ξz + ( ξ)z )dξ, we have And considering Φǧ h (z ) z z z h f(ξz + ( ξ)z )dξ + O(h 3 ). Let ẑ : z +z, f (q) : f (q) (ẑ), F : f(ξz + ( ξ)z )dξ and η ξ. Considering τ, we have τ and F f (τ)(ξz + ( ξ)z )dξ f(ξz + ( ξ)z )dξ F. We consider ηdη η 3 dη, η dη. For τ 3, we have τ and z z h f(ẑ + η(z z ))dη + O(h 3 ) hf + O(h 3 ) (4.7) or So we can obtain z z h [f(ẑ) + ηf (ẑ)(z z ) + O(h )]dη + O(h 3 ) (4.8) hf(ẑ) + O(h 3 ) hf + O(h 3 ) or z z O(h). (4.9) F f ( )(ẑ + η(z z ))dη f (ẑ + η(z z ))f (ẑ + η(z z ))f(ẑ + η(z z ))dη [f (ẑ) + ηf (ẑ)(z z ) + η! f (ẑ)(z z, z z ) + η3 3! f (4) (ẑ)(z z, z z, z z ) + O(h 4 )] [f (ẑ) + ηf (ẑ)(z z ) + η! f (ẑ)(z z, z z ) + η3 3! f (4) (ẑ)(z z, z z, z z ) + O(h 4 )] f(ẑ + η(z z ))dη

11 A Sixth Order Averaged Vector Field Method 489 {f (ẑ)f (ẑ)f(ẑ + η(z z )) + f (ẑ)[ηf (ẑ)(z z ) + η! f (ẑ)(z z, z z ) + η3 3! f (4) (ẑ)(z z, z z, z z )] [f(ẑ) + ηf (ẑ)(z z ) + η! f (ẑ)(z z, z z ) + O(h 3 )] + [ηf (ẑ)(z z ) + η! f (ẑ)(z z, z z ) + η3 3! f (4) (ẑ)(z z, z z, z z )] [f (ẑ) + ηf (ẑ)(z z ) + η! f (ẑ)(z z, z z ) + η3 3! f (4) (ẑ)(z z, z z, z z )] [f(ẑ) + ηf (ẑ)(z z ) + η! f (ẑ)(z z, z z ) + O(h 3 )]}dη + O(h 4 ) f (ẑ)f (ẑ) f(ẑ + η(z z ))dη + [η f (ẑ)f (ẑ)(z z, f (ẑ)(z z )) + η! f (ẑ)f (ẑ)(z z, z z, f(ẑ)) + η f (ẑ)(z z, f (ẑ)f (ẑ)(z z )) + η f (ẑ)(z z, f (ẑ)(z z, f(ẑ))) + η! f (ẑ)(z z, z z, f (ẑ)f(ẑ))]dη + O(h 4 ) f f F + h [f f (f f, F ) + f (f f f, F ) + f (f (f, f), F ) + f f (f, f, F ) + f (f f, f, F )] + O(h 4 ). Considering τ 5, we have F f ( )(ẑ + η(z z ))dη f (ẑ + η(z z ))f (ẑ + η(z z ))f (ẑ + η(z z ))f (ẑ + η(z z ))f(ẑ + η(z z ))dη [f (ẑ) + ηf (ẑ)(z z ) + O(h )] [f (ẑ) + ηf (ẑ)(z z ) + O(h )] [f (ẑ) + ηf (ẑ)(z z ) + O(h )] [f (ẑ) + ηf (ẑ)(z z ) + O(h )] [f(ẑ) + ηf (ẑ)(z z ) + O(h )]dη f (ẑ)f (ẑ)f (ẑ)f (ẑ)f(ẑ + η(z z ))dη + O(h ) f f f f F + O(h ) : A( )F + O(h ), where the coefficient matrix A( ) is defined by f f f f F A( )F. In the same way, we can obtain F f ( )(ẑ + η(z z ))dη f (f (F, f), f) + O(h ) : A( )F + O(h ), F f ( )(ẑ + η(z z ))dη f f f (F, f) + O(h ) : A( )F + O(h ),

12 49 H.C. LI, Y.S. WANG AND M.Z. QIN So we have F f ( )(ẑ + η(z z ))dη f (f f F, f) + O(h ) : A( )F + O(h ), F f ( )(ẑ + η(z z ))dη f f (F, f, f) + O(h ) : A( )F + O(h ), F f ( )(ẑ + η(z z ))dη f (f F, f, f) + O(h ) : A( )F + O(h ), F f ( )(ẑ + η(z z ))dη f (f F, f f) + O(h ) : A( )F + O(h ), F f ( )(ẑ + η(z z ))dη f f (F, f f) + O(h ) : A( )F + O(h ). ǧ(ξz + ( ξ)z ))dξ ǧ(ẑ + η(z z ))dη {I h h4 A( ) 7 [5A( ) + 5A( ) + 5A( ) + 5 A( ) + 5 A( )] } + h4 [6A( ) + 4A( ) + A( ) + 4A( ) + A( ) + A( ) + 3A( ) + 8A( )] F + O(h 6 ) 7 {I h h4 A( ) + 7 [6A( ) A( ) + A( ) A( ) 3 A( ) 3 A( ) } + 3A( ) + 3A( )] F + O(h 6 ). We obtain the sixth order AVF method ( ) Φ h (z ) z c( )z + h τ c(τ)a(τ) f(ξz + ( ξ)z ))dξ τ T [ z + {hi h3 h5 A( ) + 6A( ) A( ) + A( ) A( ) 3 7 A( ) 3 ]} A( ) + 3A( ) + 3A( ) f(ξz + ( ξ)z ))dξ, (4.) where A(τ) A(τ)( z+z ) are coefficient matrices of F f(ξz +( ξ)z ))dξ, and c(τ) for all τ 6. Theorem 4.3. The sixth order AVF method Φ h (z ) satisfies Proof. Now we have Φ h (z ) φ f h (z ) O(h 7 ). Φǧ h (z ) Φ h (z ) + O(h 7 ), and Φ g h (z ) φ f h (z ),

13 A Sixth Order Averaged Vector Field Method 49 so we can obtain Φ h (z ) φ f h (z ) Φ h (z ) Φ g h (z ) (Φǧ h (z ) + O(h 7 )) Φ g h (z ) (z z ) + h O(h 7 ). [ǧ(ξz + ( ξ)z )) g(ξz + ( ξ)z ))]dξ + O(h 7 ) The proof is complete. Theorem 4.4. If (4.) is a Hamiltonian system, the sixth order AVF method Φ h (z ) can preserve the discrete energy of it, i.e. Proof. (4.) can be rewritten as h (H(z ) H(z )). ż f(z) S H(z), where S denotes an arbitrary constant skew-symmetric matrix and H denotes the Hamiltonian. So the sixth order AVF method (4.) can be rewritten as z z h S H(ξz + ( ξ)z ))dξ, (4.) where S {I h h4 A( ) + [6A( ) A( ) + A( ) A( ) 7 is a skew-symmetric matrix. It is given by 3 A( ) 3 A( ) + 3A( ) + 3A( )]}S, IS S, A( )S SHSHS, A( )S SHSHSHSHS, A( )S ST ST S, A( )S SHSHST S, A( )S ST SHSHS, A( )S SHSLS, A( )S SLSHS, A( )S SRSHS, A( )S SHSRS, with the symmetric matrices H(z), T (z), L(z) and R(z) being given by H ij : L ij : H 3 H kl H, T ij : S, z i z j z i z j z k z l 4 H km H ln H S S, R ij : z i z j z k z l z m z n 3 H S kl H S z i z j z k z l z m mn H z n,

14 49 H.C. LI, Y.S. WANG AND M.Z. QIN and (SHSHST S ST SHSHS) T ST SHSHS SHSHST S (SHSHST S ST SHSHS), (SHSLS + SLSHS) T SLSHS SHSLS (SHSLS + SLSHS), (SRSHS + SHSRS) T SHSRS SRSHS (SRSHS + SHSRS). So S is also a skew-symmetric matrix and the Hamiltonian H is conserved at every time step. The proof is complete. Table 4.: Coefficients σ(τ), γ(τ), a(τ), b(τ) and c(τ) for trees of order 5. τ σ(τ) 6 γ(τ) a(τ) / /3 /4 /4 /6 /6 /8 b(τ) -/ c(τ) -/ τ σ(τ) 4 6 γ(τ) a(τ) /5 /8 / /8 / / / /6 /9 b(τ) /36 / / /9 /8 /36 / /9 c(τ) -/48 /4 -/48 /4 -/7 /7 / -/7 Remark 4.. In the same way, we can also obtain a fifth order method Φ h (z ) z z + {hi h3 h4 h5 A( ) [A( ) + A( )] + [6A( ) 6A( ) + A( ) 4 7 6A( ) 9A( ) 9A( ) + 3A( ) A( )]} f(ξz + ( ξ)z ))dξ, (4.) where A(τ) A(τ)(z ) are coefficient matrices of F f(ξz + ( ξ)z ))dξ. This method is of order five, but it can not preserve the Hamiltonian, because S which is the corresponding total coefficient matrix of F turns out to be not skew-symmetric when expanding F f (τ)(ξz +( ξ)z ) in a Taylor series about z z. Remark 4.. Omitting the items containing h 5 in (4.), the method ) z z + (hi h3 A( ) f(ξz + ( ξ)z ))dξ is the fourth order method (.4), where A( ) is defined in (4.).

15 A Sixth Order Averaged Vector Field Method 493 Remark 4.3. We recall that a method z Φ h (z ) is symmetric if exchanging z z and h h leaves the method unaltered [7]. Letting η ξ in (4.), we have f(ηz + ( η)z )dη f(ξz + ( ξ)z )dξ. (4.3) It implies that the sixth order AVF method (4.) is a symmetric method. Remark 4.4. Considering the trees of order 4 (see [8, 9, 4]) in (4.), the method is just the fourth order AVF method (.4) (see [8]). 5. Numerical Simulations We here report a few numerical experiments, in order to illustrate our results presented in the previous section. The relative energy error at t t j is defined as RH j H j H, H where H j denotes the Hamiltonian at t t j, j,,..., N. The solution error at t t N is defined as error(h) z N z(t N ), where h is the time step. We define ( ) error(h) order log, error(h/) and recall that for a p-order accurate scheme 5.. Numerical example error(h) error(h/) p (h ). First, we consider a nonlinear Hamiltonian system which has the exact solution ż J H, H(z) 4 (p + q ), (5.a) ( ) ( ) ( ) p z, z, J, (5.b) q p(t) cos(t), q(t) sin(t). We can obtain the sixth order AVF method of (5.) ( ) ( ) p p + {hi h3 h5 A( ) + [6A( ) A( ) + A( ) A( ) 7 q q 3 A( ) 3 A( ) + 3A( ) + 3A( )] }F,

16 494 H.C. LI, Y.S. WANG AND M.Z. QIN Table 5.: The convergence order of the sixth order AVF method with different time steps, h., h., h 3.5, h 4.5, h 5.5. ( ) ( ) ( ) ( ) Order log error(h ) error(h log error(h ) ) error(h 3 log error(h3 ) ) error(h 4 log error(h4 ) ) error(h 5 ) t t t t t t t where A(τ) A(τ)( z +z ) are coefficient matrices of F f(ξz + ( ξ)z ))dξ, and ( F F F ) ( f (ξz + ( ξ)z ))dξ f (ξz + ( ξ)z ))dξ ( 6 [H q(z ) + 4H q ( z+z ) + H q (z )] 6 [H p(z ) + 4H p ( z+z ) + H p (z )] To examine the performance of the proposed method in a long time computing, the problem is solved by the sixth order AVF method (AVF6) and a sixth order Runge-Kutta method with ν / (RK6) [3] till time t 4. Table 5. shows the convergence order of the AVF6 method with different time steps. We can conclude that the AVF6 method is order of six, supporting Theorem 4.3. Figs. 5. and 5. provide the numerical solutions obtained by using the AVF6 method and the RK6 method from t to t 4, respectively. As can be seen from Figs. 5.a and 5.a, the solution errors of the AVF6 method grow linearly and are less than the errors of the RK6 method. The linear error growth and Table 5. indicate that there exist a positive constant C such that z j z(t j ) Ct j h p for j,,..., N, where p 6 is the order of the AVF6 method. Figs. 5.b and 5.b show the relative energy errors of the two methods respectively. Comparing with the RK6 method, the AVF6 method can conserve the Hamiltonian up to round-off error in long time, supporting Theorem 4.4. The relative energy errors look like to show slow linear increase because of the iteration errors. ) ). 5.. Numerical example Second, we consider the Hénon-Heiles system ż J H, H(z) (q + q + p + p ) + qq 3 q3, (5.a) q. z q p, z.5, J. (5.b) p The Hénon-Heiles system has a critical energy value E c 6 at which the qualitative nature of the solution changes from bounded to unbounded orbits. In the experiment shown here, the

17 A Sixth Order Averaged Vector Field Method 495 error t RH 7 x t a Solution errors b Relative energy errors Fig. 5.. The numerical solution of the sixth order AVF method solving the example with h.6. error t RH x t a Solution errors b Relative energy errors Fig. 5.. The numerical solution of the sixth order RK method solving the example with h.6. energy of the system (5.3) is exactly E c with initial spatial coordinates (q, q ) at a point on the boundary of the critical triangular region (see [, 8]). We solve the system using the AVF6 method (see Fig. 5.3) and the RK6 method (see Fig. 5.4) with h.4 respectively. From Fig. 5.3 we can see that the solution of the AVF6 method stays within the stable zone and conserves the energy E c exactly in long time. Fig. 5.4 shows that the RK6 method can not preserve the energy and from t 366 the solution strays outside the stable zone and soon becomes completely unstable Numerical example 3 Finally, we consider the Kepler problem ż J H, z p p q q H(z) (p + p ), z.4, J, (5.3a) q + q. (5.3b) We solve the Kepler problem using the AVF6 method and the RK6 method from t to t 5 with h. respectively. The integration f(ξz + ( ξ)z ))dξ is calculated

18 496 H.C. LI, Y.S. WANG AND M.Z. QIN q q RH x t a Configuration space orbit b Relative energy errors Fig The numerical solution of the sixth order AVF method solving the example from t to t 4 with h.4. q.5.5 RH 4 x q a Configuration space orbit t b Relative energy errors Fig The numerical solution of the sixth order RK method solving the example from t to t 366 with h.4. RH x t RH t a The AVF6 method b The RK6 method Fig Relative energy errors of the numerical solution for solving the example 3 from t to t 5 with h.. using the ninth order Gauss quadrature. Fig. 5.5 shows relative energy errors of the numerical solutions of the two methods. From Fig. 5.5a we can see that the errors in the integration is up to round-off error and the AVF6 method can conserve the energy exactly in longtime. We

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